Hoermander Spaces, Interpolation, and Elliptic Problems

De Gruyter Studies in Mathematics 60

Edited byCarsten Carstensen, Berlin, GermanyNicola Fusco, Napoli, ItalyFritz Gesztesy, Columbia, Missouri, USANiels Jacob, Swansea, United KingdomKarl-Hermann Neeb, Erlangen, Germany

Vladimir A. MikhailetsAleksandr A. Murach

Hörmander Spaces, Interpolation, and Elliptic Problems

De Gruyter

Mathematics Subject Classification 2000: 46E35, 46B70, 35J30, 35J40, 35J45

AuthorsProf. Dr. Vladimir Andreevich MikhailetsNational Academy of Sciences of UkraineInstitute of MathematicsTereshchenkovskaya st.,3 KIEV-4 01601 UKRAINE [email protected]

Prof. Dr. Aleksandr Aleksandrovich MurachNational Academy of Sciences of UkraineInstitute of MathematicsTereshchenkovskaya st.,3 KIEV-4 01601 UKRAINE [email protected]

Translated by Peter V. Malyshev

ISBN 978-3-11-029685-3e-ISBN 978-3-11-029689-1Set-ISBN 978-3-11-029690-7ISSN 0179-0986

Library of Congress Cataloging-in-Publication DataA CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

© 2014 Walter de Gruyter GmbH, Berlin/Boston

Printing and binding: CPI buch bücher.de GmbH, Birkach Printed on acid-free paperPrinted in Germany

www.degruyter.com

Preface

The fundamental applications of the Sobolev spacesWm2 (G) to the investigation

of many-dimensional differential equations, in particular of the elliptic type,are well known. Without the theory of spaces of this kind, the investigationof elliptic problems is, in fact, impossible. At the same time, the theory ofHörmander spaces more general than the Sobolev spaces was developed about40 years ago. At present, there are numerous papers devoted to the applicationsof Hörmander spaces to differential equations.

However, the applications of Hörmander spaces to boundary-value problemsfor elliptic equations have been episodic up to now. The main part of the bookis devoted to a fairly systematic investigation of the applications of Hörmanderspaces to this class of problems. The authors introduce and study Hörmanderspaces of the “intermediate” type. The functions from these spaces are char-acterized by the degree of smoothness intermediate between the smoothness offunctions from the spacesWm

2 (G) andWm+12 (G), where m is an integer. As G,

we can take a domain of n-dimensional Euclidean space or a compact manifoldof dimension n. The first two chapters of the book are devoted to the detailedintroduction and study of these spaces.

In Chapters 3 and 4, the authors consider elliptic equations and homogeneousand inhomogeneous boundary-value problems for these equations. Numeroussignificant results (similar to the results known for the Sobolev spaces) areobtained for these problems in Hörmander spaces. It is possible to say that theauthors managed to transfer the classical “Sobolev” theory of boundary-valueproblems to the case of Hörmander spaces. It should also be emphasized thatsome problems posed independently of the notion of Hörmander spaces can besolved with the help of these spaces.

The last fifth chapter of the book is devoted to the transfer of the obtainedresults to the case of elliptic systems of differential equations.

I think that the book is fairly interesting and useful. It should definitelybe translated into English. In this case, the results accumulated there wouldbecome accessible for a broader circle of mathematicians. In the case of trans-lation, it would be necessary to include the proofs of various auxiliary factsmentioned in the text, which belong to the other authors. This would signifi-cantly increase the circle of possible readers of the book.

Yu. M. Berezansky,Academician of the Ukrainian National Academy of Sciences

Preface to the English edition

The English translation of the monograph slightly differs from the Russian-language edition.

Thus, in particular, we extended the list of references, corrected the detectedmisprints, and improved the presentation of some results. In addition, the bookis equipped with the index.

V. A. Mikhailets and A. A. Murach

Acknowledgements

The authors are especially grateful to Yu. M. Berezansky for his valuable adviceand great influence, which determined, to a significant extent, their scientificinterests.

We are also thankful to M. S. Agranovich, B. P. Paneyah, I. V. Skrypnik,and S. D. Eidel’man for stimulating discussions.

The support of M. L. Gorbachuk and A. M. Samoilenko, interest of B. Bo-yarskii, and kind participation of V. P. Burskii and S. D. Ivasyshen are alsohighly appreciated.

We also thank all our colleagues for their sincere interest to the new theoryand its applications.

Contents

Preface v

Preface to the English edition vi

Acknowledgements vii

Introduction 1

1 Interpolation and Hörmander spaces 9

1.1 Interpolation with function parameter . . . . . . . . . . . . . . . . . . . . . . . 91.1.1 Definition of interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Embeddings of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Reiteration property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.4 Interpolation of dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.5 Interpolation of orthogonal sums of spaces . . . . . . . . . . . . . 181.1.6 Interpolation of subspaces and factor spaces . . . . . . . . . . . 201.1.7 Interpolation of Fredholm operators . . . . . . . . . . . . . . . . . . 211.1.8 Estimate of the operator norm in interpolation spaces . . . 231.1.9 Criterion for a function to be an interpolation parameter 25

1.2 Regularly varying functions and their generalization . . . . . . . . . . 291.2.1 Regularly varying functions . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.2 Quasiregularly varying functions . . . . . . . . . . . . . . . . . . . . . 311.2.3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.3 Hörmander spaces and the refined Sobolev scale . . . . . . . . . . . . . . 381.3.1 Preliminary information and notation . . . . . . . . . . . . . . . . . 381.3.2 Hörmander spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.3 Refined Sobolev scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.3.4 Properties of the refined scale . . . . . . . . . . . . . . . . . . . . . . . . 44

1.4 Uniformly elliptic operators on the refined scale . . . . . . . . . . . . . . 471.4.1 Pseudodifferential operators . . . . . . . . . . . . . . . . . . . . . . . . . 471.4.2 A priori estimate of the solutions . . . . . . . . . . . . . . . . . . . . 501.4.3 Smoothness of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.5 Remarks and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 Hörmander spaces on closed manifolds and their applications 59

2.1 Hörmander spaces on closed manifolds . . . . . . . . . . . . . . . . . . . . . . 592.1.1 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

x Contents

2.1.2 Interpolation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.1.3 Equivalent norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.1.4 Embedding theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.2 Elliptic operators on closed manifolds . . . . . . . . . . . . . . . . . . . . . . . 782.2.1 Pseudodifferential operators on closed manifolds . . . . . . . . 792.2.2 Elliptic operators on the refined scale . . . . . . . . . . . . . . . . . 812.2.3 Smoothness of solutions to the elliptic equation . . . . . . . . 842.2.4 Parameter-elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.3 Convergence of spectral expansions . . . . . . . . . . . . . . . . . . . . . . . . . 932.3.1 Convergence almost everywhere for general orthogonal

series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.3.2 Convergence almost everywhere for spectral expansions . 952.3.3 Convergence of spectral expansions in the metric of the

space Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.4 RO-varying functions and Hörmander spaces . . . . . . . . . . . . . . . . . 982.4.1 RO-varying functions in the sense of Avakumović . . . . . . . 982.4.2 Interpolation spaces for a pair of Sobolev spaces . . . . . . . . 1002.4.3 Applications to elliptic operators . . . . . . . . . . . . . . . . . . . . . 107


3 Semihomogeneous elliptic boundary-value problems 111

3.1 Regular elliptic boundary-value problems . . . . . . . . . . . . . . . . . . . . 1113.1.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.1.2 Formally adjoint problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.2 Hörmander spaces for Euclidean domains . . . . . . . . . . . . . . . . . . . . 1143.2.1 Spaces for open domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2.2 Spaces for closed domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.3 Rigging of L2(Ω) with Hörmander spaces . . . . . . . . . . . . . . 123

3.3 Boundary-value problems for homogeneous elliptic equations . . . 1263.3.1 Main result: boundedness and Fredholm property of the

operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.3.2 A theorem on interpolation of subspaces . . . . . . . . . . . . . . 1273.3.3 Elliptic boundary-value problem in Sobolev spaces . . . . . . 1313.3.4 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.3.5 Properties of solutions to the homogeneous elliptic

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.4 Elliptic problems with homogeneous boundary conditions . . . . . . 1423.4.1 Theorem on isomorphisms for elliptic operators . . . . . . . . 1423.4.2 Interpolation and homogeneous boundary conditions . . . . 146

Contents xi

3.4.3 Proofs of theorems on isomorphisms and the Fredholmproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.4.4 Local increase in smoothness of solutions up to theboundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.5 Some properties of Hörmander spaces . . . . . . . . . . . . . . . . . . . . . . . 1583.5.1 Space Hs,ϕ

0 (Ω) and its properties . . . . . . . . . . . . . . . . . . . . . 1583.5.2 Equivalent description of Hs,ϕ(Ω) . . . . . . . . . . . . . . . . . . . . 160


4 Inhomogeneous elliptic boundary-value problems 165

4.1 Elliptic boundary-value problems in the positive one-sided scale 1654.1.1 Theorems on Fredholm property and isomorphisms . . . . . 1654.1.2 Smoothness of the solutions up to the boundary . . . . . . . . 1694.1.3 Nonregular elliptic boundary-value problems . . . . . . . . . . . 1734.1.4 Parameter-elliptic boundary-value problems . . . . . . . . . . . 1754.1.5 Formally mixed elliptic boundary-value problem . . . . . . . . 186

4.2 Elliptic boundary-value problems in the two-sided scale . . . . . . . 1884.2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.2.2 The refined scale modified in the sense of Roitberg . . . . . 1894.2.3 Roitberg-type theorems on solvability. The complete

collection of isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.2.4 Smoothness of generalized solutions up to the boundary . 2044.2.5 Interpolation in the modified refined scale . . . . . . . . . . . . . 207

4.3 Some properties of the modified refined scale . . . . . . . . . . . . . . . . . 2104.3.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.3.2 Proof of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

4.4 Generalization of the Lions–Magenes theorems . . . . . . . . . . . . . . . 2264.4.1 Lions–Magenes theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.4.2 Key individual theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.4.3 Individual theorem for Sobolev spaces . . . . . . . . . . . . . . . . 2364.4.4 Individual theorem for weight spaces . . . . . . . . . . . . . . . . . 238

4.5 Hörmander spaces and individual theorems on solvability . . . . . . 2434.5.1 Key individual theorem for the refined scale . . . . . . . . . . . 2434.5.2 Other individual theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 244

4.6 Remarks and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5 Elliptic systems 251

5.1 Uniformly elliptic systems in the refined Sobolev scale . . . . . . . . . 2515.1.1 Uniformly elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2515.1.2 A priori estimate for the solutions of the system . . . . . . . 2525.1.3 Smoothness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

xii Contents

5.2 Elliptic systems on a closed manifold . . . . . . . . . . . . . . . . . . . . . . . 2575.2.1 Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2575.2.2 Operator of the elliptic system on the refined scale . . . . . 2585.2.3 Local smoothness of solutions . . . . . . . . . . . . . . . . . . . . . . . . 2625.2.4 Parameter-elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . 264

5.3 Elliptic boundary-value problems for systems of equations . . . . . 2685.3.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695.3.2 Theorem on solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

5.4 Remarks and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Bibliography 275Index 291

Introduction

In the theory of partial differential equations, the problems of existence, unique-ness, and regularity of solutions are in the focus of investigations. As a rule, theregularity properties of solutions are formulated in terms of belonging of thesesolutions to the standard classes of function spaces. Moreover, the finer the cal-ibration of the scale of spaces, the more exact and informative the accumulatedresults.

Unlike the case of ordinary differential equations with smooth coefficients,these problems are fairly complicated. Indeed, some linear partial differentialequations with smooth coefficients and right-hand sides are known to have nosolutions in the neighborhood of a given point even in the class of distributions[113], [81, Sec. 6.0]. Moreover, some homogeneous equations (specifically, ofthe elliptic type) with smooth but not analytic coefficients admit nontrivialsolutions with compact supports [193], [85, Theorem 13.6.5]. Therefore, thenontrivial null space of this equation cannot be removed by any homogeneousboundary conditions; i.e., the operator corresponding to any boundary-valueproblem for the analyzed equation is not injective. Finally, the problem ofregularity of solutions is also quite complicated. Thus, even for the Laplaceoperator, it is known that

4u = f ∈ C(Ω) ; u ∈ C 2(Ω)

for any Euclidean domain Ω [64, Chap. 4, Notes].These problems have been most completely investigated for linear elliptic

equations, systems, and boundary-value problems. The fundamental results inthis direction were obtained in the 1950s and 1960s by S.Agmon, A.Douglis,and L.Nirenberg [4, 5, 47], M. S.Agranovich and A. S.Dynin [12], Yu.M.Bere-zansky, S.G. Krein, and Ya.A. Roitberg [22, 21, 202, 203, 209], F. E. Brow-der [28, 29], L. R. Volevich [267, 268], J.-L. Lions and E. Magenes [121, 126],L.N. Slobodetskii [240, 241], V.A. Solonnikov [245, 246, 247], L. Hörmander[81], M. Schechter [222, 224, 225], and other researchers. In the cited works, theelliptic equations and problems were studied in the classical scales of Hölderspaces (of nonintegral order) and Sobolev spaces (both of positive and negativeorders).

As a fundamental result in the theory of elliptic equations, we can mentionthe fact that they generate bounded Fredholm operators (i.e., operators withfinite index) acting between appropriate function spaces. Thus, let Au = f

2 Introduction

be a linear elliptic differential equation of order m given on a closed smoothmanifold Γ. Then

A : Hs+m(Γ)→ Hs(Γ), with s ∈ R,

is a bounded Fredholm operator. Moreover, the finite-dimensional spaces formedby the solutions of the homogeneous equations Au = 0 and A+v = 0 lie inC∞(Γ). Here, A+ is the operator formally adjoint to A, whereas Hs+m(Γ) andHs(Γ) are the Sobolev inner product spaces over Γ of the orders s+m and s,respectively. This result implies that each solution u of the elliptic equationAu = f has an important regularity property in the Sobolev scale, namely,

(f ∈ Hs(Γ) for some s ∈ R) ⇒ u ∈ Hs+m(Γ). (1)

If a manifold has an edge, then the Fredholm operator is generated by anelliptic boundary-value problem for the equation Au = f (e.g., by the Dirichletboundary-value problem).

Some of these theorems were extended by H. Triebel [258, 256] and Murach[163, 164] to the scales of Nikol’skii–Besov, Zygmund, and Lizorkin–Triebelfunction spaces.

The cited results were applied, in various ways, to the theory of differentialequations, mathematical physics, and spectral theory of differential operators(see the books by Yu. M. Berezansky [21], Yu. M. Berezansky, G. F. Us, andZ. G. Sheftel [23], O. A. Ladyzhenskaya and N. N. Ural’tseva [111], J.-L. Li-ons [118, 117], J.-L. Lions and E. Magenes [121], Ya. A. Roitberg [209, 210],I. V. Skrypnik [237], H. Triebel [258], the surveys by M. S. Agranovich [7, 10, 11],and references therein).

From the viewpoint of applications, especially to the spectral theory, the caseof Hilbert spaces is of especial importance. Note that, until recently, the scaleof Sobolev inner product spaces was the sole scale of Hilbert spaces in whichthe properties of elliptic operators were systematically studied. However, itwas shown that the Sobolev scale is insufficiently fine for various importantproblems.

We present two typical examples. The first of them deals with the smoothnessof the solution of the elliptic equation Au = f on the manifold Γ. Accordingto the Sobolev embedding theorem, we have

Hσ(Γ) ⊂ Cr(Γ) ⇔ σ > r + n/2, (2)

where r ≥ 0 is an integer and n := dim Γ. This fact, together with property (1),allow us to study the classical smoothness of the solution u. Thus, if f ∈Hs(Γ) for some s > r −m + n/2, then u ∈ Hs+m(Γ) ⊂ Cr(Γ). However, thisembedding is not true for s = r−m+n/2; i.e., the Sobolev scale cannot be usedto formulate unimprovable sufficient conditions for the inclusion u ∈ Cr(Γ).

Introduction 3

A similar situation is also encountered in the theory of elliptic boundary-valueproblems.

The second typical example corresponds to the spectral theory. We assumethat a differential operator A of order m > 0 is elliptic on Γ and self-adjoint inthe space L2(Γ). Consider the expansion of a function f ∈ L2(Γ) in the series

f =∞∑j=1

cj(f)hj , (3)

where (hj)∞j=1 is the complete orthonormal system of eigenfunctions of A and

cj(f) are the Fourier coefficients of the function f in its expansion in hj . Theeigenfunctions are enumerated so that the moduli of the corresponding eigen-values form a (nonstrictly) increasing sequence. By the Menchoff–Rademachertheorem (valid for general orthogonal series), expansion (3) converges almosteverywhere on Γ provided that

∞∑j=1

|cj(f)|2 log2(j + 1) <∞. (4)

This hypotheses cannot be reformulated in equivalent way in terms of the factthat the function f belongs to Sobolev spaces because

‖f‖2Hs(Γ) ∞∑j=1

|cj(f)|2 j2s

for any s > 0. We can only state that the condition “the fact that f ∈ Hs(Γ)for some s > 0” implies the convergence of series (3) almost everywhere on Γ.This condition does not adequately express the hypotheses (4) of the Menchoff–Rademacher theorem.

In 1963, L. Hörmander [81, Sec. 2.2] proposed a significant and useful gen-eralization of Sobolev spaces in the category of Hilbert spaces (see also [85,Sec. 10.1]). He introduced spaces parametrized by a sufficiently general weightfunction playing the role of an analog of the differentiation order (or smoothnessindex) used for the Sobolev spaces. In particular, L. Hörmander considered theHilbert spaces

B2,µ(Rn) :=u : µFu ∈ L2(Rn)

, (5)

‖u‖B2,µ(Rn) := ‖µFu‖L2(Rn),

where Fu is the Fourier transform of a tempered distribution u given in Rn,and µ is a weight function of n scalar arguments.

4 Introduction

In the case where

µ(ξ) = 〈ξ〉s, 〈ξ〉 := (1 + |ξ|2)1/2, ξ ∈ Rn, s ∈ R,

we get the Sobolev space B2,µ(Rn) = Hs(Rn) of the (differentiation) order s.In 1965, spaces (5) were independently introduced and studied by L. R. Vole-

vich and B. P. Paneah [269].The Hörmander spaces occupy an especially important place among the

spaces of generalized smoothness characterized by a function parameter in-stead of a number. These spaces serve as an object of numerous profoundinvestigations, and a good deal of work was performed in the last decades. Werefer the reader to the survey by G. A. Kalyabin and P. I. Lizorkin [90], mono-graph by H. Triebel [259, Sec. 22], and recent papers by A. M. Caetano andH.-G. Leopold [32], W. Farkas, N. Jacob, and R. L. Schilling [55], W. Farkas andH.-G. Leopold [56], P. Gurka, and B. Opic [70], D. D. Haroske and S. D. Moura[74, 75], S. D. Moura [162], B. Opic, and W. Trebels [177], and the referencestherein. Various classes of spaces of generalized smoothness naturally appearin the embedding theorems for function spaces, in the interpolation theory offunction spaces, in the approximation theory, in the theory of differential andpseudodifferential operators, and in the theory of stochastic processes; see themonographs by D. D. Haroske [73], N. Jacob [87], V. G. Maz’ya and T. O. Sha-poshnikova [132, Sec. 16], F. Nicola and L. Rodino [175], B. P. Paneah [181],and A. I. Stepanets [248, Chap. 1, § 7], [249, Part I, Chap. 3, Sec. 7.1], the pa-pers by F. Cobos and D. L. Fernandez [35], C. Merucci [136], and M. Schechter[226] devoted to the interpolation of function spaces, and also the papers byD. E. Edmunds and H. Triebel [50, 51] and V. A. Mikhailets and V. Molyboga[140, 141, 142].

As early as in 1963, L. Hörmander [81] used the spaces (5) and more generalBanach spaces Bp,µ(Rn) with 1 ≤ p ≤ ∞ to study the regularity propertiesof the solutions of partial differential equations with constant coefficients andsolutions of some classes of equations with variable coefficients given in Eu-clidean domains. However, unlike Sobolev spaces, the Hörmander spaces werenot widely applied to the general elliptic equations on manifolds and ellip-tic boundary-value problems. This is explained by the long-term absence ofa proper definition of Hörmander spaces on smooth manifolds (this definitionshould be independent of the choice of local charts covering the manifold) andthe absence of analytic tools required for the effective investigation of thesespaces.

For the Sobolev spaces, the required tool is available: this is the interpolationof spaces. Thus, every Sobolev space of fractional order can be obtained asa result of the interpolation of a certain pair of Sobolev spaces of integer order.This fact significantly facilitates both the investigation of these spaces andthe proofs of various theorems from the theory of elliptic equations because the

Introduction 5

procedure of interpolation preserves boundedness and the Fredholm property(if the defect is invariant) of linear operators.

Therefore, it seems reasonable to select Hörmander inner product spacesobtained as a result of interpolation (in this case, with a function parameter)between Sobolev inner product spaces. To this end, we introduce a class ofisotropic spaces

Hs,ϕ(Rn) := B2,µ(Rn) for µ(ξ) = 〈ξ〉sϕ(〈ξ〉). (6)

Here, s ∈ R is a numerical parameter and ϕ is a positive function parameterslowly varying at infinity in Karamata’s sense [26, 235]. (It is possible to assumethat ϕ is constant outside the neighborhood of infinity). Thus, the logarithmicfunction, its iterations, all their powers, and the products of these functionsmay play the role of ϕ.

The class of spaces (6) contains the Hilbert scale of Sobolev spaces Hs ≡Hs,1 and is attached to it by a number parameter. However, it is calibratedfiner than the Sobolev scale. Indeed,

Hs+ε(Rn) ⊂ Hs,ϕ(Rn) ⊂ Hs−ε(Rn) for any ε > 0.

Therefore, the number parameter s specifies the main (power) smoothness,while the function parameter ϕ is responsible for an additional (subpower)smoothness in the class of spaces (6). Thus, in particular, if ϕ(t) → ∞ (orϕ(t) → 0) as t → ∞, then ϕ specifies an additional positive (or negative)smoothness. In other words, the parameter ϕ refines the main smoothness s.Hence, it is natural to say that the class of spaces (6) is a refined Sobolev scaleor simply the refined scale.This scale has an important interpolation property, namely, every space

Hs,ϕ(Rn) is obtained as a result of the interpolation of a pair of Sobolev spacesHs−ε(Rn) and Hs+δ(Rn), where ε, δ > 0, with an appropriate function param-eter. This parameter is a function regularly varying at infinity (in Karamata’ssense) of the order θ ∈ (0, 1), where θ := ε/(ε + δ). Moreover, the class ofspaces (6) is closed with respect to this interpolation.

Thus, the Hörmander spaces Hs,ϕ(Rn) possess the interpolation propertywith respect to the Hilbert scale of Sobolev spaces. This means that any lin-ear operator bounded in each space Hs−ε(Rn) and Hs+δ(Rn) is also boundedin Hs,ϕ(Rn). In this case, the interpolation property plays a decisive role be-cause it allows us to establish important properties of the refined Sobolev scaleguaranteeing the possibility of efficient application of this scale to the theoryof elliptic equations. Thus, we can prove with the help of interpolation thatthe spaces Hs,ϕ(Rn), just as the Sobolev spaces, are invariant under the diffeo-morphic transformations of Rn. This enables us to correctly define the spaceHs,ϕ(Γ) over a closed smooth manifold Γ because the set of distributions and

6 Introduction

the topology in this space are independent of the choice of local charts cover-ing Γ. The spaces Hs,ϕ(Rn) and Hs,ϕ(Γ) are useful in the theory of ellipticoperators on manifolds and in the theory of elliptic boundary-value problems.Moreover, they are implicitly present in various problems of analysis.

We now present some results demonstrating advantages of the refined scaleas compared with the Sobolev scale. These results are related to the examplesconsidered above. As earlier, let A be an elliptic differential operator on Γ oforder m. Then it specifies bounded Fredholm operators

A : Hs+m,ϕ(Γ)→ Hs,ϕ(Γ) for all s ∈ R and ϕ ∈M.

Here, M is a class of slowly varying function parameters ϕ used in (6). Notethat the differential operator A leaves the functional parameter ϕ, which refinesthe main smoothness s, invariant. In addition, we have the following regularityproperty of a solution of the equation Au = f :

(f ∈ Hs,ϕ(Γ) for some s ∈ R and ϕ ∈M) ⇒ u ∈ Hs+m,ϕ(Γ).

An important sharpening of Sobolev’s Embedding Theorem is true for the re-fined scale. Let an integer r ≥ 0 and a function parameter ϕ ∈ M be given.Then the embedding Hr+n/2,ϕ(Γ) ⊂ Cr(Γ) is equivalent to the inequality

∞∫1

dt

t ϕ2(t)<∞. (7)

Therefore, if f ∈ Hr−m+n/2,ϕ(Γ) for a certain parameter ϕ ∈M satisfying (7),then the solution u ∈ Cr(Γ).

Similar results are also valid for elliptic systems and elliptic boundary-valueproblems.

We now pass to the analysis of convergence of the spectral expansion (3). Itis additionally supposed that the operator A of order m > 0 is unbounded andself-adjoint in the space L2(Γ). Condition (4) for the convergence of (3) almosteverywhere on Γ is equivalent to the inclusion

f ∈ H0,ϕ(Γ) with ϕ(t) := max1, log t.

This condition is much broader than the condition “f ∈ Hs(Γ) for some s > 0.”In a similar way, we can also represent the conditions of unconditional con-vergence of series (3) almost everywhere or in the Banach space Cr(Γ) forinteger r ≥ 0.

These and some other results show that the refined Sobolev scale can bequite useful and convenient. This scale can be also used in the other fieldsof contemporary analysis (see, e.g., the papers by M. Hegland [78, 79] andP. Mathé, U. Tautenhahn [129]).

Introduction 7

The proposed monograph gives the first systematic presentation of the theoryof elliptic (scalar and matrix) operators and elliptic boundary-value problemson the refined Sobolev scale. We also dwell upon the related topics concerningthe interpolation of Hilbert (abstract and Sobolev) spaces with function param-eter. This theory was developed by the authors in the recent papers [143–156,165–173] (see also the survey [159]). The contents and structure of the mono-graph are fairly completely reflected in the TOC.

Chapter 1

Interpolation and Hörmander spaces

1.1 Interpolation with function parameter

In the present section, we consider the interpolation of pairs of Hilbert spaceswith the help of a function parameter. This is a natural generalization of theclassical interpolation method developed by J.-L. Lions and S. G. Krein [121,Chap. 1, Sec. 5], [107, Chap. IV, § 9] to the case where a fairly general functionis taken as an interpolation parameter instead of a number. The procedure ofinterpolation with function parameter is one of the main methods used in theproofs of our results.

1.1.1 Definition of interpolation

We now present the definition of interpolation with the help of a functionparameter for pairs of Hilbert spaces and study numerous properties of thisinterpolation required in what follows. For our purposes, it is sufficient torestrict ourselves to the case of separable spaces.

Definition 1.1. An ordered pair [X0, X1] of complex Hilbert spaces X0 andX1 is called admissible, if the spaces X0, X1 are separable and the followingdense and continuous embedding X1 → X0 is true.

Let X = [X0, X1] be a given admissible pair of Hilbert spaces. It is known(see [121, Chap. 1, Sec. 2.1] and [107, Chap. IV, § 9, Sec. 1]) that there existsan isometric isomorphism J : X1 ↔ X 0 such that J is a self-adjoint positivedefinite operator with a domain X1 in the space X0. The mapping J is calleda generating operator for the pair X. This operator is uniquely determined bythe pair X. Indeed, let J1 be another generating operator for the pair X. Thenthe operators J and J1 are metrically equal:

‖Ju‖X 0 = ‖u‖X1 = ‖J1u‖X0

for any u ∈ X1. In addition, these operators are positive definite. This impliesthat they are equal: J = J1.

By B we denote the set of all functions ψ : (0,∞)→ (0,∞) such that

(a) ψ is Borel measurable on the semiaxis (0,∞);

10 Chapter 1 Interpolation and Hörmander spaces

(b) the function 1/ψ is separated from zero on every set [r,∞), with r > 0;

(c) ψ is bounded on every segment [a, b], with 0 < a < b <∞.

Let ψ ∈ B. The operator ψ(J) is defined inX0 as a function of J. By [X0, X1]ψ(or, simply, by Xψ) we denote the domain of ψ(J) equipped with the innerproduct

(u1, u2)Xψ := (ψ(J)u1, ψ(J)u2)X0

and the corresponding norm

‖u ‖Xψ = (u, u)1/2Xψ.

The space Xψ is Hilbert and separable and the following dense and continuousembedding is true: Xψ → X0. Indeed, Spec J ⊆ [r,∞) and ψ(t) ≥ c for t ≥ rand some positive numbers r and c. (As usual, Spec J denotes the spectrumof the operator J.) Hence, Specψ(J) ⊆ [c,∞), which yields the isometric iso-morphism ψ(J) : Xψ ↔ X0. Thus, we conclude that Xψ is a complete andseparable space (and also that the function ‖ · ‖Xψ is positive definite and playsthe role of norm). Since the operator ψ−1(J) is bounded in the space X0, theembedding operator

I = ψ−1(J)ψ(J) : Xψ → X0

is also bounded. This embedding is dense because the domain of ψ(J) is a denselinear manifold in the space X0.

Remark 1.1. Assume that functions ϕ, ψ ∈ B are such that ϕ ψ in thevicinity of ∞. Then, by the definition of B, we have ϕ ψ on Spec J . Hence,Xϕ = Xψ up to equivalence of norms. (As usual, the relation ϕ ψ meansthat both functions ϕ/ψ and ψ/ϕ are bounded on the corresponding set; inthis case, ϕ and ψ are assumed to be positive.)

Definition 1.2. A function ψ ∈ B is called an interpolation parameter if, forany admissible pairs X = [X0, X1] and Y = [Y0, Y1] of Hilbert spaces andevery linear mapping T given on X0, the following condition is satisfied: If therestriction of the mapping T to the space Xj is a bounded operator T : Xj → Yjfor every j ∈ 0, 1, then the restriction of the mapping T to the space Xψ isalso a bounded operator T : Xψ → Yψ.

In other words, the function ψ is an interpolation parameter if and only ifthe mapping X 7→ Xψ is an interpolation functor given on the category ofadmissible pairs X of Hilbert spaces (see [24, Sec. 2.4] and [258, Sec. 1.2.2]).If ψ is an interpolation parameter, then we say that the space Xψ is obtainedby the interpolation with function parameter ψ of an admissible pair X.

In what follows, we study the main properties of the mapping X 7→ Xψ.

Section 1.1 Interpolation with function parameter 11

1.1.2 Embeddings of spaces

We now consider some properties of interpolation related to the embeddings ofspaces.

Theorem 1.1. Let ψ ∈ B be an interpolation parameter and let X = [X0, X1]be an admissible pair of Hilbert spaces. Then the continuous and dense embed-dings X1 → Xψ → X0 are true.

Proof. By virtue of the results presented above, it remains to show thatthe continuous and dense embedding X1 → Xψ is true. Consider the boundedembedding operators I : X1 → X0 and I : X1 → X1. Since ψ is an interpolationparameter, this means that the embedding operator I : X1 → Xψ is bounded,i.e., the embedding X1 → Xψ holds and is continuous.

To prove that this embedding is dense, we choose an arbitrary u ∈ Xψ. Thenv := ψ(J)u ∈ X0, where J is the generating operator for the pair X. Since X1

is dense in X0, there exists a sequence (vk) ⊂ X1 such that vk → v in X0 ask →∞. This yields the convergence

uk := ψ−1(J)vk → ψ−1(J)v = u in Xψ as k →∞.

It remains to note that

uk = ψ−1(J)J−1Jvk = J−1ψ−1(J)Jvk ∈ X1.

Theorem 1.1 is proved.

Theorem 1.2. Let functions ψ, χ ∈ B be such that the function ψ/χ is boundedin the vicinity of∞. Then the continuous and dense embedding Xχ → Xψ holdsfor every admissible pair X = [X0, X1] of Hilbert spaces. If the embeddingX1 → X0 is compact and

ψ(t)

χ(t)→ 0 as t→∞,

then the embedding Xχ → Xψ is also compact.

Proof. Let J be the generating operator for the pair X. Note that Spec J ⊆[r,∞) for some number r > 0. By the condition, we have

ψ(t)

χ(t)≤ c

for t ≥ r and, hence,

Xχ = Domχ(J) ⊆ Domψ(J) = Xψ, ‖ψ(J)u‖X0 ≤ c ‖χ(J)u‖X0 ,


whence, by virtue of the definition of Xχ and Xψ, we conclude that the embed-ding Xχ → Xψ is continuous.

We now prove that this embedding is dense.Consider isometric isomorphisms ψ(J) : Xψ ↔ X0 and χ(J) : Xχ ↔ X0. Let

u ∈ Xψ. Then ψ(J)u ∈ X0. Since the space Xχ is densely embedded in X0,there exists a sequence (vk) ⊂ Xχ such that vk → ψ(J)u in X0 as k → ∞.Hence,

ψ−1(J) vk → u

in Xψ as k →∞. Here,

ψ−1(J) vk = ψ−1(J)χ−1(J)χ(J) vk

= χ−1(J)ψ−1(J)χ(J) vk ∈ Xχ.

This proves that the embedding Xχ → Xψ is dense.We now assume that the embedding X1 → X0 is compact and

ψ(t)/χ(t)→ 0 as t→∞.

We prove that the embedding Xχ → Xψ is compact. Let (uk) be an arbitrarybounded sequence in Xχ. Since the sequence of elements wk := J−1 χ(J)uk isbounded in X1, we can select a subsequence of elements wkn = J−1 χ(J)uknfundamental in X0. We now show that the subsequence (ukn) is fundamentalin Xψ.

Let Et with t ≥ r be the resolution of identity in X0 corresponding to theself-adjoint operatorJ. We can write

‖ukn − ukm‖2Xψ = ‖ψ(J) (ukn − ukm)‖2X0

= ‖ψ(J)χ−1(J)J (wkn − wkm)‖2X0

=

∞∫r

ψ2(t)χ−2(t) t2 d ‖Et(wkn − wkm)‖2X0. (1.1)

Further, we choose an arbitrary ε > 0 . There exists a number % = %(ε) > rsuch that

ψ(t)

χ(t)≤ (2c0)

−1ε

for t ≥ % andc0 := sup ‖wk‖X1 : k ∈ N <∞.


Thus, for any numbers n,m, we can write∞∫%

ψ2(t)χ−2(t) t2 d ‖Et(wkn − wkm)‖2X0

≤ (2c0)−2 ε2

∞∫%

t2 d ‖Et(wkn − wkm)‖2X0

≤ (2c0)−2 ε2 ‖J (wkn − wkm)‖2X0

= (2c0)−2 ε2 ‖wkn − wkm‖2X1

≤ ε2. (1.2)

Moreover, in view of the inequality

ψ(t)

χ(t)≤ c

for t ≥ r, we get%∫r

ψ2(t)χ−2(t) t2 d ‖Et(wkn − wkm)‖2X0

≤ c2%2%∫r

d ‖Et(wkn − wkm)‖2X0

≤ c2%2 ‖wkn − wkm‖2X0→ 0 for n,m→∞. (1.3)

Relations (1.1)–(1.3) now imply the inequality ‖ukn − ukm‖Xψ ≤ 2ε for suffi-ciently large numbers n and m. Hence, the subsequence (ukn) is fundamentalin the space Xψ, which means that the embedding Xχ → Xψ is compact.


1.1.3 Reiteration property

This property can be formulated as follows: The repeated application of theinterpolation with function parameter also gives an interpolation with a certainfunction parameter.

Theorem 1.3. Let f, g, ψ ∈ B and let the function f/g be bounded in the vicin-ity of ∞. Then [Xf , Xg]ψ = Xω with the equality of norms for every admissiblepair X of Hilbert spaces. Here, the function ω ∈ B is defined by the formula

ω(t) := f(t)ψ

(g(t)

f(t)

)


for t > 0. If f, g, and ψ are interpolation parameters, then ω is also an inter-polation parameter.

Proof. Since the function f/g is bounded in the vicinity of ∞, the pair[Xf , Xg] is admissible by Theorem 1.2 and ω ∈ B. Thus, the spaces [Xf , Xg]ψand Xω are defined. We now prove their equality.

Let J be the generating operator for the pair X = [X0, X1], where Spec J ⊆[r,∞) for some number r > 0. We have the isometric isomorphisms

f(J) : Xf ↔ X0, g(J) : Xg ↔ X0,

B := f−1(J) g(J) : Xg ↔ Xf .

Consider B as a closed operator in the space Xf with the domain Xg. Theoperator B is the generating operator for the pair [Xf , Xg] because it is positivedefinite and self-adjoint in Xf . The first property follows from the conditionf(t)/g(t) ≤ c for t ≥ r, namely,

(Bu, u)Xf = (g(J)u, f(J)u)X0

≥ c−1 (f(J)u, f(J)u)X0 = c−1 ‖u‖2Xf , u ∈ Xf .

The second property now follows from the fact that 0 is a regular point for B.With the help of the spectral theorem, we can reduce the operator J self-

adjoint in the space X0 to the operator of multiplication by a function. Namely,we can write J = I−1(α · I), where I : X0 ↔ L2(U, dµ) is an isometric isomor-phism, (U, µ) is a space with finite measure, and α : U → [r,∞) is a measurablefunction. The isometric isomorphism If(J) : Xf ↔ L2(U, dµ) reduces the op-erator B, self-adjoint in Xf to the operator of multiplication by a function(g/f) α. Indeed,

If(J)B u = Ig(J)u = (g α) Iu

= (g α) If−1(J)f(J)u = ((g/f) α) If(J)u, u ∈ Xg.

Hence, for any u ∈ Xψ, we can write

‖ψ(B)u‖Xf = ‖(ψ (g/f) α) · (If(J)u)‖L2(U,dµ)

= |(ω α) · (Iu)‖L2(U,dµ) = ‖ω(J)u‖X0 .

Note that the function f/ω is bounded in the vicinity of ∞. Hence, Xω → Xf

and therefore, f(J)u is defined. Thus, the equality [Xf , Xg]ψ = Xω is proved.We now assume that f, g, and ψ are interpolation parameters and show that ω

is an interpolation parameter. Let admissible pairs X = [X0, X1], Y = [Y0, Y1],


and a linear mapping T be the same as in Definition 1.2. Thus, the operatorsT : Xf → Yf and T : Xg → Yg are bounded which means that the operatorT : [Xf , Xg]ψ → [Yf , Yg]ψ is also bounded. As shown above, [Xf , Xg]ψ = Xω

and [Yf , Yg]ψ = Yω. Hence, the operator T : Xω → Yω is defined and bounded;i.e., ω is an interpolation parameter.


1.1.4 Interpolation of dual spaces

Let H be a Hilbert space. As usual, H ′ denotes the space dual to H; i.e., H ′ isthe normed linear space of all linear continuous functionals l : H → C. By theRiesz theorem, the mapping S : v 7→ ( ·, v)H , where v ∈ H, defines an antilinearisometric isomorphism S : H ↔ H ′. This means that H ′ is a Hilbert space; theHilbert norm in H ′ is induced by the inner product

(l,m)H′ := (S−1l, S−1m)H .

Note that we do not identify H and H ′ with the help of the isomorphism S.

Theorem 1.4. Let ψ ∈ B be a function such that the function ψ(t)/t is boundedin the vicinity of ∞. Then, for any admissible pair [X0, X1] of Hilbert spaces,the equality

[X ′1, X′0]ψ = [X0, X1]

′χ

holds with equality of the norms. Here, the function χ ∈ B is given by theformula χ(t) := t/ψ(t) for t > 0. If ψ is an interpolation parameter, then χ isalso an interpolation parameter.

Proof. Note that the pair [X ′1, X ′0] is admissible for the natural identificationof functionals from X ′0 with their restrictions to the space X1. The conditionof the theorem implies that χ ∈ B. Hence, the Hilbert spaces [X ′1, X

′0]ψ and

[X0, X1]′χ are defined. We now prove that they are equal.

Let J : X1 ↔ X0 be the generating operator for the pair [X0, X1]. Considerthe isometric isomorphisms Sj : Xj ↔ X ′j , j = 0, 1, appearing in the Riesztheorem. The operator J ′ adjoint to J satisfies the equality J ′ = S1J

−1S−10 .This is a consequence of the following chain of relations:

(J ′l)u = l(Ju) = (Ju, S−10 l)X0 = (u, J−1S−10 l)X1

= (S1J−1S−10 l)u for all l ∈ X ′0, u ∈ X1.

Therefore, the operator J ′ realizes an isometric isomorphism

J ′ = S1J−1S−10 : X ′0 ↔ X ′1. (1.4)


Note that the equalities

(u, JS−11 l)X0 = (J−1u, S−11 l)X1 = l(J−1u),

(u, J−1S−10 l)X0 = (J−1u, S−10 l)X0 = l(J−1u),

valid for any l ∈ X ′0 → X ′1 and u ∈ X0 yield the following property:

JS−11 l = J−1S−10 l ∈ X1 for all l ∈ X ′0. (1.5)

We consider J ′ as a closed operator in the space X ′1 with the domain X ′0.The mapping J ′ is the generating operator for the pair [X ′1, X

′0] because J ′

is positive definite and self-adjoint in X ′1. The first property follows from thepositive definiteness of the operator J in the space X0 and property (1.5).Indeed,

(J ′l, l)X′1 = (S1J−1S−10 l, l)X′1 = (J−1S−10 l, S−11 l)X1

= (JJ−1S−10 l, JS−11 l)X0 = (JJS−11 l, JS−11 l)X0

≥ c ‖JS−11 l‖2X0= c ‖S−11 l‖2X1

= c ‖l‖2X′1 .

Here, the number c > 0 is independent of l ∈ X ′0. The second property nowfollows from the fact that 0 is a regular point for the operator J ′ by virtue ofrelation (1.4).

We now use the reduction of the operator J to the form J = I−1(α · I) ofmultiplication by a function. This reduction has already been considered in theproof of Theorem 1.3. The isometric isomorphism

IJS−11 : X ′1 ↔ L2(U, dµ) (1.6)

reduces the operator J ′ to the form of multiplication by the same function α.Indeed,

(IJS−11 )J ′l = IS−10 l = IJJ−1S−10 l

= α · IJ−1S−10 l = α · IJS−11 l for any l ∈ X ′0[here, we have used relations (1.4) and (1.5)].

By virtue of Theorem 1.2, we get the following continuous and dense em-beddings: X ′0 → [X ′1, X

′0]ψ and [X0, X1]χ → X0. The latter implies that the

embedding X ′0 → [X0, X1]′χ is also continuous and dense. We now show that

the norms in the spaces [X ′1, X′0]ψ and [X0, X1]

′χ coincide on the dense sub-

set X ′0. For any l ∈ X ′0 and u ∈ [X0, X1]χ, we can write

l(u) = (u, S−10 l)X0 = (χ(J)u, χ−1(J)S−10 l)X0

= (v, χ−1(J)S−10 l)X0 ,


wherev := χ(J)u ∈ X0.

This yields

‖l‖ [X0,X1]′χ= sup |l(u)| / ‖u‖ [X0,X1]χ : u ∈ [X0, X1]χ, u 6= 0

= sup |(v, χ−1(J)S−10 l)X0 | / ‖v‖X0 : v ∈ X0, v 6= 0

= ‖χ−1(J)S−10 l‖X0 = ‖Iχ−1(J)S−10 l‖L2(U,dµ)

= ‖(χ−1 α) · IS−10 l‖L2(U,dµ).

On the other hand, applying the isomorphisms (1.6) and (1.4), we obtain

‖l‖ [X′1,X′0]ψ = ‖ψ(J ′)l‖X′1 = ‖χ−1(J ′)J ′l‖X′1

= ‖(IJS−11 )χ−1(J ′)J ′l‖L2(U,dµ)

= ‖(χ−1 α) · (IJS−11 )J ′l‖L2(U,dµ)

= ‖(χ−1 α) · IS−10 l‖L2(U,dµ).

Thus, the norms in the spaces [X ′1, X′0]ψ and [X0, X1]

′χ coincide on the dense

subset X ′0. Therefore, these spaces coincide.Assume that ψ is an interpolation parameter. It is necessary to show that

χ is also an interpolation parameter. Let admissible pairs X = [X0, X1], Y =[Y0, Y1] and a linear mapping T be the same as in Definition 1.2. Passing to theadjoint operator T ′, we get bounded operators T ′ : Y ′j → X ′j for each j ∈ 0, 1.Since ψ is an interpolation parameter, we obtain a bounded operator

T ′ : [Y ′1 , Y′0 ]ψ → [X ′1, X

′0]ψ.

As shown above, [X ′1, X ′0]ψ = [X0, X1]′χ and [Y ′1 , Y

′0 ]ψ = [Y0, Y1]

′χ with equality

of the norms. Hence, the operator T ′ : [Y0, Y1]′χ → [X0, X1]

′χ is bounded.

Passing to the second adjoint operator T ′′, we get a bounded operator

T ′′ : [X0, X1]′′χ → [Y0, Y1]

′′χ.

Identifying the second dual spaces with the original Hilbert spaces, we obtaina bounded operator T : [X0, X1]χ → [Y0, Y1]χ. This means that χ is an inter-polation parameter.


Note that, for a sufficiently broad class of function interpolation parameters,Theorem 1.4 was proved by G. Shlenzak [231, Theorem 2].


In some applications, it is convenient to consider the space of all anti linearcontinuous functionals on H as the dual space H ′. For this interpretation of H ′,Theorem 1.4 is obviously valid. In what follows, it is always clear from thecontext functionals of what kind (linear or antilinear) are used to form thespace H ′.

1.1.5 Interpolation of orthogonal sums of spaces

Recall that an orthogonal sum of finitely or countably many separable Hilbertspaces is also a separable Hilbert space.

Theorem 1.5. Let X(k) := [X(k)0 , X

(k)1 ], k ∈ ω, be a given finite (or countable)

set of admissible pairs of Hilbert spaces. Assume that the set of norms of theembedding operators X(k)

1 → X(k)0 , k ∈ ω, is bounded. Then, for any function

parameter ψ ∈ B, we have[⊕k∈ω

X(k)0 ,

⊕k∈ω

X(k)1

]ψ

=⊕k∈ω

[X

(k)0 , X

(k)1

]ψ,

with equality of the norms.

Proof. Let ω = N (the case of a finite set ω is studied similarly and ina simpler way [231, Theorem 4]). Both the spaces

X0 :=∞⊕k=1

X(k)0 and X1 :=

∞⊕k=1

X(k)1

are Hilbert and separable. The continuity of the embeddingX1 → X0 is obviousin view of the condition of the theorem. We now show that this embedding isdense. Let u := (u1, u2, . . .) ∈ X0. For any numbers n and k, there exists anelement vn,k ∈ X

(k)1 such that

‖uk − vn,k‖X(k)0

< 1/n.

We now compose a sequence of vectors

v(n) := (vn,1, . . . , vn,n, 0, 0, . . .) ∈ X1.

Thus, we get

‖u− v(n)‖2X0=

n∑k=1

‖uk − vn,k‖2X0+

∞∑k=n+1

‖uk‖2X0

≤ n

n2+

∞∑k=n+1

‖uk‖2X0→ 0 as n→∞.


Hence X1 is dense in the space X0 and the pair X := [X0, X1] is admissible.Let Jk be the generating operator for the pairX(k).We can directly show that

J := (J1, J2, . . .) is the generating operator for the pair X. Further, we showthat ψ(J) = (ψ(J1), ψ(J2), . . .), where Domψ(J) =

⊕∞k=1X

(k)ψ . The operator

Jk is now reduced to the form IkJk = αk · Ik of multiplication by a function.Here, Ik : X(k)

0 ↔ L2(Vk, dµk) is an isometric isomorphism, Vk is a space withfinite measure µk, and αk : Vk → (0,∞) is a measurable function. Without lossof generality, we can assume that the sets Vk are pairwise disjoint. We set

V :=∞⋃k=1

Vk.

A subset Ω ⊆ V is called measurable if the set Ω∩Vk is µk-measurable for everynumber k. In the σ-algebra of measurable sets Ω ⊆ V, we introduce a σ-finitemeasure

µ(Ω) :=∞∑k=1

µk(Ω ∩ Vk).

For any vector u := (u1, u2, . . .) ∈ X0, we define measurable functions Iuand α on the set V by the formulas (Iu)(λ) := (Ikuk)(λ) and α(λ) := αk(λ)provided that λ ∈ Vk. We have an isometric isomorphism I : X0 ↔ L2(V, dµ).This isomorphism reduces the operator J to the form of multiplication by thefunction α because

(IJu)(λ) = (IkJkuk)(λ) = αk(λ)(Ikuk)(λ) = α(λ)(Iu)(λ)

for any u ∈ X1 and λ ∈ Vk. This enables us to write

Xψ = Domψ(J) = u ∈ X0 : (ψ α) · (Iu) ∈ L2(V, dµ)

=

u ∈ X0 :

∞∑k=1

‖(ψ αk) · (Ikuk)‖2L2(Vk,dµk)<∞

=

u : uk ∈ Domψ(Jk),

∞∑k=1

‖ψ(Jk)uk‖2X

(k)0

<∞

=∞⊕k=1

X(k)ψ .

Moreover, for any u ∈ Domψ(J), we find

(Iψ(J)u)(λ) = ψ(α(λ)) (Iu)(λ) = ψ(αk(λ)) (Ikuk)(λ)

= (Ikψ(Jk)uk)(λ) =(I(ψ(J1)u1, ψ(J2)u2, . . .)

)(λ)

provided that λ ∈ Vk. Hence, ψ(J)u = (ψ(J1)u1, ψ(J2)u2, . . .). This yields

‖u‖2Xψ = ‖ψ(J)u‖2X0=

∞∑k=1

‖ψ(Jk)uk‖2X

(k)0

=

∞∑k=1

‖uk‖2X

(k)ψ

.



1.1.6 Interpolation of subspaces and factor spaces

Recall that, by definition, a subspace of a Hilbert space is closed. In whatfollows, we consider, generally speaking, nonorthogonal projectors onto a sub-space.

Theorem 1.6. Let X = [X0, X1] be an admissible pair of Hilbert spaces andlet Y0 be a subspace of X0. Then Y1 := X1∩Y0 is a subspace of X1. Assume thatthere exists a linear mapping P such that P is a projector of the space Xj ontothe subspace Yj for any j ∈ 0, 1. Then the pairs [Y0, Y1] and [X0/Y0, X1/Y1]are admissible and, for any interpolation parameter ψ ∈ B, the following spacesare equal (with equivalence of the norms):

[Y0, Y1]ψ = Xψ ∩ Y0, [X0/Y0, X1/Y1]ψ = Xψ/(Xψ ∩ Y0). (1.7)

Here, Xψ ∩ Y0 is a subspace of Xψ.

Proof. Since the embeddings X1 → X0 and Xψ → X0 are continuous, thelinear manifolds Y1 = X1 ∩ Y0 and Xψ ∩ Y0 are closed in the spaces X1 andXψ respectively. The pairs Y := [Y0, Y1] and [X0/Y0, X1/Y1] are admissiblebecause Y1 is a dense subset of the space Y0. This fact is proved as follows: Forany u ∈ Y0, there exists a sequence of elements uk ∈ X1 such that uk → u in X0

as k → ∞. By the condition of the theorem, this implies that Puk → Pu = uin X0 as k →∞, where Puk ∈ Y1. This means that Y1 is indeed a dense subsetof the space Y0. Thus, the left- and right-hand sides of equalities (1.7) are welldefined.

We prove the first equality in (1.7). Suppose that ψ ∈ B is an interpolationparameter. Then it follows from the boundedness of the projectors P : X0 → Y0and P : X1 → Y1 that the restriction of P to Xψ is a bounded operatorP : Xψ → Yψ. Hence,

Xψ ∩ Y0 = P (Xψ ∩ Y0) ⊆ Yψ.

Moreover, since the embedding operators Y0 → X0 and Y1 → X1 are contin-uous, we get the continuous embedding Yψ → Xψ. Thus, Yψ = Xψ ∩ Y0 andthe norms in Yψ and Xψ ∩ Y0 are equivalent in view of the Banach theorem oninverse operator.

We now prove the second equality in (1.7). Given any j ∈ 0, 1, we considerbounded linear operators R : Xj → Xj/Yj and T : Xj/Yj → Xj defined bythe formulas Rx := [x] and T [x] := x − Px for any x ∈ Xj , where [x] denotesthe coset x+ y : y ∈ Yj ∈ Xj/Yj . Note that RT = I is the identity operatoron Xj/Yj . Since ψ ∈ B is an interpolation parameter, we conclude that thecorresponding restrictions of R and T specify the bounded operators

R : Xψ →[X0/Y0, X1/Y1

]ψ

and T :[X0/Y0, X1/Y1

]ψ→ Xψ.


Moreover, we have the following bounded operators:

R : Xψ → Xψ/Yψ and T : Xψ/Yψ → Xψ.

Further, we consider the identity mapping I as the product of the second andthird operators and arrive at the continuous embedding

I = RT :[X0/Y0, X1/Y1

]ψ→ Xψ/Yψ.

Moreover, multiplying the fourth operator by the first operator, we arrive at adifferent continuous embedding

I = RT : Xψ/Yψ →[X0/Y0, X1/Y1

]ψ.

Therefore, [X0/Y0, X1/Y1]ψ = Xψ/Yψ up to equivalence of norms with Yψ =Xψ ∩ Y0, as shown earlier.


1.1.7 Interpolation of Fredholm operators

First, we recall the following definition:

Definition 1.3. A linear bounded operator T : X → Y, where X and Yare Banach spaces, is called a Fredholm operator if both its kernel kerT andcokernel cokerT := Y/T (X) are finite-dimensional. The index of a Fredholmoperator T is defined by the formula

indT := dim kerT − dim(Y/T (X)).

Note that the domain of a Fredholm operator is closed [86, Lemma 19.1.1].

Theorem 1.7. Let X = [X 0, X1] and Y = [Y 0, Y1] be admissible pairs ofHilbert spaces. Suppose that a linear mapping T is given on X 0 and specifiesbounded Fredholm operators T : Xj → Yj, with j ∈ 0, 1, common kernelN, and the same index κ. Then, for any interpolation parameter ψ ∈ B, thebounded operator T : Xψ → Yψ is a Fredholm operator with the kernel N,domain Yψ ∩ T (X 0), and the same index κ.

Proof. By the condition, we have the following bounded operators: Tj :=T : Xj → Yj and Tψ := T : Xψ → Yψ. Consider the bounded operatorsT ′j : Y ′j → X ′j and T ′ψ : Y ′ψ → X ′ψ adjoint to these operators. By Theorem 1.1,we have the following continuous and dense embeddings: X1 → Xψ → X0 andY1 → Yψ → Y0. Therefore,

kerT1 ⊆ kerTψ ⊆ kerT0 and kerT ′0 ⊆ kerT ′ψ ⊆ kerT ′1.


However, by the condition, kerTj = N and

dim kerT ′j = dimYj/T (Xj) = −κ + dimN.

Hence, kerTψ = N andkerT ′ψ = kerT ′j =: M,

where dimM = −κ + dimN. Thus, the operator T : Xψ → Yψ has the finite-dimensional kernel N and the defect subspace M with dimN − dimM = κ.It remains to verify that T (Xψ) = Yψ ∩ T (X0) because, in this case, T (Xψ) isclosed in the space Yψ and dimYψ/T (Xψ) = dimM.

For this purpose, we consider the isomorphisms

T : Xj/N ↔ T (Xj) for j ∈ 0, 1. (1.8)

They are canonically generated by the bounded Fredholm operators Tj . Recallthat T (Xj) is a subspace of Yj .We now apply the interpolation with parameterψ to (1.8). This gives another isomorphism

T : [X0/N,X1/N ]ψ ↔ [T (X0), T (X1)]ψ. (1.9)

Note that the pair [X0/N,X1/N ] is obviously admissible. By virtue of (1.8),this implies that the pair [T (X0), T (X1)] is admissible. We now describe theinterpolation spaces appearing in (1.9).

Consider the orthogonal sum X0 = N ⊕ E. Its restriction to the space X1

is the direct sum X1 = N u (E ∩ X1) of subspaces. By P we denote theorthoprojector of the space X0 onto N. Then the restriction of P to the spaceX1 is the projector of this space onto N corresponding to the second sum.Hence, by virtue of Theorem 1.6 and the equality Xψ ∩N = N, the relation

[X0/N,X1/N ]ψ = Xψ/N (1.10)

holds with equivalence of the norms.Further, we consider the orthogonal sum Y0 = T (X0)⊕Z0, where dimZ0<∞.

Since Y1 is dense in the space Y0, we obtain the decomposition of the spaceY0 into the direct sum Y0 = T (X0) u Z1, where Z1 is a finite-dimensionalsubspace of Y1. Here, we refer the reader to Lemma 2.1 in [65]. The restrictionof this sum to the space Y1 has the form Y1 = (T (X0) ∩ Y1) u Z1. Note thatT (X0)∩Y1 = T (X1). This equality follows from the representation of the closedrange of Tj in the form

T (Xj) = f ∈ Yj : l(f) = 0 for l ∈M

for j ∈ 0, 1. Thus, we have the direct sums Y0 = T (X0) u Z1 and Y1 =T (X1) u Z1. Moreover, the second sum is a restriction of the first sum. By Q


we denote the projector of the space Y0 onto the subspace T (X0) correspondingto the first sum. Then the restriction of Q to Y1 is the projector of the spaceY1 onto the subspace T (X1) = Y1 ∩ T (X0) corresponding to the second sum.Hence, by Theorem 1.4, we can write

[T (X0), T (X1)]ψ = Yψ ∩ T (X0) (1.11)

with equivalence of the norms.Applying equalities (1.10) and (1.11) to isomorphism (1.9), we obtain the

isomorphism T : Xψ/N ↔ Yψ ∩ T (X0). This yields the equality

T (Xψ) = T (Xψ/N) = Yψ ∩ T (X0),

which completes the proof of Theorem 1.7.

Note that an analog of Theorem 1.7 was proved by G. Geymonat [63, p. 281]for any interpolation functor given on the category of all compatible pairs ofBanach spaces. The proof presented in the cited work is similar to our proof.

1.1.8 Estimate of the operator norm in interpolation spaces

We now prove the following result:

Theorem 1.8. Let an interpolation parameter ψ ∈ B and a number m > 0 begiven. Then there exists a number c = c(ψ,m) > 0 such that

‖T‖Xψ→Yψ ≤ cmax‖T‖Xj→Yj : j = 0, 1

. (1.12)

Here, X = [X0, X1] and Y = [Y0, Y1] are arbitrary admissible pairs of Hilbertspaces for which the norms of the embedding operators X1 → X0 and Y1 → Y0do not exceed the number m and T is an arbitrary linear mapping given in thespace X0 and specifying the bounded operators T : Xj → Yj for each j ∈ 0, 1.The constant c is independent of X, Y, and T.

Proof. Assume that the theorem is not true. Then

‖Tk‖X(k)ψ →Y

(k)ψ

> kmk for every k ∈ N. (1.13)

Here, X(k) := [X(k)0 , X

(k)1 ] and Y (k) := [Y

(k)0 , Y

(k)1 ] are admissible pairs of

Hilbert spaces for which the norms of the embedding operators X(k)1 → X

(k)0

and Y (k)1 → Y

(k)0 do not exceed the number m and Tk is a linear mapping given

in the spaceX(k)0 and specifying the bounded operators Tk : X(k)

j → Y(k)j , where

j ∈ 0, 1. In this case, we use the notation

mk := max‖Tk‖X(k)

0 →Y(k)0

, ‖Tk‖X(k)1 →Y

(k)1

> 0.


Consider bounded operators

T : u = (u1, u2, . . .) 7→ (m−11 T1 u1, m−12 T2 u2, . . .),

T :∞⊕k=1

X(k)j →

∞⊕k=1

Y(k)j , j ∈ 0, 1. (1.14)

Their boundedness follows from the inequalities∞∑k=1

‖m−1k Tkuk‖2Y

(k)j

≤∞∑k=1

m−2k ‖Tk‖2

X(k)j →Y

(k)j

‖uk‖2X

(k)j

≤∞∑k=1

‖uk‖2X

(k)j

.

Since ψ is an interpolation parameter, the boundedness of the operators (1.14)implies that the following operator is defined and bounded:

T :[ ∞⊕k=1

X(k)0 ,

∞⊕k=1

X(k)1

]ψ

→[ ∞⊕k=1

Y(k)0 ,

∞⊕k=1

Y(k)1

]ψ

.

By Theorem 1.5, this yields the boundedness of the operator

T :∞⊕k=1

X(k)ψ →

∞⊕k=1

Y(k)ψ .

Let c0 be the norm of this operator. For any number k, we consider a vectoru(k) := (u1, . . . , uk, . . .) such that uk ∈ X

(k)ψ and uj = 0 for j 6= k. We have

‖Tkuk‖Y (k)ψ

= mk ‖Tu(k)‖⊕∞j=1 Y

(j)ψ

≤ mk c0 ‖u(k)‖⊕∞j=1X

(j)ψ

= mk c0 ‖uk‖X(k)ψ

for all uk ∈ X(k)ψ . Hence,

‖Tk‖X(k)ψ →Y

(k)ψ

≤ c0mk for any number k,

which contradicts condition (1.13). Thus, our assumption is false. This meansthat Theorem 1.8 is true.

Note that inequality (1.12), where the constant c is independent of T (butmay depend on X and Y ) is satisfied for any interpolation functor given on thecategory of pairs of Hilbert or Banach spaces; see [24, Theorem 2.4.2] or [109,Chap. 1, Lemma 4.3]. Theorem 1.8 sharpens this fact for the interpolationfunctor X 7→ Xψ.

The admissible pair [X0, X1] of Hilbert spaces is called normal if ‖u‖X0 ≤‖u‖X1 for every u ∈ X1. By Theorem 1.8, the constant c in inequality (1.12) is


independent of the admissible pairs X, Y and the operator T whenever thesepairs are normal. We note that any admissible pair [X0, X1] can be madenormal if we change, e.g., the norm ‖u‖X0 in the space X0 by the proportionalnorm k ‖u‖X0 , where 0 < k ≤ m−1, and m is the norm of the embeddingoperator X1 → X0.

1.1.9 Criterion for a function to be an interpolationparameter

On the basis of the results obtained by J. Peetre [188, 189] (see also [24, The-orem 5.4.4]), we establish the following criterion for a function ψ ∈ B to be aninterpolation parameter.

Definition 1.4. Let a function ψ : (0,∞) → (0,∞) and a number r ≥ 0be given. The function ψ is called pseudoconcave on the half line (r,∞) ifthere exists a concave function ψ1 : (r,∞) → (0,∞) such that ψ(t) ψ1(t)for t > r. The function ψ is called pseudoconcave in the vicinity of ∞ if ψ ispseudoconcave in a half line (r,∞), where r is a sufficiently large number.

Theorem 1.9. A function ψ ∈ B is an interpolation parameter if and only ifψ is pseudoconcave in the vicinity of ∞.

Prior to proving this theorem, we establish two lemmas.

Lemma 1.1. Assume that a function ψ belongs to the set B and is pseudocon-cave in the vicinity of ∞. Then there exists a concave function ψ0 : (0,∞) →(0,∞) such that, for any number ε > 0, the relation ψ(t) ψ0(t) holds for allt ≥ ε.

Proof of Lemma 1.1. By the condition, there exist a number r 1 anda concave function ψ1 : (r,∞) → (0,∞) such that ψ(t) ψ1(t) for t > r.Since the function ψ1 is concave and positive on the semiaxis (r,∞), it is (non-strictly) increasing on this semiaxis. In addition, for any fixed point t0 ∈ (r,∞),the slope function (ψ1(t)− ψ1(t0))/(t− t0), t ∈ (r,∞) \ t0, (nonstrictly) de-creases. Hence, the function ψ1 has the right tangent at the point r+1, whichmakes an acute (or zero angle) with the abscissa axis. We now specify a func-tion ψ2 on the semiaxis [0,∞) in such a way that its graph coincides withthe indicated tangent in the interval [0, r + 1) and with the graph of the func-tion ψ1 on the semiaxis [r + 1,∞). The function ψ2 increases on [0,∞) and isconcave on (0,∞). The latter follows from the fact that, for every fixed pointt0 ∈ (0,∞), the slope function (ψ2(t) − ψ2(t0))/(t − t0) of t ∈ (0,∞) \ t0decreases. We set

ψ0(t) := ψ2(t) + |ψ2(0)|+ 1.


The function ψ0 is positive, increases, and is concave on the semiaxis (0,∞).We choose an arbitrary number ε > 0. Note that ψ(t) 1 ψ0(t) for t ∈[ε, r + 1 + ε]. Further, since the function ψ2 increases and is positive on thesemiaxis [r + 1,∞), we get

|ψ2(0)|+ 1 ≤ c ψ2(t)

for t ≥ r + 1, where

c :=|ψ2(0)|+ 1

ψ2(r + 1)> 0.

This yields the relation

ψ(t) ψ1(t) = ψ2(t) ψ0(t)

for t ≥ r + 1. Thus, ψ(t) ψ0(t) for t ≥ ε, Q.E.D.

Lemma 1.2. Let a function ψ ∈ B and a number r ≥ 0 be given. The functionψ is pseudoconcave on the semiaxis (r,∞) if and only if there exists a numberc > 0 such that

ψ(t)

ψ(s)≤ c max

1,t

s

for any t, s > r.

Proof of Lemma 1.2. For r = 0, this lemma was proved by J. Peetre [189](in this case, the condition ψ ∈ B is not necessary; see also [24, Lemma 5.4.3]).For r > 0, the sufficiency is proved similarly. The necessity is reduced to thecase r = 0 with the help of Lemma 1.1. Indeed, assume that ψ is pseudoconcaveon (r,∞) and consider the function ψ0 from this lemma, where ε = r. Then

ψ(t)

ψ(s) ψ0(t)

ψ0(s)≤ c0 max

1,t

s

for any t, s > r. (In fact, c0 = 1 for the concave function ψ0 [189]). Lemma 1.2is proved.

Proof of Theorem 1.9. Sufficiency. Assume that a function ψ ∈ B is pseu-doconcave in the vicinity of ∞. We prove that this function is an interpolationparameter.

We arbitrarily choose the same admissible pairs of Hilbert spaces X =[X0, X1] and Y = [Y0, Y1] and a linear mapping T as in Definition 1.2. LetJX : X1 ↔ X0 and JY : Y1 ↔ Y0 be the generating operators for the pairs Xand Y, respectively. With the help of the spectral theorem, we reduce theseoperators self-adjoint in X0 and Y0 to the form of multiplication by a function;namely,

JX = I−1X (α · IX) and JY = I−1Y (β · IY ). (1.15)


Here, IX : X0 ↔ L2(U, dµ) and IY : Y0 ↔ L2(V, dν) are isometric isomor-phisms, (U, µ) and (V, ν) are spaces with finite measures, and α : U → (0,∞)and β : V → (0,∞) are measurable functions. Since both operators T : X0 →Y0 and T : X1 → Y1 are bounded, we conclude that the following operators arealso bounded:

IY T I−1X : L2(U, dµ)→ L2(V, dν), (1.16)

IY JY T J−1X I−1X : L2(U, dµ)→ L2(V, dν). (1.17)

By virtue of (1.15), we find

IY JY T J−1X I−1X = (β · IY )T I−1X (α−1·).

Hence, (1.17) implies that the operator

IY T I−1X = β−1 · (IY JY T J−1X I−1X )(α·) : L2(U,α

2dµ)→ L2(V, β2dν) (1.18)

is bounded.Let ψ0 : (0,∞) → (0,∞) be the same concave function as in Lemma 1.1.

Note that ψ0 ∈ B and

Xψ = Xψ0 , Yψ = Yψ0 with equivalence of the norms (1.19)

(see Remark 1.1). J. Peetre proved [189] that a positive function is pseudocon-cave on (0,∞) if and only if it is an interpolation function of power p > 0 (seealso [24, Theorem 5.4.4]). For the concave function ψ0 and the case p = 2, hisresult means that the boundedness of operators (1.16) and (1.18) implies theboundedness of the operator

IY T I−1X : L2(U, (ψ0 α2) dµ)→ L2(V, (ψ0 β2) dν). (1.20)

We now pass from operator (1.20) to the operator T : Xψ0 → Yψ0 by usingthe isometric isomorphisms ψ0(JX) : Xψ0 ↔ X0 and ψ0(JY ) : Yψ0 ↔ Y0. Wereduce these isomorphisms to the form of multiplication by a function

IX ψ0(JX) = (ψ0 α) · IX : Xψ0 ↔ L2(U, dµ),

IY ψ0(JY ) = (ψ0 β) · IY : Yψ0 ↔ L2(V, dν).

As a result, we obtain the isometric isomorphisms

IX = (ψ−10 α) · (IX ψ0(JX)) : Xψ0 ↔ L2(U, (ψ2 α) dµ),

IY = (ψ−10 β) · (IY ψ0(JY )) : Yψ0 ↔ L2(V, (ψ2 β) dν).


Together with (1.20), they imply the boundedness of the operator

T = I−1Y (IY T I−1X )IX : Xψ0 → Yψ0 .

Thus, in view of equalities (1.19), we find

(T : Xj → Yj , j = 0, 1) ⇒ (T : Xψ0 → Yψ0) ⇒ (T : Xψ → Yψ),

where the linear operators are bounded. Hence, by Definition 1.2, the functionψ is an interpolation parameter. Sufficiency is proved.

Necessity. Assume that a function ψ ∈ B is an interpolation parameter. Letus show that ψ is pseudoconcave in the vicinity of ∞. We proceed by analogywith [189] and [24, Sec. 5.4, p. 117].

Consider a space L2(U, dµ), where U = 0, 1, µ(0) = µ(1) = 1, anddefine a linear mapping T on this space by the formulas (Tu)(0) = 0 and(Tu)(1) = u(0), where u ∈ L2(U, dµ). We choose arbitrary numbers s, t > 1and set ω(0) := s2, ω(1) := t2. Thus, we get an admissible pair of spacesX := [L2(U, dµ), L2(U, ω dµ)] and bounded operators

T : L2(U, dµ)→ L2(U, dµ) and T : L2(U, ω dµ)→ L2(U, ω dµ)

whose norms are equal to 1 and t/s respectively. Since ψ is an interpolationparameter, we get a bounded operator T : Xψ → Xψ whose norm satisfies theinequality

‖T‖Xψ→Xψ ≤ c max1,t

s

(1.21)

by virtue of Theorem 1.8, where we take Y = X and m = 1. Here, the numberc > 0 is independent of t, s > 1.

One can easily compute the norm in the space Xψ. Indeed, the operator J ofmultiplication by the function ω1/2 is the generating operator for the pair X.Thus, since ψ(J) is the operator of multiplication by the function ψ ω1/2, wecan write

‖u‖2Xψ = ‖(ψ ω1/2) · u‖2L2(U,dµ)= ψ2(s) |u(0)|2 + ψ2(t) |u(1)|2,

‖Tu‖2Xψ = ψ2(t) |u(0)|2.

This yields

‖T‖Xψ→Xψ =ψ(t)

ψ(s). (1.22)

Relations (1.21) and (1.22) now imply the inequality

ψ(t)

ψ(s)≤ c max

1,t

s

for any t, s > 1.

Section 1.2 Regularly varying functions and their generalization 29

By Lemma 1.2, this is equivalent to the pseudoconcavity of the function ψ onthe semiaxis (1,∞). Necessity is proved.


At the end of this subsection, we give a description of all Hilbert interpolationspaces for a given admissible pair of Hilbert spaces.

Definition 1.5. A Hilbert space H is called an interpolation space for anadmissible pair of Hilbert spaces [X0, X1] if

(i) the continuous embeddings X1 → H → X0 are true;

(ii) any linear operator T : X0 → X0 bounded on each space X0 and X1 isalso a bounded operator on the space H.

The following important result is due to Ovchinnikov [179, p. 511, Theorem11.4.1].

Proposition 1.1. Let X = [X0, X1] be an arbitrary admissible pair of Hilbertspaces. If a Hilbert space H is an interpolation space for this pair, then thereexists a function ψ ∈ B pseudoconcave in the vicinity of ∞ such that the spacesH and Xψ coincide up to equivalence of norms.

Proposition 1.1 and Theorem 1.9 immediately yield the following assertion:

Corollary 1.1. Let X = [X0, X1] be an arbitrary admissible pair of Hilbertspaces. The class of all Hilbert spaces that are interpolation spaces for X coin-cides (up to equivalence of norms) with the class of all spaces Xψ, where ψ ∈ Bis an arbitrary function pseudoconcave in the vicinity of ∞.

1.2 Regularly varying functions and theirgeneralization

Regularly varying functions (and the functions weakly equivalent to them)play a fundamental role in our investigation. We use these functions bothto parametrize the Hörmander spaces and as interpolation parameters.

1.2.1 Regularly varying functions

The following definition is important for our presentation.

Definition 1.6. A positive function ψ given on a real semiaxis [b,∞) is saidto be regularly varying of order θ ∈ R at ∞ if ψ is Borel measurable on [b0,∞)for some number b0 ≥ b and satisfies the condition

limt→∞

ψ(λ t)

ψ(t)= λθ for any λ > 0. (1.23)


A positive function is called slowly varying at ∞ if it is a regularly varyingfunction of order zero at ∞.

The notion of regularly varying function was introduced by J. Karamata [91](for the class of continuous functions). Regularly varying functions are closeto the power functions. They are well studied and have numerous applicationsmainly due to the special role played by these functions in the Tauberian-typetheorems (see the books [26, 128, 199, 235].

By SV we denote the set of all functions slowly varying at ∞. It is clear thatψ is a regularly varying function of order θ at ∞ if and only if ψ(t) = tθϕ(t)for t 1 and a certain function ϕ ∈ SV. Therefore, in the study of regularlyvarying functions, it is sufficient to restrict ourselves to slowly varying functions.

We now present the most known (standard) example [26, Sec.1.3.3] of a slowlyvarying function.

Example 1.1. Let r1, r2, . . . , and rk be k ∈ N given real numbers. We set

ϕ(t) = (log t)r1 (log log t)r2 . . . (log . . . log t)rk for t 1.

Then ϕ ∈ SV.

The functions considered in this example form the so-called logarithmic mul-tiscale, which has numerous applications in the theory of function spaces. Someother examples of functions from the class SV are presented in what follows.

We now formulate two fundamental properties of slowly varying functions.They were proved by J. Karamata [91, 93] for continuous functions and (some-what later) by numerous researchers for measurable functions (see the books[26, Sec. 1.2, 1.3], [235, Sec. 1.2], and the references therein).

Proposition 1.2 (Uniform Convergence Theorem). Let ϕ ∈ SV. Then, for anyfixed compact interval [a, b] with 0 < a < b < ∞, the ratio ϕ(λt)/ϕ(t) tendsto 1 as t→∞ uniformly in λ ∈ [a, b].

Proposition 1.3 (Representation Theorem). Let ϕ ∈ SV. Then

ϕ(t) = exp

(β(t) +

t∫b

α(τ)

τdτ

), whenever t ≥ b (1.24)

for some number b > 0, some continuous function α : [b,∞) → R approachingzero at ∞, and some Borel measurable bounded function β : [b,∞) → R withfinite limit at ∞. The converse assertion is also true: each function of the form(1.24) belongs to the class SV.


Remark 1.2. A function of the form (1.24) with β(t) = const is called a nor-malized slowly varying function. For this function,

α(τ) =τϕ′(τ)

ϕ(τ)

(see [26, Sec. 1.3.2].)

Proposition 1.3 yields the following useful sufficient condition for a functionto be slowly varying (see, e.g., [235, Sec. 1.2]):

Proposition 1.4. Assume that a differentiable function ϕ : (b,∞) → (0,∞)satisfies the condition tϕ ′(t)/ϕ(t)→ 0 as t→∞. Then ϕ ∈ SV.

Proposition 1.4 leads to the following three interesting examples of slowlyvarying functions:

Example 1.2. Let ψ(t) := expϕ(t) for t 1, where the function ϕ is takenfrom Example 1.1 in which r1 < 1. Then ψ ∈ SV.

Example 1.3. Let α, β, γ ∈ R with β 6= 0 and 0 < γ < 1. We set

ω(t) := α+ β sin lnγ t and ϕ(t) := (ln t)ω(t)

for t > 1. Then ϕ ∈ SV.

Example 1.4. Let α, β, γ ∈ R with α 6= 0 and 0 < γ < β < 1. We set

r(t) := α(ln t)−β sin lnγ t and ϕ(t) := t r(t)

for t > 1. Then ϕ ∈ SV.

The last two examples show that a function ϕ slowly varying at∞ may haveinfinite oscillation, i.e.,

lim inft→∞

ϕ(t) = 0 and lim supt→∞

ϕ(t) =∞.

1.2.2 Quasiregularly varying functions

We use regularly varying functions of order θ ∈ (0, 1) as interpolation parame-ters. According to Remark 1.1, the property to be an interpolation parameteris inherited under the transition to a (weakly) equivalent function. Hence, thefollowing generalization of the notion of regularly varying function is useful:


Definition 1.7. A positive function ψ given on the real semiaxis [b,∞) is calleda quasiregularly varying function of order θ ∈ R at ∞ if there exist a numberb1 ≥ b and a regularly varying function ψ1 : [b1,∞) → (0,∞) of order θ at ∞such that ψ(t) ψ1(t) for t ≥ b1.

A positive function is called quasislowly varying at∞ if it is a quasiregularlyvarying function of order zero at ∞.

By QSV we denote the set of all functions quasislowly varying at ∞.

Theorem 1.10. The class QSV consists of all functions of the form (1.24),where b is a positive number, α : [b,∞)→ R is a continuous function approach-ing zero at ∞, and β : [b,∞)→ R is a bounded function.

Proof. By Definition 1.7, ϕ ∈ QSV if and only if ϕ(t) = ω(t)ϕ1(t) for t 1,where ϕ1 ∈ SV, and ω is a positive function such that both ω and 1/ω arebounded in the vicinity of ∞. Therefore, according to Proposition 1.3,

ϕ ∈ QSV ⇔ ϕ(t) = exp

(logω(t) + β(t) +

t∫b

α(τ)

τdτ

)for t ≥ b,

where the functions α and β satisfy the condition of this proposition and thenumber b 1. This yields Theorem 1.10 because the function logω + β isbounded on the semiaxis [b,∞).


The following interpolation property of quasiregularly varying functions playsa fundamental role in our subsequent investigations:

Theorem 1.11. Assume that ψ ∈ B is a quasiregularly varying function oforder θ at ∞ with 0 < θ < 1. Then ψ is an interpolation parameter.

Proof. We write ψ(t) = tθϕ(t) for t > 0, where ϕ ∈ QSV. By using Theo-rem 1.10, we represent the function ϕ in the form (1.24), where the functionsα and β satisfy the condition of this theorem. We set

ε := minθ, 1− θ > 0

and choose a number bε ≥ b such that |α(t)| < ε, whenever t > bε. For anyt, s > bε, in view of (1.24), we get

ϕ(t)

ϕ(s)= exp

(β(t)− β(s) +

t∫s

α(τ)

τdτ

)

≤ c exp∣∣∣ t∫s

ε

τdτ∣∣∣ = cmax

( ts

)ε,(st

)ε.


Here, the number c > 0 is independent of t and s because the function β isbounded. Further, since 0 ≤ θ ± ε ≤ 1, we obtain

ψ(t)

ψ(s)=tθϕ(t)

sθϕ(s)≤ cmax

( ts

)θ+ε,( ts

)θ−ε≤ cmax

1,t

s

for t, s > bε.

Hence, by virtue of Lemma 1.2, the function ψ ∈ B is pseudoconcave in thevicinity of∞. According to Theorem 1.9, this is equivalent to the assertion thatψ is an interpolation parameter. Theorem 1.11 is proved.

Remark 1.3. The direct proof of Theorem 1.11 (without using Theorem 1.9)can be found in [144, Sec. 2].

Remark 1.4. Theorem 1.11 is not true in the limiting cases θ = 0 and θ = 1even if we additionally assume that ψ(t)→∞ as t→∞ for θ = 0 or ψ(t)/t→ 0as t→∞ for θ = 1.

We now present the corresponding examples.

Example 1.5. Case θ = 0. Let

h(t) := (ln t)−1/2 sin ln1/4 t

for t > 1. We define the function ψ as follows: ψ(t) := th(t) + ln t for t ≥ 3,and ψ(t) := 1 for 0 < t < 3. Note that ψ ∈ B and ψ(t) → ∞ as t → ∞. Weimmediately establish the convergence tψ′(t)/ψ(t) → 0 as t → ∞. By virtueof Proposition 1.4, this means that the function ψ is slowly varying at ∞.Let us show that ψ is not pseudoconcave in the vicinity of ∞ and, hence,by Theorem 1.9, it is not an interpolation parameter. Consider sequences ofnumbers

tk := exp((2πk + π/2)4) and sk := exp((2πk + π)4),

where k ∈ N. Further, we compute h(tk) = (2πk + π/2)−2 and h(sk) = 0. Thisyields

lnψ(tk) ≥ h(tk) ln tk = (2πk + π/2)2,

ψ(sk) = 1 + (2πk + π)4.

Thus,ψ(tk)

ψ(sk)≥ exp((2πk + π/2)2)

(1 + (2πk + π)4)→∞ as k →∞.

However tk < sk and, hence, by Lemma 1.2, we conclude that the function ψcannot be pseudoconcave in the vicinity of ∞.


Example 1.6. Case θ = 1. By using the function ψ from Example 1.5, we setψ1(t) := t/ψ(t) for t > 0. By the definition, ψ1 is a regularly varying functionof order θ = 1 at ∞. Note that ψ1 ∈ B and ψ1(t)/t = 1/ψ(t) → 0 as t → ∞.By Theorem 1.4, the function ψ1 is not an interpolation parameter because,otherwise, the function ψ(t) = t/ψ1(t) must be an interpolation parameter.However, as shown in the previous example, this is impossible.

Further, we use the following properties of the class QSV.

Theorem 1.12. Let ϕ, χ ∈ QSV. The following assertions are true:

(i) there exists a positive function ϕ1 ∈ C∞((0;∞)) ∩ SV such that ϕ(t) ϕ1(t) for t 1;

(ii) for any number θ > 0, t−θϕ(t)→ 0 and tθϕ(t)→∞ as t→∞;

(iii) the functions ϕ+χ, ϕχ, ϕ/χ, and ϕσ with σ ∈ R belong to the class QSV;

(iv) let θ ≥ 0 be an arbitrary number and, in addition, ϕ(t) → ∞ as t → ∞for θ = 0; then the composite function χ(tθϕ(t)) of argument t belongs tothe class QSV.

Proof. If we additionally assume that ϕ, χ ∈ SV, then we get the well-known[235, Sec. 1.5] properties of slowly varying functions. (Under this assumption,we get even the relation of strong equivalence ϕ(t) ∼ ϕ1(t) as t → ∞ inassertion (i).) This immediately yields assertions (i), (ii), and (iii) for thefunctions ϕ, χ ∈ QSV.

We now prove assertion (iv). Let λ > 0 be an arbitrary number. Since ϕ ∈QSV, the quantities ϕ(λt)/ϕ(t) and ϕ(t)/ϕ(λt) are both bounded as functionsof t 1. Hence, for any positive function χ1 ∈ SV satisfying the conditionχ1(τ) χ(τ) whenever τ 1, by virtue of Proposition 1.2, we get

χ1

((λt)θϕ(λt)

)χ1

(tθϕ(t)

) =χ1

(λθϕ(λt)ϕ(t) tθϕ(t)

)χ1

(tθϕ(t)

) → 1 as t→∞.

Here, we have used the fact that tθϕ(t) → ∞ as t → ∞. Hence, the func-tion χ1(t

θϕ(t)) slowly varies at ∞. At the same time, χ(tθϕ(t)) χ1(tθϕ(t))

whenever t 1. Thus, the function χ(tθϕ(t)) of t belongs to the class QSV.Assertion (iv) and, therefore, Theorem 1.12 are proved.

In what follows, we also need the following generalization of assertion (iv) inTheorem 1.12 to the case where θ = 0 and ϕ(t) 9∞ as t→∞.

Theorem 1.13. Suppose that ϕ : [b,∞)→ (0,∞) and χ : (0,∞)→ (0,∞) arefunctions from the class QSV. Assume that the function 1/ϕ is bounded on thesemiaxis [b,∞) and that the functions χ and 1/χ are bounded on every segment


[a1, a2], where 0 < a1 < a2 < ∞. Then the composite function χ(ϕ(t)) of theargument t belongs to the class QSV.

Proof. By the condition, ϕ(t) ≥ r for all t ≥ b and some r > 0. According toTheorem 1.12(i), there exist a number b1 ≥ maxb, r and continuous functionsϕ1, χ1 : [b1,∞) → (0,∞) slowly varying at ∞ and such that ϕ(t) ϕ1(t)and χ(t) χ1(t) whenever t ≥ b1. In this case, it is possible to assume thatϕ1(t) ≥ r for t ≥ b1. In addition, let ϕ1(t) := ϕ1(b1) and χ1(t) := χ1(b1) for0 < t < b1. Then ϕ1, χ1 ∈ C((0,∞)) and ϕ1(t) ≥ r for t > 0 and, moreover,χ(t) χ1(t) for t ≥ r in view of the condition of this theorem. We now showthat χ(ϕ(t)) χ1(ϕ1(t)) whenever t ≥ b1 and that χ1(ϕ1(t)) is a functionslowly varying at ∞. This would imply the required property χ ϕ ∈ QSV.

Let t ≥ b1. We havec−1 ≤ ϕ1(t)/ϕ(t) ≤ c

for some number c ≥ 1. Thus, by Proposition 1.2, the function χ1 ∈ SV pos-sesses the property

χ1(ϕ1(t))

χ1(ϕ(t))=χ1

(ϕ1(t)ϕ(t) ϕ(t)

)χ1(ϕ(t))

→ 1 as ϕ(t)→∞.

Hence, there exists a number % ≥ r such that χ1(ϕ(t)) χ1(ϕ1(t)) wheneverϕ(t) ≥ %. Moreover,

χ1(ϕ(t)) χ(ϕ(t)) 1 χ(ϕ1(t)) χ1(ϕ1(t))

provided that r ≤ ϕ(t) ≤ %. Hence,

χ(ϕ(t)) χ1(ϕ(t)) χ1(ϕ1(t)) whenever t ≥ b1.

We fix a number λ > 0 and prove that

χ1(ϕ1(λt))

χ1(ϕ1(t))→ 1 as t→∞.

We choose an arbitrary number ε > 0. Since ϕ1 ∈ SV, the function βλ(t) :=ϕ1(λt)/ϕ1(t) → 1 as t → ∞. In particular, 1/2 ≤ βλ(t) ≤ 2 for t ≥ tλ > 0.Therefore, by virtue of Proposition 1.2, for the function χ1 ∈ SV, there existsa number k = k(ε) > r such that∣∣∣∣χ1(ϕ1(λt))

χ1(ϕ1(t))− 1

∣∣∣∣ = ∣∣∣∣χ1(βλ(t)ϕ1(t))

χ1(ϕ1(t))− 1

∣∣∣∣ < ε (1.25)

whenever t ≥ tλ and ϕ1(t) > k. In addition, since the function χ1 > 0 isuniformly continuous on the segment [r, k + 1], there exists a number m =


m(ε) > 0 such that

|χ1(ϕ1(λt))− χ1(ϕ1(t))| = |χ1(βλ(t)ϕ1(t))− χ1(ϕ1(t))|

< ε minχ1(τ) : r ≤ τ ≤ k

whenever t ≥ m and r ≤ ϕ1(t) ≤ k. Hence,∣∣∣∣χ1(ϕ1(λt))

χ1(ϕ1(t))− 1

∣∣∣∣ = |χ1(ϕ1(λt))− χ1(ϕ1(t))|χ1(ϕ1(t))

< ε (1.26)

for t ≥ m and r ≤ ϕ1(t) ≤ k. Relations (1.25) and (1.26) now yield theinequality ∣∣∣∣χ1(ϕ1(λt))

χ1(ϕ1(t))− 1

∣∣∣∣ < ε for t ≥ maxtλ,m.

Since the positive number ε is chosen arbitrarily, this inequality means that

χ1(ϕ1(λt))

χ1(ϕ1(t))→ 1 as t→∞.

Hence, the function χ1(ϕ1(t)) is slowly varying at ∞.Theorem 1.13 is proved.

1.2.3 Auxiliary results

In the present subsection, we prove two auxiliary assertions about the propertiesof the class QSV. They will be used in what follows.

Lemma 1.3. Suppose that a function ϕ ∈ QSV is positive on the semiaxis[1,∞) and bounded, together with the function 1/ϕ, on every segment [1, b],where 1 < b <∞. Then, for any number ε > 0, there exists a number c(ε) > 0such that

ϕ(t)

ϕ(s)≤ c(ε) (1 + |t− s|)ε for any t ≥ 1 and s ≥ 1. (1.27)

Proof. Without loss of generality, we can assume that 0 < ε < 1. Ac-cording to Theorem 1.10, we represent ϕ in the form (1.24). Since, in thisrepresentation, we have α(τ) → 0 as τ → ∞, there exists a number bε ≥ 1such that |α(τ)| ≤ ε for τ ≥ bε. We choose arbitrary numbers t ≥ 1 and s ≥ 1.In our proof, c1, c2, and c3 are finite positive constants independent of t and s.We prove (1.27) separately for four possible cases of location of the numbers tand s relative to bε.


In the first case, t ≥ bε and s ≥ bε. By virtue of (1.24), we find

ϕ(t)

ϕ(s)= exp

t∫s

α(τ)

τdτ ≤ exp

∣∣∣∣t∫

s

ε

τdτ

∣∣∣∣ = max( t

s

) ε,(st

) ε

= max(

1 +t− ss

) ε,(1 +

s− tt

) ε≤ (1 + |t− s|) ε.

In the second case, t ≥ bε and 1 ≤ s ≤ bε. By virtue of Theorem 1.12(ii) andthe condition of the lemma, we get ϕ(t) ≤ c1 t ε and 1/ϕ(s) ≤ c1. This yields

ϕ(t)

ϕ(s)≤ c 21 tε = c 21 (s+ (t− s)) ε

≤ c 21 (bε + |t− s|) ε ≤ c 21 bε(1 + |t− s|) ε.

In the third case, 1 ≤ t ≤ bε and s ≥ bε. By analogy with the previous case, weobtain 1/ϕ(s) ≤ c2 s ε, ϕ(t) ≤ c2, and hence,

ϕ(t)

ϕ(s)≤ c 22 s ε = c 22 (t+ (s− t)) ε

≤ c 22 (bε + |s− t|) ε ≤ c 22 bε(1 + |t− s|) ε.

In the fourth case, 1 ≤ t ≤ bε and 1 ≤ s ≤ bε. This case is trivial. Indeed,

ϕ(t)

ϕ(s)≤ c3 ≤ c3(1 + |t− s|)ε.

Thus, inequality (1.27) holds for any t ≥ 1 and s ≥ 1.Lemma 1.3 is proved.

Lemma 1.4. Let ψ1 ∈ QSV be a function positive and continuous on the semi-axis [1,∞) and satisfying the condition

I1 :=

∞∫1

d t

t ψ1(t)<∞. (1.28)

Then there exists a positive continuous function ψ0 ∈ SV given on [1,∞) suchthat ψ0(t)/ψ1(t)→ 0 as t→∞ and

∞∫1

d t

t ψ0(t)<∞. (1.29)


Proof. In view of Theorem 1.12(i), without loss of generality, we can assumethat ψ1 ∈ C((0,∞)) ∩ SV. We set

ϕ(t) :=

∞∫t

d t

t ψ1(t)and ψ0(t) := ψ1(t)

√ϕ(t) for t ≥ 1. (1.30)

By the condition, ϕ is a finite positive function on [1,∞) such that ϕ(t) → 0as t → ∞. Moreover, ϕ has the continuous derivative ϕ ′(t) = −(t ψ1(t))

−1,whenever t ≥ 1. Thus, in view of the inclusion ψ1 ∈ SV, we get, by using theL’Hospital rule,

limt→∞

ϕ(λ t)

ϕ(t)= lim

t→∞

λ (λ tψ1(λ t))−1

(t ψ1(t))−1= lim

t→∞

ψ1(t)

ψ1(λ t)= 1

for any λ > 0. Hence, ϕ ∈ SV. We now consider the function ψ 0. It is positiveand continuous on [1,∞). Since ψ1, ϕ ∈ SV, the function ψ0 ∈ SV is slowlyvarying at ∞ (by definition). Furthermore, ψ 0(t)/ψ1(t) =

√ϕ(t) → 0 as

t→∞ and∞∫1

d t

t ψ0(t)=

∞∫1

d t

t ψ1(t)√ϕ(t)

= −∞∫1

dϕ(t)√ϕ(t)

= −0∫

I1

d τ√τ<∞.

Thus, ψ0 satisfies all conditions of the lemma.Lemma 1.4 is proved.

1.3 Hörmander spaces and the refined Sobolev scale

In the present section, we consider an important class of Hörmander innerproduct spaces parametrized with the help of regularly varying functions. Thisis the main class of function spaces in which we study elliptic operators. It iscalled the refined (Sobolev) scale.

1.3.1 Preliminary information and notation

We use the following generally accepted notation for complex linear topologicalspaces of test functions and distributions given in the Euclidean space Rn:

• C∞0 (Rn) and D(Rn) denote the space of all infinitely differentiable func-tions u : Rn → C with compact support;

• S(Rn) is the Schwartz space of all infinitely differentiable functions u :Rn → C such that

max(1 + |x|)m |∂αxu(x)| : x ∈ Rn, |α| ≤ m

<∞

Section 1.3 Hörmander spaces and the refined Sobolev scale 39

for any integer m ≥ 0; here and in what follows α = (α1, . . . , αn) isa multiindex (a vector with nonnegative integer coordinates), |α| := α1 +. . .+ αn, and

∂αxu(x) :=∂|α|u(x)

∂xα11 . . . ∂xαnn

is the partial derivative corresponding to the multiindex α;

• S ′(Rn) is the space of all tempered distributions given in Rn; this space isdual to S(Rn);

• D′(Rn) is the space of all distributions given in Rn; this space is dual toD(Rn).

From the viewpoint of applications to differential operators, it is convenientto interpret distributions as antilinear functionals. Thus, we consider antilinearcontinuous functionals on S(Rn) or D(Rn) as elements of the dual spaces S ′(Rn)or D′(Rn). The mutual duality of the spaces of test functions and distributionsin Rn is considered with respect to the expansion by continuity of a sesquilinearform

(u, v)Rn :=∫Rn

u(x) v(x) dx.

This expansion is also denoted by (u, v)Rn . It is equal to the value of thedistribution u on the test function v.

Every locally Lebesgue integrable function u : Rn → C is identified witha distribution (antilinear functional) v 7→ (u, v)Rn given on the test functionsv ∈ D(Rn). This distribution is called regular. In this sense, we have thefollowing dense and continuous embeddings:

D(Rn) → S(Rn) → S ′(Rn) → D′(Rn).

The Fourier transform of an arbitrary distribution u ∈ S ′(Rn) is denoted byFu or simply by u. If u ∈ S(Rn), then we use the following formula for theFourier transform:

(Fu)(ξ) = u(ξ) :=∫Rn

eix·ξ u(x) dx, ξ ∈ Rn.

Here, as usual, i is the imaginary unit and x · ξ := x1ξ1+ . . .+xnξn is the innerproduct of vectors x, ξ ∈ Rn. The Fourier transform is an isomorphism of thespace S(Rn) onto itself. The inverse Fourier transform is given by the formula

u(x) = (F−1u)(x) = (2π)−n∫Rn

e−ix·ξ u(ξ) dξ, x ∈ Rn.


The Fourier transform is uniquely continued to an isomorphism of the spaceS ′(Rn) onto itself. In this case, the Parseval equality is preserved, namely,

(u, v)Rn = (2π)n(u, v)Rn for all u ∈ S ′(Rn), v ∈ S(Rn).

This equality can be used as a definition of the Fourier transform of an arbitrarydistribution u ∈ S ′(Rn).

We now recall some additional standard notation for function spaces. Thespace Lp(Rn) consists of all Lebesgue measurable functions u : Rn→ C such that

‖u‖pLp(Rn) :=∫Rn

|u(x)|p dx <∞ for 1 < p <∞,

‖u‖L∞(Rn) := ess sup |u(x)| : x ∈ Rn <∞ for p =∞.

The space Lp(Rn) with 1 ≤ p ≤ ∞ is a Banach space with respect to the norm‖u‖Lp(Rn). Naturally, we identify the functions that coincide almost everywhereon Rn as elements of Lp(Rn). In the case p = 2, this space turns into a Hilbertspace. The norm in this space is generated by the inner product (u, v)Rn . TheFourier transform multiplied by (2π)−n/2 is an isometric isomorphism of thespace L2(Rn) onto itself.

As usual, Ck(Rn), where k ≥ 0 is an integer, denotes the space of all functionsu : Rn → C with continuous partial derivatives of any order ≤ k. The subspaceCkb (Rn) is formed by the functions u ∈ Ck(Rn) for which all partial derivativesof the orders ≤ k are bounded on Rn. The space Ckb (Rn) is Banach with respectto the norm

sup|∂αxu(x)| : x ∈ Rn, |α| ≤ k

.

We also use the spaces

C∞(Rn) :=⋂k≥0

Ck(Rn) and C∞b (Rn) :=⋂k≥0

Ckb (Rn).

1.3.2 Hörmander spaces

We now present the definition of function spaces introduced and investigatedby L. Hörmander in [81, Sec. 2.2] (see also [85, Sec. 10.1]). These spaces areformed by the distributions in Rn, where n ∈ N, They are denoted by Bp,µ(Rn).In the present subsection, the scalar index p satisfies the inequality 1 ≤ p ≤ ∞.Moreover, the index µ is a continuous positive function µ = µ(ξ) of ξ ∈ Rnplaying the role of a weight function in the following sense:

Definition 1.8. A function µ : Rn → (0,∞) is called a weight function if thereexist numbers c ≥ 1 and l > 0 such that

µ(ξ)

µ(η)≤ c (1 + |ξ − η|)l for any ξ, η ∈ Rn. (1.31)


The following definition is basic for our presentation.

Definition 1.9. A Hörmander space Bp,µ(Rn) is defined as a linear space ofall tempered distributions u ∈ S ′(Rn) such that their Fourier transform u islocally Lebesgue integrable on Rn and satisfies the inclusion µ u ∈ Lp(Rn). Thenorm in the linear space Bp,µ(Rn) is introduced by the formula

‖u‖Bp,µ(Rn) := ‖µ u‖Lp(Rn).

The space Bp,µ(Rn) is complete with respect to this norm and is continuouslyembedded in S ′(Rn). If 1 ≤ p < ∞, then the space Bp,µ(Rn) is separable andthe set C∞0 (Rn) is dense in this space [81, Theorem 2.2.1]. The case p = 2 inwhich Bp,µ(Rn) becomes a Hilbert space is of especial interest .

Remark 1.5. L. Hörmander [81, Definition 2.1.1] first supposed that the func-tion µ must satisfy a stronger condition than (1.31); namely, that there existpositive numbers c and l such that

µ(ξ)

µ(η)≤ (1 + c |ξ − η|)l for each ξ, η ∈ Rn. (1.32)

However, later he understood [81, see Remark at the end of Sec. 2.1] that thesets of functions satisfying conditions (1.31) or (1.32) lead to the same class ofspaces Bp,µ(Rn).

Remark 1.6. L. R. Volevich and B. P. Paneah [269] introduced and studiedspaces Hµ

p (Rn), 1 < p <∞, closely related to the Hörmander spaces, namely,

Hµp (Rn) :=

u ∈ S ′(Rn) : F−1(µ u) ∈ Lp(Rn)

.

Thus, for p 6= 2, some additional conditions were imposed on the weight func-tion µ. In the Hilbert case p = 2, the spaces Bp,µ(Rn) and Hµ

p (Rn) coincide.

Among the properties of Hörmander spaces, we mention the following im-portant embedding theorem [81, Theorem 2.2.7].:

Proposition 1.5. Let p, q ∈ [1,∞], 1/p+ 1/q = 1, and let µ : Rn → (0,∞) bea continuous weight function. Assume that k ≥ 0 is an arbitrary integer. Thenthe condition

(1 + |ξ|)k µ−1(ξ) ∈ Lq(Rnξ ) (1.33)

implies the continuous embedding Bp,µ(Rn) → Ckb (Rn). Conversely, if

u ∈ Bp,µ(Rn) : suppu ⊂ V ⊂ Ck(Rn)

for some open nonempty set V ⊆ Rn, then condition (1.33) is satisfied.


1.3.3 Refined Sobolev scale

From the viewpoint of applications to elliptic operators, it is reasonable torestrict ourselves to the isotropic Hörmander inner product spaces B2,µ(Rn),where µ(ξ) = 〈ξ〉sϕ(〈ξ〉), s ∈ R, and ϕ ∈ QSV. Here and in what follows,

〈ξ〉 := (1 + ξ21 + . . .+ ξ2n)1/2

is the smoothed modulus of a vector ξ = (ξ1, . . . , ξn) ∈ Rn. This space isdenoted by Hs,ϕ(Rn). Making the choice of a function parameter ϕ somewhatmore specific, we give the definition of the space Hs,ϕ(Rn) :

ByM we denote the set of all functions ϕ : [1;∞)→ (0;∞) such that

(i) ϕ is Borel measurable on the semiaxis [1;∞);

(ii) the functions ϕ and 1/ϕ are bounded on every compact interval [1; b],where 1 < b <∞;

(iii) ϕ ∈ QSV.

Theorem 1.10 immediately yields the following description of the classM:

ϕ ∈M ⇔ ϕ(t) = exp

(β(t) +

t∫1

α(τ)

τdτ

)for t ≥ 1.

Here, α is a continuous function such that α(τ)→ 0 as τ →∞. Moreover, β isa Borel measurable function bounded on the semiaxis [1,∞).

Let s ∈ R and let ϕ ∈M.

Definition 1.10. The linear space Hs,ϕ(Rn) consists of all tempered distri-butions u ∈ S ′(Rn) such that their Fourier transforms u are locally Lebesguesummable on Rn and satisfy the condition∫

Rn

〈ξ〉2sϕ2(〈ξ〉) |u(ξ)|2 dξ <∞.

The inner product in the space Hs,ϕ(Rn) is defined by the formula

(u1, u2)Hs,ϕ(Rn) :=∫Rn

〈ξ〉2sϕ2(〈ξ〉) u1(ξ) u2(ξ) dξ

and induces the norm in this space in a standard way.

By Lemma 1.3, µ(ξ) = 〈ξ〉sϕ(〈ξ〉) is a weight function. Indeed, for anyξ, η ∈ Rn, we get

µ(ξ)

µ(η)=

(〈ξ〉〈η〉

)s ϕ(〈ξ〉)ϕ(〈η〉)

≤ c (1 + |〈ξ〉 − 〈η〉|)|s|+1 ≤ c (1 + |ξ − η|)|s|+1,


where c > 0 is a constant independent of ξ and η. If the function ϕ is contin-uous, then the space Hs,ϕ(Rn) = B2,µ(Rn) is a special case of the Hörmanderspace. Note that the replacement of the continuity condition by a weaker con-dition of Borel measurability realized in the definition of the setM of functionparameters ϕ does not lead to new spaces. This follows from Theorem 1.12(i).Indeed, for any ϕ ∈ M, there exists a function ϕ1 ∈ C∞([1,∞)) ∩ M suchthat ϕ ϕ1 on the semiaxis [1,∞). Hence, the spaces Hs,ϕ(Rn) and Hs,ϕ1(Rn)coincide up to equivalence of norms.

In a special case where ϕ ≡ 1, the space Hs,ϕ(Rn) is equal to the Sobolevinner product space Hs(Rn) of order s. In the general case, we get the followinglemma:

Lemma 1.5. Let s ∈ R and let ϕ ∈M. Then the following continuous embed-dings are true:

Hs+ε(Rn) → Hs,ϕ(Rn) → Hs−ε(Rn) for each ε > 0. (1.34)

Proof. Let ε > 0. Since ϕ ∈ M ⊂ QSV, it follows from Theorem 1.12(ii)that t−ε ≤ ϕ(t) ≤ t ε for t 1. In view of the definition ofM, this means thatthere exists a number c > 1 such that

c−1t−ε ≤ ϕ(t) ≤ c t ε

for all t ≥ 1. Therefore,

c−1〈ξ〉s−ε ≤ 〈ξ〉sϕ(〈ξ〉) ≤ c〈ξ〉s+ε for any ξ ∈ Rn.

This relation immediately yields the continuous embeddings(1.34).Lemma 1.5 is proved.

It is useful to represent embeddings (1.34) in the form⋃ε>0

Hs+ε(Rn) =: Hs+(Rn) ⊂ Hs,ϕ(Rn) ⊂ Hs−(Rn) :=⋂ε>0

Hs−ε(Rn). (1.35)

It is easy to see that the number parameter s specifies the main (power) smooth-ness in the class of spaces

Hs,ϕ(Rn) : s ∈ R, ϕ ∈M, (1.36)

whereas the function parameter ϕ specifies an additional smoothness subordi-nated to the main smoothness. Depending on the convergence ϕ(t) → ∞ orϕ(t) → 0 as t → ∞, the parameter ϕ specifies either the positive or nega-tive additional smoothness. In other words, the parameter ϕ refines the mains-smoothness. Therefore, it is natural to give the following definition:

Definition 1.11. The class of function spaces (1.36) is called the refined Sobolevscale or simply the refined scale given over Rn.


1.3.4 Properties of the refined scale

The relationship between the refined scale and the Sobolev scale is not ex-hausted by embeddings (1.35). It turns out that every space on the refinedscale can be obtained by the interpolation of a pair of Sobolev inner productspaces with an appropriate function parameter.

Theorem 1.14. Let a function ϕ ∈M and positive numbers ε and δ be given.Also let

ψ(t) :=

t ε/(ε+δ) ϕ(t1/(ε+δ)) for t ≥ 1,

ϕ(1) for 0 < t < 1.

Then

(i) the function ψ belongs to the set B and is an interpolation parameter;

(ii) for any s ∈ R, [Hs−ε(Rn), Hs+δ(Rn)

]ψ= Hs,ϕ(Rn)

with equality of the norms.

Proof. (i) By virtue of Theorem 1.12(ii) and (iv), the function ψ belongsto B and is quasiregularly varying at ∞ of the order θ = ε/(ε + δ) ∈ (0, 1).Hence, in view of Theorem 1.11, ψ is an interpolation parameter. Assertion (i)is proved.

(ii) Let s ∈ R. The pair of the Sobolev spaces Hs−ε(Rn) and Hs+δ(Rn)is admissible and the pseudodifferential operator with symbol 〈ξ〉ε+δ is thegenerating operator J for this pair. By using the Fourier transform

F : Hs−ε(Rn)↔ L2

(Rn, 〈ξ〉2(s−ε)dξ

),

the operator is reduced J to the form of multiplication by the function 〈ξ〉ε+δof the argument ξ ∈ Rn. Hence, the operator ψ(J) is reduced to the form ofmultiplication by the function

ψ(〈ξ〉ε+δ) = 〈ξ〉εϕ(〈ξ〉).

Thus, in view of (1.35), we obtain[Hs−ε(Rn), Hs+δ(Rn)

]ψ

=u ∈ Hs−ε(Rn) : 〈ξ〉εϕ(〈ξ〉) u(ξ) ∈ L2

(Rn, 〈ξ〉2(s−ε)dξ

)


=

u ∈ Hs−ε(Rn) :

∫Rn

〈ξ〉2sϕ2(〈ξ〉) |u(ξ)|2 dξ <∞

= Hs−ε(Rn) ∩Hs,ϕ(Rn) = Hs,ϕ(Rn).

Moreover, the norm in the space[Hs−ε(Rn), Hs+δ(Rn)

]ψis equal to

‖ψ(J)u‖Hs−ε(Rn) =

( ∫Rn

|〈ξ〉εϕ(〈ξ〉) u(ξ)|2 〈ξ〉2(s−ε) dξ

)1/2

= ‖u‖Hs,ϕ(Rn).

Assertion (ii) is proved.Theorem 1.14 is proved.

The properties of the refined scale over Rn required for our subsequent pre-sentation are collected in the following theorem:

Theorem 1.15. Let s ∈ R and let ϕ,ϕ1 ∈M. The following assertions hold:

(i) For any ε > 0, the following continuous and dense embedding is true:

Hs+ε,ϕ1(Rn) → Hs,ϕ(Rn)

(ii) The function ϕ(t)/ϕ1(t) is bounded in the vicinity of infinity if and only ifHs,ϕ1(Rn) → Hs,ϕ(Rn). This embedding is continuous and dense.

(iii) For a given integer k ≥ 0, the condition

∞∫1

dt

t ϕ 2(t)<∞ (1.37)

is equivalent to the embedding

Hk+n/2,ϕ(Rn) → Ckb (Rn). (1.38)

This embedding is continuous.

(iv) The spaces Hs,ϕ(Rn) and H−s,1/ϕ(Rn) are mutually dual with respect tothe expansion by continuity of the inner product in L2(Rn).

Proof. (i) By virtue of Lemma 1.5, we have the following continuous em-beddings:

Hs+ε,ϕ1(Rn) → Hs+ε/2(Rn) → Hs,ϕ(Rn) for any ε > 0.


They are dense because the set C∞0 (Rn) is dense in each of these spaces. As-sertion (i) is proved.

(ii) Since ϕ,ϕ1 ∈M, the function ϕ/ϕ1 is bounded in the vicinity of infinityif and only if µ(ξ) ≤ cµ1(ξ) for any ξ ∈ Rn. Here,

µ(ξ) := 〈ξ〉sϕ(〈ξ〉), µ1(ξ) := 〈ξ〉sϕ1(〈ξ〉),

and the constant c > 0 is independent of ξ. Hörmander [81, Theorem 2.2.2]proved that the inequality µ(ξ) ≤ cµ1(ξ) is equivalent to the continuous em-bedding

Hs,ϕ1(Rn) = B2,µ1(Rn) → B2,µ(Rn) = Hs,ϕ(Rn).

This embedding is dense because the set C∞0 (Rn) is dense in these spaces.Assertion (ii) is proved.

(iii) For a given integer k ≥ 0, by virtue of Proposition 1.5 in which we set

µ(ξ) := 〈ξ〉k+n/2 ϕ(〈ξ〉)

and p = 2, it is possible to conclude that∫Rn

dξ

〈ξ〉n ϕ2(〈ξ〉)<∞ ⇔ Hk+n/2,ϕ(Rn) → Ckb (Rn). (1.39)

In this case, the embedding is continuous. Passing to the spherical coordinatesin the integral, after the change of variables t = (1 + r2)1/2, we find

∫Rn

dξ

〈ξ〉n ϕ2(〈ξ〉)= c

∞∫1

rn−1 dr

(1 + r2)n/2 ϕ2((1 + r2)1/2)

= c

∞∫1

(t2 − 1)(n−1)/2

tn ϕ2(t)

t dt

(t2 − 1)1/2= c

∞∫1

ωn(t) dt

t ϕ2(t). (1.40)

Here, c > 0 is a number and

ωn(t) := (√t2 − 1/t)n−2.

Note that the functions ωn and 1/ωn are bounded on the semiaxis [2,∞).Moreover, since n ≥ 1, we get

2∫1

ωn(t)dt <∞.

Section 1.4 Uniformly elliptic operators on the refined scale 47

In view of the definition of the set M, the function ϕ−2(t) is bounded on thesegment [1, 2]. Hence,

∞∫1

dt

tϕ 2(t)<∞⇔

∞∫2

dt

tϕ 2(t)<∞

⇔∞∫2

ωn(t)dt

tϕ2(t)<∞⇔

∞∫1

ωn(t)dt

tϕ2(t)<∞.

By virtue of (1.40), this yields

∞∫1

dt

tϕ 2(t)<∞ ⇔

∫Rn

dξ

〈ξ〉n ϕ2(〈ξ〉)<∞. (1.41)

Relations (1.39) and (1.41) now imply assertion (iii).

(iv) Assertion (iv) is a special case of Hörmander’s result [81, Theorem 2.2.9].Note that ϕ ∈ M ⇔ 1/ϕ ∈ M. Hence, the space H−s,1/ϕ(Rn) belongs to therefined scale.


1.4 Uniformly elliptic operators on the refined scale

In the present section, we study uniformly elliptic pseudodifferential operatorsgiven on the Euclidean space Rn. We obtain an a priori estimate for the solu-tions of an elliptic equation considered on the refined Sobolev scale and studytheir internal smoothness.

1.4.1 Pseudodifferential operators

The detailed presentation of the theory of pseudodifferential operators (PsDOs)can be found, e.g., in the monographs [86, 232, 253, 254] and in the survey [10].For the sake of convenience, we recall the definition of pseudodifferential op-erators on Rn and discuss some related notions necessary for what follows.We mainly use the terminology and notation from the survey by M. S. Agra-novich [10, § 1, 3].

Let m ∈ R. By Sm(R2n) we denote the set of all functions a ∈ C∞(R2n)satisfying the condition: For any multiindices α and β, there exists a numbercα,β > 0 such that

| ∂αx ∂βξ a(x, ξ) | ≤ cα,β 〈ξ〉m−|β| for any x, ξ ∈ Rn.


Definition 1.12. A pseudodifferential operator (PsDO) A on Rn with symbola ∈ Sm(R2n) is defined by the formula

(Au)(x) := (2π)−n∫Rn

e−ix·ξ a(x, ξ) u(ξ) dξ, x ∈ Rn;

here, u is an arbitrary function from the space S(Rn).

By Ψm(Rn) we denote the class of all PsDOs on Rn with symbols fromSm(R2n).

An important example of PsDO in Ψm(Rn) is given by a linear differentialoperator

A(x,D) :=∑|µ|≤m

aµ(x)Dµ (1.42)

of order m with coefficients aµ ∈ Cb(Rn). Its symbol is

a(x, ξ) =∑|µ|≤m

aµ(x) ξµ, x, ξ ∈ Rn.

Here, µ = (µ1, . . . , µn) is a multiindex,

Dµ := i|µ| ∂µx

and, as usual, ξµ := ξµ11 . . . ξµnn for a vector ξ = (ξ1, . . . , ξn). (The Fouriertransformation transforms the differentiation operator Dµ into the operator ofmultiplication by ξµ.)

Note that, as the parameter m increases, the class Ψm(Rn) is extended.We set

Ψ−∞(Rn) :=

⋂m∈R

Ψm(Rn), Ψ

∞(Rn) :=⋃m∈R

Ψm(Rn).

Every PsDO A ∈ Ψ∞(Rn) is continuous in S(Rn) and can be uniquely extendedto a linear continuous operator on S ′(Rn). This operator is also denoted by A.

We now consider a narrower family of polyhomogeneous (or classical) PsDOsmost important for applications. First, we introduce a set of homogeneoussymbols of order m ∈ R.

By Smh (R2n) we denote the set of all infinitely differentiable complex-valuedfunctions b(x, ξ) of x, ξ ∈ Rn with ξ 6= 0 satisfying the following conditions:

(i) b(x, λξ) = λmb(x, ξ) for any λ > 0;

(ii) for any multiindices α and β, there exists a number cα,β > 0 such that

| ∂αx ∂βξ b(x, ξ) | ≤ cα,β |ξ|m−|β| for any x, ξ ∈ Rn with ξ 6= 0.


Definition 1.13. Let A be a PsDO in Ψm(Rn) with symbol a(x, ξ). The op-erator A is called a polyhomogeneous (or classical) PsDO of order m if thereexist a (strictly) decreasing sequence of real numbers (mj)

∞j=0 with m0 = m

approaching −∞ and a sequence of homogeneous symbols (aj)∞j=0 ⊂ Smjh (R2n)

such that

a(x, ξ)− θ(ξ)k∑j=0

aj(x, ξ) ∈ Smk+1(R2n(x,ξ))

for every integer k ≥ 0. In this case, the function a0(x, ξ) is assumed to bedifferent from the zero function. Moreover, the function θ ∈ C∞b (Rn) is setequal to 1 outside a certain neighborhood of the origin and equal to 0 in itssomewhat smaller neighborhood. The function a0(x, ξ) is called the principalsymbol of the polyhomogeneous PsDO A.

By Ψmph(Rn) we denote the class of all polyhomogeneous PsDOs of order m

on Rn. This class is independent of the indicated choice of θ. Since the principalsymbol of a polyhomogeneous PsDO is assumed to be not identically equal tozero, we note that Ψm

ph(Rn) ∩Ψrph(Rn) = ∅ for m 6= r.

The differential operator (1.42) belongs to the class Ψmph(Rn) if at least one

coefficient aµ(x) with |µ| = m is not identically zero. The principal symbol ofthis operator has the form

a0(x, ξ) =∑|µ|=m

aµ(x) ξµ for all x, ξ ∈ Rn.

We are interested in polyhomogeneous PsDOs uniformly elliptic on Rn. Werecall their definition.

Definition 1.14. A PsDO A ∈ Ψmph(Rn) and its principal symbol a0(x, ξ)

are called uniformly elliptic on Rn if there exists a number c > 0 such that|a0(x, ξ)| ≥ c for any x, ξ ∈ Rn with |ξ| = 1.

It is worth noting that every uniformly elliptic PsDO A has a parametrix B;i.e., the following proposition is true [10, Theorem 1.8.3]:

Proposition 1.6. Let a PsDO A ∈ Ψmph(Rn) be uniformly elliptic on Rn. Then

there exists a PsDO B ∈ Ψ−mph (Rn) uniformly elliptic on Rn and such that

BA = I + T1, AB = I + T2, (1.43)

where T1 and T2 are some PsDOs from Ψ−∞(Rn) and I is the identity operatoron S ′(Rn).


1.4.2 A priori estimate of the solutions

Consider a PsDO A ∈ Ψmph(Rn) with m ∈ R. In this and next subsections,

we assume that the PsDO A is uniformly elliptic on Rn. Consider the equationAu = f in Rn. For its solution u, we establish an a priori estimate in the refinedSobolev scale.

Prior to do this, we prove the following lemma on the boundedness of PsDOsin the refined scale.

Lemma 1.6. Let G be a PsDO in Ψr(Rn) with r ∈ R. Then the restriction ofthe mapping u 7→ Gu, u ∈ S ′(Rn), is a bounded linear operator

G : Hσ,ϕ(Rn)→ Hσ−r,ϕ(Rn) (1.44)

for arbitrary parameters σ ∈ R and ϕ ∈M.

Proof. In the Sobolev case with ϕ ≡ 1, this lemma is known (see, e.g.,[10, Theorem 1.1.2] or [86, Theorem 18.1.13]). We choose arbitrary σ ∈ R andϕ ∈M . Consider bounded linear operators

G : Hσ∓1(Rn)→ Hσ∓1−r(Rn).

We apply the interpolation with the function parameter ψ from Theorem 1.14,where we set ε = δ = 1. According to assertion (i) of this theorem, we get thebounded operator

G :[Hσ−1(Rn), Hσ+1(Rn)

]ψ→[Hσ−r−1(Rn), Hσ−r+1(Rn)

]ψ.

By virtue of assertion (ii), this implies that the PsDO G defines the boundedoperator (1.44).

Lemma 1.6 is proved.

It is easy to see that each PsDO in Ψr(Rn) decreases the main smoothness σby r and preserves invariant the additional smoothness ϕ in the spaceHσ,ϕ(Rn).

By virtue of Lemma 1.6, the operator

A : Hs+m,ϕ(Rn)→ Hs,ϕ(Rn)

is bounded for all s ∈ R and ϕ ∈M.

Theorem 1.16. Let s ∈ R, σ > 0, and ϕ ∈ M. There exists a number c =c(s, σ, ϕ) > 0 such that, for any distribution u ∈ Hs+m,ϕ(Rn), the followinga priori estimate is true:

‖u‖Hs+m,ϕ(Rn) ≤ c(‖Au‖Hs,ϕ(Rn) + ‖u‖Hs+m−σ,ϕ(Rn)

). (1.45)


Proof. We use Proposition 1.6. By virtue of the first equality in (1.43), wecan write u = BAu− T1u. This yields estimate (1.45). Indeed,

‖u‖Hs+m,ϕ(Rn) = ‖BAu− T1u‖Hs+m,ϕ(Rn)

≤ ‖BAu‖Hs+m,ϕ(Rn) + ‖T1u‖Hs+m,ϕ(Rn)

≤ c ‖Au‖Hs,ϕ(Rn) + c ‖u‖Hs+m−σ,ϕ(Rn).

Here, c is the maximum of norms of the operators

B : Hs,ϕ(Rn)→ Hs+m,ϕ(Rn), (1.46)

T1 : Hs+m−σ,ϕ(Rn)→ Hs+m,ϕ(Rn). (1.47)

These operators are bounded by Lemma 1.6 and Proposition 1.6.Theorem 1.16 is proved.This theorem improves the well-known a priori estimate for the solutions of

uniformly elliptic equations considered on the Sobolev scale (the case ϕ ≡ 1)[10, Theorem 1.8.4].

1.4.3 Smoothness of the solutions

Assume that the right-hand side of the equation Au = f is characterized bya certain internal smoothness on a given open nonempty set V ⊆ Rn withrespect to the refined Sobolev scale. We study the internal smoothness of thesolution u on this set. First, we consider the case where V = Rn. By H−∞(Rn)we denote the union of all spaces Hs,ϕ(Rn) with s ∈ R and ϕ ∈M. The linearspace H−∞(Rn) is endowed with the topology of inductive limit of spaces [23,Chap. 14, Sec. 2.3].

Theorem 1.17. Assume that u ∈ H−∞(Rn) is a solution of the equationAu = f on Rn, where f ∈ Hs,ϕ(Rn) for certain parameters s ∈ R and ϕ ∈ M.Then u ∈ Hs+m,ϕ(Rn).

Proof. By Theorem 1.15(i), there exists a number σ > 0 such that

u ∈ Hs+m−σ,ϕ(Rn). (1.48)

Thus, the required property follows from this inclusion, the condition of thetheorem, and relations (1.43), (1.46), and (1.47):

u = BAu− T1u = Bf − T1u ∈ Hs+m,ϕ(Rn).



We now consider the general case where V is an arbitrary open nonemptysubset of Rn. We set

Hσ,ϕint (V ) :=

w ∈ H−∞(Rn) : χw ∈ Hσ,ϕ(Rn)

for all χ ∈ C∞b (Rn), suppχ ⊂ V, dist(suppχ, ∂V ) > 0,

where σ ∈ R and ϕ ∈ M. The topology in the space Hσ,ϕint (V ) is given by the

seminorms w 7→ ‖χw‖Hσ,ϕ(Rn), where the functions χ are the same as in thedefinition of this space.

Theorem 1.18. Assume that u ∈ H−∞(Rn) is a solution of the equationAu = f on the set V, where f ∈ Hs,ϕ

int (V ) for some parameters s ∈ R andϕ ∈M. Then u ∈ Hs+m,ϕ

int (V ).

Proof. It is necessary to show that the condition f ∈ Hs,ϕint (V ) yields the

following property of increase in the internal smoothness of the solutions ofequation Au = f : for every number r ≥ 1,

u ∈ Hs−r+m,ϕint (V ) ⇒ u ∈ Hs−r+1+m,ϕ

int (V ). (1.49)

We arbitrarily choose a function χ ∈ C∞b (Rn) such that

suppχ ⊂ V and dist(suppχ, ∂V ) > 0. (1.50)

For this function, there exists a function η ∈ C∞b (Rn) such that

supp η ⊂ V, dist(supp η, ∂V ) > 0,

and η = 1 in the vicinity of suppχ.(1.51)

Indeed, we can define this function with the help of the operation of convolution,namely, η := χ2ε∗ωε, where ε := dist(suppχ, ∂V )/4, χ2ε is the indicator functionfor the 2ε-vicinity of suppχ, and ωε ∈ C∞(Rn) is a function satisfying theconditions

ωε ≥ 0, suppωε ⊆ x ∈ Rn : ‖x‖ ≤ ε, and∫Rn

ωε(x) dx = 1.

We can directly verify that the function η belongs to C∞b (Rn) and has thefollowing properties: η ≡ 1 in the ε-vicinity of the set suppχ, and η ≡ 0 outsidethe 3ε-vicinity of suppχ. Hence, η satisfies conditions (1.51).

As a result of permutation of the PsDO A and the operator of multiplicationby the function χ, we can write

Aχu = Aχηu = χAηu+A′ηu

= χAu+ χA(η − 1)u+A′ηu

= χf + χA(η − 1)u+A′ηu on Rn. (1.52)


Here, the PsDO A′ ∈ Ψm−1(Rn) is the commutator of A and the operator ofmultiplication by χ. By Lemma 1.6, we get the bounded operator

A′ : Hs−r+m,ϕ(Rn)→ Hs−r+1,ϕ(Rn).

Hence,u ∈ Hs−r+m,ϕ

int (V ) ⇒ A′ηu ∈ Hs−r+1,ϕ(Rn). (1.53)

Further, in view of the condition f ∈ Hs,ϕint (V ) and the inequality r ≥ 1, we find

χf ∈ Hs,ϕ(Rn) → Hs−r+1,ϕ(Rn). (1.54)

In addition, since the supports of the functions χ and η − 1 are disjoint, thePsDO χA(η − 1) belongs to Ψ−∞(Rn). This immediately follows from theformula for the symbol of composition of the following two PsDOs: χA andthe operator of multiplication by the function η − 1 (see [10, Theorem 1.2.4]).Since relation (1.48) holds for u ∈ H−∞(Rn) and some σ > 0, by virtue ofLemma 1.6, we arrive at the inclusion

χA(η − 1)u ∈ Hs−r+1,ϕ(Rn) (1.55)

In view of relations (1.52)–(1.55) and Theorem 1.17, we conclude that

u ∈ Hs−r+m,ϕint (V ) ⇒ Aχu ∈ Hs−r+1,ϕ(Rn) ⇒ χu ∈ Hs−r+1+m,ϕ(Rn).

This proves implication (1.49) in view of the arbitrariness of the choice of thefunction χ ∈ C∞b (Rn) satisfying (1.50).

Further, by using implication (1.49), one can readily deduce the inclusionu ∈ Hs,ϕ

int (V ). It is possible to assume that the number σ > 0 in (1.48) isinteger. Hence, u ∈ Hs−σ+m,ϕ

int (V ). We now successively apply implication(1.49) for r = σ, r = σ − 1,..., and r = 1. This enables us to deduce therequired inclusion, namely,

u ∈ Hs−σ+m,ϕint (V ) ⇒ u ∈ Hs−σ+1+m,ϕ

int (V ) ⇒ . . .⇒ u ∈ Hs+m,ϕint (V ).


As applied to the spaces Hs,ϕ(Rn), Theorem 1.18 refines the well-known as-sertions on the increase in internal smoothness of the solutions of linear ellipticequations considered in the Sobolev scale (see, e.g., [21, Chap. III, Theorem 4.3],[52, Chap. II, Lemma 3.2], and [253, Chap. III, Corollary 1.5]). It is easy tosee that the refined smoothness ϕ of the right-hand side of an elliptic equationis inherited by its solutions. If the operator A is differential and the set V isbounded, then Theorem 1.18 is contained in Hörmander’s theorem [81, Theo-rem 7.4.1] on the regularity of solutions of hypoelliptic equations.


Remark 1.7. It is necessary to distinguish internal smoothness from the localsmoothness in an open set V ⊂ Rn. The space of distributions with given localsmoothness on this set is defined as follows:

Hσ,ϕloc (V ) :=

w ∈ H−∞(Rn) :

χw ∈ Hσ,ϕ(Rn) for all χ ∈ C∞0 (Rn), suppχ ⊂ V.

In the case where the set V is bounded, the spaces Hσ,ϕint (V ) and Hσ,ϕ

loc (V )coincide. At the same time, if V is unbounded, then we may get the strictinclusion Hσ,ϕ

int (V ) ⊂ Hσ,ϕloc (V ). An analog of Theorem 1.18 is true for the local

refined smoothness. One must only replace int with loc in the notation of thespaces. This analog readily follows from Theorem 1.18.

By using Theorems 1.18 and 1.15(iii), we can establish the existence of con-tinuous generalized partial derivatives of the solutions of equation Au = f.

Theorem 1.19. Let r ≥ 0 be a given integer and let ϕ ∈M be a function sat-isfying condition (1.37). Assume that a distribution u ∈ H−∞(Rn) is a solutionof the equation Au = f on the open set V ⊆ Rn and that

f ∈ Hr−m+n/2,ϕint (V ). (1.56)

Then the solution u has continuous partial derivatives up to the order r, inclu-sively, on the set V and these derivatives are bounded on each set V0 ⊂ V suchthat dist(V0, ∂V ) > 0. In particular, if V = Rn, then u ∈ Crb (Rn).

Proof. The inclusion u ∈ Hr+n/2,ϕint (V ) holds by virtue of Theorem 1.18 in

which we set s := r−m+n/2. Let a function η ∈ C∞b (Rn) satisfy the conditions

supp η ⊂ V, dist(supp η, ∂V ) > 0, and η = 1 in the vicinity of V0.

This function can be constructed in exactly the same way as in the proof ofTheorem 1.18 if we replace the set suppχ by V0. According to Theorem 1.15(iii),the distribution ηu satisfies the inclusion

ηu ∈ Hr+n/2,ϕ(Rn) → Crb (Rn).

This implies that all generalized partial derivatives of the function u up to theorder r, inclusively, are continuous and bounded in a certain neighborhood ofthe set V0. Hence, these derivatives are also continuous on the set V becausewe can take V0 := x0 for any point x0 ∈ V.


Section 1.5 Remarks and comments 55

Remark 1.8. If we use an analog of Theorem 1.19 for the Sobolev scale, then,instead of (1.56), it is necessary to demand that

f ∈ Hr−m+n/2+ε,1int (V ) for some ε > 0.

This condition is much stronger than (1.56).

Remark 1.9. Condition (1.37) is not only sufficient in Theorem 1.19 but alsonecessary in the class of all solutions of the equation Au = f . Namely, (1.37)is equivalent to the implication(

u ∈ H−∞(Rn) and f := Au ∈ Hr−m+n/2,ϕint (V )

)⇒ u ∈ Cr(V ). (1.57)

Indeed, if u ∈ Hr+n/2,ϕint (V ), then

f = Au ∈ Hr−m+n/2,ϕint (V ),

whence it follows that u ∈ Cr(Ω) provided that (1.57) is true. Therefore, (1.57)implies (1.37) in view of Proposition 1.5 and relation (1.41).

1.5 Remarks and comments

Section 1.1. The first method of interpolation of spaces was independentlyproposed by J.-L. Lions [114] and S. G. Krein [105]. This was the interpolationof pairs of Hilbert spaces with power parameter in which the exponent was usedas a parameter of interpolation. This method was extended to pairs of normedspaces by J.-L. Lions and J. Peetre [115, 122, 187] (real interpolation) and byS. G. Krein [105, 106], J.-L. Lions [116], A. P. Calderon [33], and M. Schechter[226] (complex or holomorphic interpolation). Generally speaking, the complexand real methods of interpolation lead to different interpolation spaces. Theprinciples of construction of general interpolation methods were developed byE. Gagliardo [59].

At present, we have various real and complex interpolation methods fornormed and more general topological spaces with finite collections of numbersplaying the role of interpolation parameters; see the monographs by K. Ben-net and R. Sharpley [20], J. Bergh and J. Löfström [24], Yu. A. Brudnyi andN. Ya. Krugljak [30], S. G. Krein, Yu. I. Petunin, and E. M. Semenov [109],J.-L. Lions and E. Magenes [121], V. I. Ovchinnikov [179], L. Tartar [258], andH. Triebel [258] and the great number of references therein.

A method of interpolation of normed spaces with function parameter wasfirst introduced by C. Foiaş and J.-L. Lions in [57], where the Hilbert casewas considered separately. The interpolation of Hilbert spaces with func-tion parameter was investigated by W.F. Donoghue [46], G. Shlenzak [231],


E. I. Pustyl’nik [195], V. I. Ovchinnikov [179], and the authors [144, 146, 153].Ovchinnikov [179] described (in terms of interpolation with a function parame-ter) all Hilbert spaces obtained as interpolation spaces with respect to a givenpair of Hilbert spaces. Note that, in some applications, this interpolation iscalled the method of variable Hilbert scales; see, e.g., the works by M. Hegland[78, 79], and P. Mathé and U. Tautenhahn [129].

Various methods of interpolation of normed spaces with general function pa-rameters were introduced and studied by T. F. Kalugina [89], J. Gustavsson [71],S. Janson [88], C. Merucci [135], L.-E. Persson [190], N. Ya. Krugljak [110], andV. I. Ovchinnikov [180] for the case of real interpolation and by M. J. Carro andJ. Cerdà [34] for the case of complex interpolation. Note that the interpolationmethods proposed by S. Janson [88] involve a fairly broad class of interpolationparameters, namely, arbitrary positive pseudoconcave functions.

All theorems presented in Section 1.1 were proved by the authors. Thus, The-orems 1.1–1.5, 1.8, and 1.9 were proved in [153, Sec. 2] and Theorems 1.6 and1.7 in [146, Sec. 3]. The proof of Theorem 1.6 is similar to the proof presentedin [258, Sec. 1.17] and the proof of Theorem 1.7 is close to the proof proposedby Geymonat [63, p. 133]. Triebel and Geymonat studied interpolation functorson the category of all compatible pairs of Banach spaces.

Section 1.2. The notion of regularly varying function was introduced byJ. Karamata [91] (in the case of continuous functions). He also established[92, 93] the main properties of regularly varying functions. The theory ofthese functions and its various applications can be found in the monographs byN. H. Bingham, C. M. Goldie, and J. L. Teugels [26], J. L. Geluk and L. de Haan[62], L. de Haan [72], V. Maric [128], S. I. Reshnick [199], and E. Seneta [235].

The notion of quasiregularly varying function was introduced in [146, p. 15]and [153, p. 90]. This notion is convenient in the theory of interpolation ofspaces. All theorems of Section 1.2 (except Theorem 1.13) were proved in [153,Sec. 3.1]. The proof of Theorem 1.13 has never been presented earlier. Thedirect proof of the important interpolation theorem 1.11 was given in [144,Sec. 2]. This proof is based on a modification [231, p. 49] of the interpolationmethod of traces proposed in [121, Chap. 1, § 3 and 5]. The auxiliary lemmas 1.3and 1.4 were established in [144, Sec. 1].

Section 1.3. The theory of distributions can be traced back to S. L. Sobolevand L. Schwartz. Sobolev introduced important Banach spaces of distributions,which are named after him and play a key role in the modern theory of partialdifferential equations. The theory of distributions and Sobolev spaces is pre-sented, e.g., in the monographs by S. L. Sobolev [242, 243], L. Schwartz [228,229], R. A. Adams [1], I. M. Gel’fand and G. E. Shilov [60], S. G. Mikhlin [161],L. Tartar [252], and V. S. Vladimirov [266]. Various applications of Sobolevspaces stimulated their profound investigation, which led to the construction


of new important classes of the spaces of distributions, such as the Nikol’skiispaces, the Besov spaces, the scale of Lizorkin–Triebel spaces, and their variousweight and anisotropic analogs; see the monographs by O. V. Besov, V. P. Il’in,and S. M. Nikol’skii [25], S. M. Nikol’skii [176], H. Triebel [256, 257, 258, 260]and the references therein. These spaces are parametrized by finite collectionsof numbers.

The subsequent generalization of the Sobolev spaces was obtained as a re-sult of the transition from numerical to function parameters. The functionparameters enable one to give a finer description of the regularity propertiesof distributions contained in these spaces as compared with the number pa-rameters. This generalization was realized by B. Malgrange [127] and, sys-tematically, by L. Hörmander in his monograph [81, Chap. II] and L. R. Vole-vich and B. P. Paneah in [269] who introduced and investigated various spacesparametrized with the help of fairly general function parameters. The Hörman-der spaces coincide with the Volevich–Paneah spaces in the Hilbert case. Someapplications of these spaces to the theory of partial differential equations canbe found in the monographs by L. Hörmander [81, 85] and B. P. Paneah [181].

The Hörmander and Volevich–Paneah spaces occupy a central place in thecollection of spaces of generalized smoothness. In the last decades, these spacesserve as the object of numerous profound investigations; see the surveys byG. A. Kalyabin and P. I. Lizorkin [90], the monographs by N. Jacob [87],F. Nicola and L. Rodino [175], and H. Triebel [259, Chapt. III], recent pa-pers by V. I. Burenkov [31], A. M. Caetano and H.-G. Leopold [32], D. E. Ed-munds, P. Gurka, and B. Opic [48], D. E. Edmunds and D. D. Haroske [49],W. Farkas, N. Jacob, and R. L. Schilling [54, 55], W. Farkas and H.-G. Leopold[56], P. Gurka and B. Opic [70], D. D. Haroske and S. D. Moura [74, 75],H.-G. Leopold [112], S. D. Moura [162], and B. Opic and W. Trebels [177], andthe references therein. Various analogs of the Nikol’skii–Besov and Lizorkin–Triebel spaces parametrized with the help of function parameters were con-structed. The exact embedding theorems, extension theorems, theorems ontraces, and other results were established for these spaces. The interpolationproperties of the spaces of generalized smoothness were studied by M. Schechter[226], C. Merucci [136], and F. Cobos and D. L. Fernandez [35].

The definition of the refined Sobolev scale on Rn was given in [143]; itsproperties (Theorems 1.14 and 1.15) were established in [145, Sec. 3].

The term “refined scale” was earlier used by G. Shlenzak [231] for a differ-ent class of Hörmander inner product spaces. This class has no constructivedescription and is not attached to the Sobolev scale. The corresponding classof interpolation parameters is quite narrow and obeys a redundant conditionimposed on the behavior of the parameters near the origin.

Triebel [259, Chapt. III] and Haroske and Moura [74] introduced and studiedsome analogs of the Nikol’skii–Besov and Lizorkin–Triebel normed spaces in


which the regularity of distributions is characterized, as for the refined Sobolevscale, by two parameters, namely, by the main numerical parameter and by anadditional function parameter. In his works, Triebel used logarithmic functionparameters, whereas Haroske and Moura applied more general slowly varyingfunction parameters.

Section 1.4. The algebra of pseudodifferential operators (PsDOs) was con-structed and investigated mainly by J. J. Kohn, L. Nirenberg [96], and L. Hör-mander [82, 84]. Elliptic operators form a very important class of PsDOs andhave various applications to the theory of elliptic boundary-value problems fordifferential equations, spectral theory of differential operators, theory of func-tion spaces, etc. The detailed presentation of the theory of PsDOs can be found,e.g., in the monographs by L. Hörmander [86], M. A. Shubin [232], M. Taylor[253], and F. Treves [254] and in the surveys by M. S. Agranovich [7, 10]. In ourpresentation, we mainly follow the notation and terminology used in [10].

For elliptic differential equations, we know internal a priori estimates of thesolutions in suitable pairs of Hölder spaces (of fractional positive order) orSobolev spaces (of arbitrary real order) and theorems on the local regularity(or smoothness) of the solutions. The presentation of these results and the cor-responding references can be found in Yu. M. Berezansky’s book [21, Chap. III,§ 4]. The regularity of solutions of hypoelliptic differential equations in Hör-mander spaces was investigated by Hörmander in [81, Theorems 4.1.5 and 7.4.1]and [85, Theorems 11.1.8 and 13.4.1]. For elliptic pseudodifferential equationson the Sobolev scale, the internal a priori estimates of the solutions and theo-rems on the local regularity of solutions are presented in the cited papers andbooks devoted to the theory of PsDOs. If a PsDO is uniformly elliptic on Rn,then the a priori estimate holds in the entire Rn. This is also true for the asser-tion concerning the increase in the regularity of solutions; see, e.g., the surveyby Agranovich [10, Sec. 1.8].

All theorems in Section 1.4 were proved in [172] for more general matrixPsDOs uniformly elliptic in Petrovskii’s sense.

Chapter 2

Hörmander spaces on closed manifoldsand their applications

2.1 Hörmander spaces on closed manifolds

In the present section, we consider Hörmander spaces from the refined Sobolevscale on a smooth closed manifold and present the equivalent definitions ofthese spaces similar to the definitions used for the Sobolev spaces.

2.1.1 Equivalent definitions

Throughout the chapter, Γ denotes a closed (i.e., compact and without bound-ary) infinitely smooth oriented manifold of dimension n ≥ 1. Assume thata C∞-density dx is given on Γ. Recall that D′(Γ) is the topological linear spaceof all distributions on Γ. This space is dual to the space C∞(Γ) with respect tothe extension by continuity of the inner product in the space L2(Γ, dx) =: L2(Γ)of square integrable functions f : Γ→ C. We denote this extension by (f, w)Γ,where f ∈ D′(Γ) and w ∈ C∞(Γ).

Let s ∈ R and ϕ ∈ M. We now give three equivalent definitions of theHörmander space Hs,ϕ(Γ).

The first definition characterizes Hs,ϕ(Γ) in terms of the local properties ofdistributions f ∈ D′(Γ).We choose an arbitrary finite atlas from the C∞-struc-ture on Γ. Assume that this atlas is formed by local charts αj : Rn ↔ Γj , wherej = 1, . . . , r. Here, the open sets Γj form a finite covering of the manifold Γ.Wealso choose an arbitrary finite collection of functions χj ∈ C∞(Γ), j = 1, . . . , r,such that 0 ≤ χj(x) ≤ 1 and

∑rj=1 χj(x) = 1 for each x ∈ Γ and, in addition,

suppχj ⊂ Γj . These functions form a decomposition of unit on Γ.

Definition 2.1 (local). By definition, the linear space Hs,ϕ(Γ) consists of alldistributions f ∈ D′(Γ) such that

(χjf) αj ∈ Hs,ϕ(Rn)

for any j ∈ 1, . . . , r. Here, (χjf) αj is the representation of the distributionχjf in the local chart αj . The inner product inHs,ϕ(Γ) is defined by the formula

(f1, f2)Hs,ϕ(Γ) :=r∑j=1

((χjf1) αj , (χj f2) αj)Hs,ϕ(Rn)

for all f1, f2 ∈ Hs,ϕ(Γ). This inner product induces the Hilbert norm in a stan-dard way.

60 Chapter 2 Hörmander spaces on closed manifolds and their applications

In the important special case ϕ ≡ 1, the space Hs,ϕ(Γ) coincides with theSobolev space Hs(Γ) of order s. The latter is complete and, up to equivalence ofnorms, is independent of the indicated choice of an atlas and the decompositionof unit on Γ (see, e.g., [81, Sec. 2.6] or [232, Sec. 7.5]).

The second definition connects the space Hs,ϕ(Γ) with the Sobolev spaces bymeans of interpolation and shows thatHs,ϕ(Γ) is also complete and independentof the choice of an atlas and the decomposition of unit mentioned above.

Definition 2.2 (via interpolation). Let k0 and k1 be two integers such thatk0 < s < k1. By definition,

Hs,ϕ(Γ) :=[Hk0(Γ), Hk1(Γ)

]ψ,

where the interpolation parameter ψ is given by the formula

ψ(t) =

t (s−k0)/(k1−k0) ϕ(t1/(k1−k0)) for t ≥ 1,

ϕ(1) for 0 < t < 1.

Remark 2.1. The function ψ in Definition 2.2 is an interpolation parameterby virtue of Theorem 1.11 because ψ is a regularly varying function at infinityof the order θ = (s− k0)/(k1 − k0) ∈ (0, 1).

The third definition of the space Hs,ϕ(Γ) is useful for the spectral theoryof differential operators. This definition connects the norm in Hs,ϕ(Γ) witha function of 1− ∆Γ, where ∆Γ is the Beltrami–Laplace operator on Γ. (In thiscase, the Riemannian metric is introduced on the manifold Γ; see, e.g., [232,Sec. 22.1] or [255, Chap. XII, § 1].)

Definition 2.3 (via the operator). The Hilbert space Hs,ϕ(Γ) is defined as thecompletion of C∞(Γ) with respect to the norm

f 7→ ‖(1− ∆Γ)s/2ϕ((1− ∆Γ)

1/2) f‖L2(Γ), f ∈ C∞(Γ).

Theorem 2.1. Definitions 2.1, 2.2, and 2.3 are equivalent; namely, they specifythe same Hilbert space Hs,ϕ(Γ) up to equivalence of norms.

We prove Theorem 2.1 in Subsections 2.1.2 and 2.1.3. In connection withthis theorem, it is reasonable to give the following definition.

Definition 2.4. A family of Hilbert spaces

Hs,ϕ(Γ) : s ∈ R, ϕ ∈M

is called the refined Sobolev scale or simply the refined scale over the closedmanifold Γ.

Section 2.1 Hörmander spaces on closed manifolds 61

2.1.2 Interpolation properties

In this subsection,we consider the interpolation properties of the refined Sobolevscale on Γ and obtain, as a consequence, the proof of the equivalence of Defi-nitions 2.1 and 2.2. We use the local definition 2.1 as the original definition ofthe space Hs,ϕ(Γ).

Theorem 2.2. Let ϕ ∈M be an arbitrary function and let ε and δ be arbitrarypositive numbers. Then, for any s ∈ R,[

Hs−ε(Γ), Hs+δ(Γ)]ψ= Hs,ϕ(Γ) (2.1)

with the equivalence of norms. Here, ψ is the interpolation parameter in Theo-rem 1.14.

Proof. The pair of Sobolev spaces appearing on the left-hand side of equal-ity (2.1) is admissible (see, e.g., [232, Proposition 7.4 and Theorem 7.4]). Wededuce this equality from Theorem 1.14 with the help of the well-known pro-cedures of “flattening” and “sewing” of the manifold Γ. By Definition 2.1, thelinear “flattening” mapping

T : f 7→ ( (χ1f) α1, . . . , (χrf) αr ), f ∈ D′(Γ),

specifies the isometric operators

T : Hσ(Γ)→ (Hσ(Rn))r, σ ∈ R, (2.2)

T : Hs,ϕ(Γ)→ (Hs,ϕ(Rn))r. (2.3)

Since the function ψ is an interpolation parameter, the boundedness of op-erators (2.2), where σ ∈ s− ε, s+ δ, implies the boundedness of the operator

T :[Hs−ε(Γ), Hs+δ(Γ)

]ψ→[(Hs−ε(Rn))r, (Hs+δ(Rn))r

]ψ. (2.4)

By virtue of Theorems 1.5 and 1.14, we get the following equalities for thespaces and norms in these spaces:[

(Hs−ε(Rn))r, (Hs+δ(Rn))r]ψ=( [Hs−ε(Rn), Hs+δ(Rn)

]ψ

)r= (Hs,ϕ(Rn))r. (2.5)

Hence, the boundedness of operator (2.4) implies the boundedness of the oper-ator

T :[Hs−ε(Γ), Hs+δ(Γ)

]ψ→ (Hs,ϕ(Rn))r. (2.6)


Further, we construct the left inverse “sewing” operator K for T. For any j ∈1, . . . , r, we choose a function ηj ∈ C∞0 (Rn) such that ηj = 1 on the setα−1j (suppχj). Consider a linear mapping

K : (h1, . . . , hr) 7→r∑j=1

Θj

((ηjhj) α−1j

), h1, . . . , hr ∈ S ′(Rn).

Here, (ηjhj)α−1j is a distribution defined in the open set Γj ⊆ Γ and satisfyingthe following condition: the representative of this distribution in the local chartαj has the form ηjhj . In addition, Θj denotes the operator of extension ofa function/distribution by zero from Γj onto the entire manifold Γ. The operatorΘj is well defined on the distributions whose support lies in Γj . By the choiceof χj and ηj , we can write

KTf =r∑j=1

Θj

((ηj ((χjf) αj)

) α−1j

)

=

r∑j=1

Θj

((χjf) αj α−1j

)=

r∑j=1

χjf = f.

Thus,KTf = f for any f ∈ D′(Γ). (2.7)

We now show that the linear mapping K defines a bounded operator

K : (Hs,ϕ(Rn))r → Hs,ϕ(Γ). (2.8)

For every vector h = (h1, . . . , hr) ∈ (Hs,ϕ(Rn))r, we can write

∥∥Kh∥∥2Hs,ϕ(Γ)

=r∑l=1

∥∥(χlKh) α l∥∥2Hs,ϕ(Rn)

=r∑l=1

∥∥∥(χ l r∑j=1

Θj

((ηjhj) α−1j

)) α l

∥∥∥2H s,ϕ(Rn)

=

r∑l=1

∥∥∥ r∑j=1

(ηj,l hj) β j,l∥∥∥2Hs,ϕ(Rn)

≤r∑l=1

( r∑j=1

∥∥(ηj,l hj) βj,l ∥∥H s,ϕ(Rn)

)2. (2.9)


Here,ηj,l := (χ l αj) ηj ∈ C∞0 (Rn)

and βj,l : Rn ↔ Rn is a C∞-diffeomorphism such that

βj,l = α−1j αl

in the vicinity of the set supp ηj,l and βj,l(x) = x for all x ∈ Rn sufficiently largein the absolute value. It is well known that the operator of multiplication bya function in C∞0 (Rn) and the operator of the change of variables u 7→ u βj,lare bounded in each space Hσ(Rn) with σ ∈ R (see, e.g., [86, Theorems B.1.7and B.1.8]). Therefore, the linear operator v 7→ (ηj,l v) βj,l is bounded inthe space Hσ(Rn). By Theorem 1.14, this implies its boundedness in the spaceHs,ϕ(Rn). Hence, relations (2.9) yield the estimate

∥∥Kh∥∥2Hs,ϕ(Γ)

≤ cr∑j=1

∥∥hj∥∥2Hs,ϕ(Rn),

where the number c > 0 is independent of h = (h1, . . . , hr). Thus, operator(2.8) is bounded for all s ∈ R and ϕ ∈M.

Specifically, the operators K : (Hσ(Rn))r → Hσ(Γ) with σ ∈ R are bounded.Taking the values σ ∈ s− ε, s+ δ and using the interpolation with a param-eter ψ, in view of relation (2.5), we conclude that the operator

K : (Hs,ϕ(Rn))r →[Hs−ε(Γ), Hs+δ(Γ)

]ψ

(2.10)

is bounded. Relations (2.3), (2.10), and (2.7) now imply that the identity oper-ator KT realizes a continuous embedding of the space Hs,ϕ(Γ) in the interpo-lation space [Hs−ε(Γ), Hs+δ(Γ)]ψ. Moreover, it follows from relations (2.6) and(2.8) that the same operator KT realizes the inverse continuous embedding.


In the case where the numbers k0 := s − ε and k1 := s + δ in Theorem 2.2are integer, we obtain the following important result:

Corollary 2.1. Definitions 2.1 and 2.2 are equivalent for any parameters s ∈ Rand ϕ ∈M.

Applying Theorem 2.2 and the properties of interpolation, we get the follow-ing properties of the refined Sobolev scale over the manifold Γ :

Theorem 2.3. Let s ∈ R and let ϕ,ϕ1 ∈ M. The following assertions aretrue:

(i) The space Hs,ϕ(Γ) is complete (Hilbert) and, up to equivalence of norms,is independent of the choice of an atlas of the manifold Γ and the decom-position of unit used in Definition 2.1.


(ii) The set C∞(Γ) is dense in the space Hs,ϕ(Γ).

(iii) The compact and dense embedding Hs+ε,ϕ1(Γ) → Hs,ϕ(Γ) is true for anynumber ε > 0.

(iv) The function ϕ/ϕ1 is bounded in the vicinity of∞ if and only if Hs,ϕ1(Γ) →Hs,ϕ(Γ). This embedding is dense and continuous. It is compact if and onlyif ϕ(t)/ϕ1(t)→ 0 as t→∞.

(v) The spaces Hs,ϕ(Γ) and H−s,1/ϕ(Γ) are mutually dual up to equivalence ofnorms with respect to the extension by continuity of the inner product inL2(Γ).

Proof. (i) By Theorem 2.2, the space Hs,ϕ(Γ) is complete because it is ob-tained as a result of the interpolation of Sobolev inner product spaces. We ar-bitrarily choose two pairs A1 and A2 each of which consists of a finite atlas of Γ

and a relevant decomposition of unit on Γ used in Definition 2.1. ByHs,ϕ(Γ,Aj)we denote the space Hs,ϕ(Γ) corresponding to the pair Aj with j ∈ 1, 2. Simi-larly, Hσ(Γ,Aj) stands for the Sobolev space Hσ(Γ) corresponding to this pair.For the Sobolev spaces, the identity mapping defines an isomorphism

I : Hσ(Γ,A1)↔ Hσ(Γ,A2)

for every σ ∈ R. We now take σ := s∓ 1 and apply the interpolation with thesame parameter ψ as in Theorem 1.14. According to Theorem 2.2, we get theisomorphism

I : Hs,ϕ(Γ,A1)↔ Hs,ϕ(Γ,A2).

This means that the space Hs,ϕ(Γ) is independent of the indicated choice ofthe atlas and the decomposition of unity.

Assertion (i) is proved.

(ii) By virtue of Theorems 1.1 and 2.2, the embedding Hs+δ(Γ) → Hs,ϕ(Γ) iscontinuous and dense. It is known that the set C∞(Γ) is dense in the Sobolevspace Hs+δ(Γ) (see, e.g., [232, Proposition 7.4]). Hence, this set is dense inHs,ϕ(Γ).

Assertion (ii) is proved.

(iii) Let ε > 0. According to Theorem 2.2 there exist interpolation parametersχ, η ∈ B such that [

Hs+ε/2(Γ), Hs+2ε(Γ)]χ= Hs+ε,ϕ1(Γ),

[Hs−ε(Γ), Hs+ε/3(Γ)

]η= Hs,ϕ(Γ)


up to equivalence of norms. Hence, by Theorem 1.1, we get a chain of continuousembeddings

Hs+ε,ϕ1(Γ) → Hs+ε/2(Γ) → Hs+ε/3(Γ) → Hs,ϕ(Γ),

where the central embedding of Sobolev spaces is compact (see, e.g., [232,Theorem 7.4]). Thus, the embedding Hs+ε,ϕ1(Γ) → Hs,ϕ(Γ) is also compact.It is dense in view of assertion (ii).

Assertion (iii) is proved.

(iv) Assume that the function ϕ/ϕ1 is bounded in the vicinity of ∞. Ac-cording to Theorem 2.2, we arrive at the following equalities of spaces up toequivalence of norms: [

Hs−1(Γ), Hs+1(Γ)]ψ= Hs,ϕ(Γ),[

Hs−1(Γ), Hs+1(Γ)]ψ1

= Hs,ϕ1(Γ).

Here, the interpolation parameters ψ,ψ1 ∈ B satisfy the condition

ψ(t)

ψ1(t)=

ϕ(t1/2)

ϕ1(t1/2)

for t ≥ 1. Hence, the function ψ/ψ1 is bounded in the vicinity of ∞ and theembedding Hs,ϕ1(Γ) → Hs,ϕ(Γ) is continuous and dense by virtue of Theo-rem 1.2. Further, if ϕ(t)/ϕ1(t)→ 0 as t→∞, then ψ(t)/ψ1(t)→ 0 as t→∞.Therefore, it follows from the compactness of the embedding of Sobolev spacesHs+1(Γ) → Hs−1(Γ) that the embedding Hs,ϕ1(Γ) → Hs,ϕ(Γ) is compact byTheorem 1.2 provided that ϕ(t)/ϕ1(t)→ 0 as t→∞.

We now show that the embedding Hs,ϕ1(Γ) → Hs,ϕ(Γ) implies that thefunction ϕ/ϕ1 is bounded on the semiaxis [1,∞). Assume that this embedding istrue. We use the local definition 2.1. It is possible to assume that the conditionsU ⊂ Γ1 and U ∩ Γj = ∅ with j 6= 1 are satisfied for some open nonemptyset U ⊂ Γ. For any distribution w ∈ Hs,ϕ1(Rn) such that suppw ⊂ α−11 (U), wefind

Θ(w α−11 ) ∈ Hs,ϕ1(Γ) → Hs,ϕ(Γ).

Here, Θ is the operator of extension of a distribution by zero from the set Uonto the entire Γ. Thus,

w =(χ1(Θ(w α−11 ))

)α1 ∈ Hs,ϕ(Rn).

Hence, according to Hörmander’s result [81, Theorem 2.2.2], the function

ϕ(〈ξ〉)/ϕ1(〈ξ〉)


of ξ ∈ Rn is bounded. Therefore, the function ϕ/ϕ1 is bounded on the semi-axis [1,∞).

Finally, we show that the compactness of the embeddingHs,ϕ1(Γ) → Hs,ϕ(Γ)

implies the convergence ϕ(t)/ϕ1(t)→ 0 as t→∞. If this embedding is compact,then the embedding operator

w ∈ Hs,ϕ1(Rn) : suppw ⊂ α−11 (U) → Hs,ϕ(Rn)

is also compact. Thus, by virtue of the above-mentioned result [81, Theorem2.2.3], we conclude that ϕ(〈ξ〉)/ϕ1(〈ξ〉)→ 0 as |ξ| → ∞. Hence, ϕ(t)/ϕ1(t)→ 0

as t→∞.Assertion (iv) is proved.

(v) Assertion (v) is well known in the Sobolev case ϕ ≡ 1 (see, e.g., [232,Theorem 7.7]). Thus, the Sobolev spaces Hs±1(Γ) and H−s∓1(Γ) are mutuallydual with respect to the extension of the inner product in L2(Γ) by continuity.This means that the linear mapping Q : w 7→ (w, ·)Γ, where w ∈ C∞(Γ), canbe extended by continuity to the isomorphisms

Q : Hs∓1(Γ)↔ (H−s±1(Γ))′.

Applying the interpolation with the same parameter ψ as in Theorem 2.2 forε = δ = 1, we get one more isomorphism

Q :[Hs−1(Γ), Hs+1(Γ)

]ψ↔[(H−s+1(Γ))′, (H−s−1(Γ))′

]ψ. (2.11)

Here, the interpolation space on the left-hand side is Hs,ϕ(Γ), whereas theinterpolation space on the right-hand side can be represented (in view of The-orem 1.4) in the form[

(H−s+1(Γ))′, (H−s−1(Γ))′]ψ=[H−s−1(Γ), H−s+1(Γ)

]′χ= (H−s,1/ϕ(Γ))′.

Note that the last equality is true because

χ(t) :=t

ψ(t)=

t1/2

ϕ(t1/2)for t ≥ 1.

Thus, relation (2.11) implies the isomorphism

Q : Hs,ϕ(Γ)↔ (H−s,1/ϕ(Γ))′,

i.e., the indicated mutual duality of the spaces Hs,ϕ(Γ) and H−s,1/ϕ(Γ).Assertion (v) is proved.


Theorem 2.2 and the reiteration property (Theorem 1.3) now imply the fol-lowing assertion showing that the refined Sobolev scale over Γ is closed withrespect to the interpolation with function parameters quasiregularly varyingat ∞.


Theorem 2.4. Let s0, s1 ∈ R, s0 ≤ s1, and let ϕ0, ϕ1 ∈ M. Assume thatthe function ϕ0/ϕ1 is bounded in the vicinity of ∞ for s0 = s1. Suppose thatψ ∈ B is a function quasiregularly varying at∞ of order θ, where 0 < θ < 1. ByTheorem 1.11, this function is an interpolation parameter. It can be representedin the form ψ(t) = tθχ(t) for some χ ∈ QSV. Also let s := (1− θ)s0 + θs1 andlet

ϕ(t) := ϕ1−θ0 (t)ϕθ1(t)χ

(ts1−s0ϕ1(t)/ϕ0(t)

)for t ≥ 1. (2.12)

Then ϕ ∈M and[Hs0,ϕ0(Γ)Hs1,ϕ1(Γ) ]ψ = Hs,ϕ(Γ) (2.13)

with equivalence of the norms.

Proof. The positive function ϕ is Borel measurable on the set [1,∞) andbounded, together with the function 1/ϕ, on every segment [1, b], 1 < b <∞,because the functions ϕ0, ϕ1, and χ have similar properties. Moreover, thecondition ϕ0, ϕ1, χ ∈ QSV implies the inclusion ϕ ∈ QSV by virtue of Theo-rem 1.12(iii) and (iv) for s0 < s1 or Theorem 1.13 for s0 = s1. Thus, ϕ ∈ M.

Further, according to Theorem 2.3(iii) and (iv), the pair of spaces

[Hs0,ϕ0(Γ), Hs1,ϕ1(Γ)]

is admissible.We now prove equality (2.13). We set

% := s1 − s0 + 1 εj := sj − s+ %, and δj := s− sj + %

for any j ∈ 0, 1. The numbers %, εj , and δj are positive because s0 ≤ s ≤ s1.Note that they have the following properties:

εj + δj = 2%, ε1 − ε0 = s1 − s0, (1− θ)ε0 + θε1 = %. (2.14)

By virtue of Theorem 2.2, we get the equalities[Hsj−εj (Γ), Hsj+δj (Γ)

]ψj

= Hsj ,ϕj (Γ) for each j ∈ 0, 1

with equivalence of the norms. Here, the interpolation parameter ψj is givenby the formula

ψj(t) :=

t εj/(εj+δj) ϕj(t

1/(εj+δj)) for t ≥ 1,

ϕj(1) for 0 < t < 1.(2.15)

Since ψ is an interpolation parameter, sj − εj = s− %, and sj + δj = s+ %, thisyields[

Hs0,ϕ0(Γ), Hs1,ϕ1(Γ)]ψ

=[ [Hs−%(Γ), Hs+%(Γ)

]ψ0,[Hs−%(Γ), Hs+%(Γ)

]ψ1

]ψ

(2.16)


up to equivalence of norms. Note (see (2.14)) that the function

ψ0(t)

ψ1(t)=t(s0−s1)/(2%)ϕ0(t

1/(2%))

ϕ1(t1/(2%))of t ≥ 1

is bounded in the vicinity of∞ by virtue of Theorem 1.12(ii) in the case wheres0 < s1 and by virtue of the condition for s0 = s1. Applying the reiterationtheorem 1.3 to (2.16), we get

[Hs0,ϕ0(Γ), Hs1,ϕ1(Γ) ]ψ =[Hs−%(Γ), Hs+%(Γ)

]ω

(2.17)

up to equivalence of norms. Here, the interpolation parameter ω is given bythe formula

ω(t) := ψ0(t)ψ(ψ1(t)/ψ0(t)) for t > 0.

By virtue of relations (2.14), (2.15), and (2.12), after elementary calculations,we obtain the equalities:

ω(t) = t1/2ϕ(t1/(2%))

for t ≥ 1 and ω(t) = ϕ(1) for 0 < t < 1. Thus, according to Theorem 2.2, wefind [

Hs−%(Γ), Hs+%(Γ)]ω= Hs,ϕ(Γ) (2.18)

up to equivalence of norms. Relations (2.17) and (2.18) now yield (2.13).Theorem 2.4 is proved.

Remark 2.2. Theorem 2.4 is true in the limiting cases θ = 0 or θ = 1 if weadditionally assume that ψ is a pseudoconcave function in the vicinity of ∞.Hence, by Theorem 1.9, the function ψ is an interpolation parameter, and theproof presented above remains valid. Thus, Theorem 2.4 remains true for eachfunction

ψ(t) := lnr t and ψ(t) := t/ lnr t,

where t 1 and r > 0.

Remark 2.3. Theorem 2.4 remains true if we replace Γ in its formulationby Rn. The proof is similar to the proof presented above. Thus, the refinedSobolev scale over Rn (just as its analog over Γ) is closed with respect to theinterpolation with function parameters quasiregularly varying at ∞.

2.1.3 Equivalent norms

In this subsection, we construct equivalent norms in the space Hs,ϕ(Γ) withthe help of elliptic positive-definite PsDOs. As a consequence, we prove thatDefinitions 2.1 and 2.3 are equivalent.


Let A be an elliptic polyhomogeneous PsDO of order m > 0 on the mani-fold Γ. (The definition of PsDO on closed manifolds and all related notions arepresented in Subsection 2.2.1.) Suppose that the operator A : C∞(Γ)→ C∞(Γ)

is positive definite in the space L2(Γ), i.e., there exists a number κ > 0 suchthat

(Au, u)Γ ≥ κ (u, u)Γ for any u ∈ C∞(Γ). (2.19)

By A0 we denote the closure of the operator A : C∞(Γ)→ C∞(Γ) in the spaceL2(Γ). Since PsDO A is elliptic on Γ, this closure exists and is defined onHm(Γ).Condition (2.19) implies that the PsDO A is formally self-adjoint. Therefore,A0 is an unbounded self-adjoint operator in the space L2(Γ) with SpecA0 ⊆[κ,∞). (See, e.g., [10, Theorems 2.3.5 and 2.3.7] or [232, Proposition 8.4 andTheorem 8.3]).

Let s ∈ R and ϕ ∈M. We set

ϕs,m(t) :=

ts/mϕ(t1/m) for t ≥ 1,

ϕ(1) for 0 < t < 1.(2.20)

Since the function ϕs,m is positive and Borel measurable on the semiaxis (0,∞),

the unbounded self-adjoint operator ϕs,m(A0) is defined as a function of A0 ona certain linear manifold in L2(Γ).

Lemma 2.1. The following assertions are true:

(i) The domain of the operator ϕs,m(A0) contains C∞(Γ);

(ii) The mappingf 7→ ‖ϕs,m(A0)f‖L2(Γ), f ∈ C∞(Γ), (2.21)

is a norm in the space C∞(Γ).

Proof. (i) We choose an integer k > s/m. Since ϕ ∈ M, the function ϕs,mis bounded on every compact subset of the semiaxis (0,∞), and, moreover,t−kϕs,m(t) → 0 as t → ∞ by Theorem 1.12(ii) and (iv). Hence, there exists anumber c > 0 such that ϕs,m(t) ≤ c tk for all t ≥ κ. Consider an unboundedoperator Ak0 acting in the space L2(Γ). Since A : C∞(Γ) → C∞(Γ), we canwrite

C∞(Γ) ⊂ DomAk0 ⊂ Domϕs,m(A0).


(ii) According to assertion (i), mapping (2.21) is well defined. For this map-ping, all properties of the norm are obvious except the property of positivedefiniteness.


We now establish this property. In view of the spectral theorem, for anyfunction f ∈ C∞(Γ), we find

‖ϕs,m(A0)f‖2L2(Γ)=

∞∫κ

ϕ2s,m(t) d(Etf, f)Γ, (2.22)

‖f‖2L2(Γ)=

∞∫κ

d(Etf, f)Γ. (2.23)

Here, Et with t ≥ κ, is the resolution of the identity in L2(Γ) corresponding tothe self-adjoint operator A0. Further, if

‖ϕs,m(A0)f‖2L2(Γ)= 0,

then it follows from relation (2.22) and the positiveness of the function ϕs,mthat the measure (E(·)f, f)Γ of the set [κ,∞) is equal to zero. Hence, by virtueof (2.23), we arrive at the equality f = 0 on Γ.

Assertion (ii) is proved.Lemma 2.1 is proved.

Theorem 2.5. For all s ∈ R and ϕ ∈M, the norm (2.21) and the norm in thespace Hs,ϕ(Γ) are equivalent on C∞(Γ). Thus, the space Hs,ϕ(Γ) coincides (upto equivalence of norms) with the completion of the linear space C∞(Γ) withrespect to the norm (2.21).

Proof. First, we assume that s > 0. We choose an integer k ≥ 1 suchthat km > s. Since the operator Ak0 is closed and positive definite on L2(Γ), itsdomain DomAk0 is a Hilbert space with respect to the inner product (Ak0f,Ak0g)Γ

of functions f, g. In this case, the pair of spaces [L2(Γ),DomAk0] is admissibleand Ak0 is a generating operator for this pair. In addition, since Ak0 is the closureof the elliptic PsDO Ak acting in L2(Γ), the spaces DomAk0 and Hkm(Γ) areequal up to equivalence of norms. Let ψ be the interpolation function parameterused in Theorems 1.14 and 2.2, where we set ε = s and δ = km− s. Then

ψ(tk) = ϕs,m(t)

for all t > 0 and, by virtue of Theorem 2.2, we get

‖f‖Hs,ϕ(Γ) ‖f‖[H0(Γ), Hkm(Γ)]ψ ‖f‖[L2(Γ),DomAk0 ]ψ

= ‖ψ(Ak0)f‖L2(Γ) = ‖ϕs,m(A0)f‖L2(Γ),

where f ∈ C∞(Γ).


Now let the real number s be arbitrary. We choose an integer k ≥ 1 suchthat s+ km > 0. In the previous paragraph, we have proved that∥∥g∥∥

Hs+km,ϕ(Γ)∥∥ϕs+km,m(A0) g

∥∥L2(Γ)

, g ∈ C∞(Γ). (2.24)

The PsDO Ak realizes the isomorphisms

Ak : Hσ+km(Γ)↔ Hσ(Γ), σ ∈ R, (2.25)

Ak : Hs+km,ϕ(Γ)↔ Hs,ϕ(Γ). (2.26)

This is proved in the next paragraph. By A−k we denote the operator inverseto Ak. In view of (2.25), we find

g := A−kf ∈⋂σ∈R

Hσ+km(Γ) = C∞(Γ) and Ak0A−kf = f

for any function f ∈ C∞(Γ). By virtue of (2.26) and (2.24), this yields therequired equivalence of norms:∥∥f∥∥

Hs,ϕ(Γ)∥∥A−kf∥∥

Hs+km,ϕ(Γ)∥∥ϕs+km,m(A0)A

−kf∥∥L2(Γ)

=∥∥ϕs,m(A0)A

k0A−kf

∥∥L2(Γ)

=∥∥ϕs,m(A0)f

∥∥L2(Γ)

, f ∈ C∞(Γ).

It remains to show that Ak realizes isomorphisms (2.25) and (2.26). Since thePsDO A is elliptic on Γ, it specifies a bounded Fredholm operator

A : Hσ+m(Γ)→ Hσ(Γ)

for any σ ∈ R. Both the kernel and the index of this operator are independentof σ (see, e.g., [86, Theorem 19.2.1] or [232, Theorem 8.1]). Since 0 /∈ SpecA0,

the self-adjoint operator A0 isomorphically maps its own domain Hm(Γ) ontothe space H0(Γ) = L2(Γ). Hence, the kernel is trivial and the index is equal tozero. Thus, the PsDO A establishes the isomorphisms

A : Hσ+m(Γ)↔ Hσ(Γ), σ ∈ R,

whence, as a result of k iterations, we arrive at isomorphism (2.25). Finally, weset σ = s ∓ 1 in (2.25) and apply the interpolation Theorem 2.2. This yieldsisomorphism (2.26).


For A0 = 1− ∆Γ, Theorem 2.5 gives the following important result:

Corollary 2.2. Definitions 2.1 and 2.3 are equivalent for all parameters s ∈ Rand ϕ ∈M.


It is worth to distinguish the case where the space Hs,ϕ(Γ) coincides withthe domain of the operator ϕs,m(A0).

Theorem 2.6. Let s ≥ 0 and let ϕ ∈ M. In the case where s = 0, we ad-ditionally assume that the function 1/ϕ is bounded in the vicinity of ∞. Thenthe space Hs,ϕ(Γ) coincides with the domain of the operator ϕs,m(A0) and thenorm in the space Hs,ϕ(Γ) is equivalent to the graph norm of the unboundedoperator ϕs,m(A0) acting in L2(Γ).

Proof. The domain Domϕs,m(A0) of the closed operator ϕs,m(A0) is aHilbert space with respect to the inner product of the graph of this operator. Weprove that the graph norm of the operator ϕs,m(A0) is equivalent to norm (2.21)on the linear manifold C∞(Γ) and that this manifold is dense in the spaceDomϕs(A0). By virtue of Theorem 2.5, this yields Theorem 2.6.

By the condition and Theorem 1.12(ii), there exists a number c > 0 suchthat ϕs,m(t) ≥ c for any t > 0. Therefore,∥∥ϕs,m(A0)f

∥∥L2(Γ)

≥ c∥∥f∥∥

L2(Γ)for any f ∈ C∞(Γ).

This yields the indicated equivalence of norms. It remains to show that the setC∞(Γ) is dense in the space Domϕs,m(A0).

Let f ∈ Domϕs,m(A0). Since ϕs,m(A0)f ∈ L2(Γ), there exists a sequence offunctions hj ∈ C∞(Γ) such that hj → ϕs,m(A0)f in L2(Γ) as j → ∞. Since1/ϕs,m(t) ≤ 1/c for every t > 0, the operator ϕ−1s,m(A0) is bounded in thespace L2(Γ). Hence,

fj := ϕ−1s,m(A0)hj → f,

ϕs,m(A0)fj = hj → ϕs,m(A0)f in L2(Γ) as j →∞.

In other words, fj → f with respect to the graph norm of the operator ϕs,m(A0).

In addition, since hj ∈ C∞(Γ), the relation

fj = A−k0 ϕ−1s,m(A0)Ak0 hj ∈ Hkm(Γ)

holds for any k ∈ N. Thus, fj ∈ C∞(Γ). Hence, we have shown that the setC∞(Γ) is dense in the Hilbert space Domϕs,m(A0).


At the end of this subsection, we endow the space Hs,ϕ(Γ) with an equivalentnorm expressed in terms of sequences.

Since the PsDO A is elliptic on Γ and positive definite on L2(Γ), it has thefollowing spectral properties (see, e.g., [10, Sec. 6.1], [253, Chap. 2, § 2], and[232, Sec. 8.3 and 15.2]). The space L2(Γ) has an orthonormal basis (hj)

∞j=1


formed by the eigenfunctions hj ∈ C∞(Γ) of the operator A0. In this case,Ahj = λjhj and the sequence of eigenvalues (λj)

∞j=1 consists of positive num-

bers, is nonstrictly increasing, and approaches ∞. The spectrum of the opera-tor A0 coincides with the set of all its eigenvalues λj : j ∈ N. The followingasymptotic relation is true:

λj ∼ c jm/n as j →∞, (2.27)

where c is a positive number depending on A and n = dim Γ. An arbitrarydistribution f ∈ D′(Γ) can be expanded in the Fourier series

f =∞∑j=1

cj(f)hj convergent in D′(Γ), (2.28)

where cj(f) := (f, hj)Γ are the Fourier coefficients of f with respect to hj .

Theorem 2.7. Let s ∈ R, and let ϕ ∈M. Then

Hs,ϕ(Γ) =

f ∈ D′(Γ) :

∞∑j=1

j 2s/n ϕ2(j 1/n) |cj(f)|2 <∞, (2.29)

‖f‖Hs,ϕ(Γ) ( ∞∑

j=1

j 2s/n ϕ2(j 1/n) |cj(f)|2)1/2

. (2.30)

The proof of this theorem is preceded by three lemmas. As above, in theselemmas, s ∈ R and ϕ ∈M.

Lemma 2.2. For any function f ∈ Dom (ϕs,m(A0)), the following equality istrue:

‖ϕs,m(A0)f‖2L2(Γ)=

∞∑j=1

ϕ2s,m(λj) |cj(f)|2. (2.31)

Proof. By Pj we denote the orthoprojector of the space L2(Γ) onto its one-dimensional subspace lhj : l ∈ C. Then Pjf = cj(f)hj for any f ∈ L2(Γ).

SinceSpecA0 = λj : j ∈ N,

by virtue of the spectral theorem, we obtain

ϕs,m(A0) f =∞∑j=1

ϕs,m(λj)Pjf =∞∑j=1

ϕs,m(λj)cj(f)hj

for every f ∈ Dom (ϕs,m(A0)). These series are convergent in L2(Γ). Thus,in view of the Parseval equality, we get (2.31).



Lemma 2.3. For any f ∈ D′(Γ), the inclusion f ∈ Hs,ϕ(Γ) is equivalent tothe inequality

∞∑j=1

ϕ2s,m(λj) |cj(f)|2 <∞. (2.32)

Proof. Let f ∈ Hs,ϕ(Γ). We now prove inequality (2.32). We choose a se-quence (fk) ⊂ C∞(Γ) such that fk → f in Hs,ϕ(Γ) as k → ∞. According toLemma 2.2 and Theorem 2.5, we get

∞∑j=1

ϕ2s,m(λj) |cj(fk)|2 = ‖ϕs,m(A0)fk‖2L2(Γ)

‖fk‖2Hs,ϕ(Γ) ≤(1 + ‖f‖2Hs,ϕ(Γ)

)for any k ∈ N. In addition, it follows from the continuity of the embeddingHs,ϕ(Γ) → D′(Γ) that

cj(fk) = (fk, hj)Γ → (f, hj)Γ = cj(f) as k →∞

for any j ∈ N. Therefore, by the Fatou lemma for positive series (see, e.g., [23,Vol. 1, Chap. 3, Theorem 6.2]), we obtain

∞∑j=1

ϕ2s,m(λj) |cj(f)|2 ≤ lim inf

k→∞

∞∑j=1

ϕ2s,m(λj) |cj(fk)|2 <∞.

Thus, the inclusion f ∈ Hs,ϕ(Γ) yields inequality (2.32).We now suppose that inequality (2.32) holds and prove the inclusion f ∈

Hs,ϕ(Γ). By virtue of (2.32), the orthogonal series∞∑j=1

ϕs,m(λj) cj(f)hj =: h ∈ L2(Γ) (2.33)

is convergent in the space L2(Γ). Consider its partial sum

wk :=k∑j=1

ϕs,m(λj) cj(f)hj ∈ C∞(Γ).

In view of (2.33), we conclude that

wk → h in L2(Γ) as k →∞. (2.34)

This means that the sequence (ϕ−1s,m(A0)wk)∞k=1 is fundamental in the space

Hs,ϕ(Γ). Indeed, since A0hj = λjhj , we get

ϕ−1s,m(A0)hj = ϕ−1s,m(λj)hj


and

ϕ−1s,m(A0)wk =k∑j=1

ϕs,m(λj) cj(f)ϕ−1s,m(A0)hj

=k∑j=1

cj(f)hj ∈ C∞(Γ).

Hence, by virtue of Theorem 2.5 and relation (2.34), we find

‖ϕ−1s,m(A0)wk − ϕ−1s,m(A0)wp‖Hs,ϕ(Γ) ‖wk − wp‖L2(Γ) → 0 as k, p→∞,

i.e., the sequence (ϕ−1s,m(A0)wk)∞k=1 is fundamental in the space Hs,ϕ(Γ). Its

limit is denoted by g. This enables us to write

g = limk→∞

ϕ−1s,m(A0)wk =∞∑j=1

cj(f)hj in Hs,ϕ(Γ), (2.35)

whence, in view of inequality (2.28), we conclude that f = g ∈ Hs,ϕ(Γ). Thus,inequality (2.32) implies the inclusion f ∈ Hs,ϕ(Γ).


Remark 2.4. It follows from relation (2.35) that, for each f ∈ Hs,ϕ(Γ), theseries (2.28) converges to f in the space Hs,ϕ(Γ).

Lemma 2.4. The following equivalence of norms is true:

‖f‖Hs,ϕ(Γ) ( ∞∑

j=1

ϕ2s,m(λj) |cj(f)|2

)1/2

, f ∈ Hs,ϕ(Γ). (2.36)

Proof. We use the reasoning and notation from the proof of Lemma 2.3.Let f ∈ Hs,ϕ(Γ) and let g be defined by relation (2.35). Then f = g and

‖f‖Hs,ϕ(Γ) = limk→∞

‖ϕ−1s,m(A0)wk‖Hs,ϕ(Γ). (2.37)

Here, we recall that ϕ−1s,m(A0)wk ∈ C∞(Γ). Therefore, by Theorem 2.5, thereexists a number c ≥ 1 independent of f and such that

c−1 ‖wk‖L2(Γ) ≤ ‖ϕ−1s,m(A0)wk‖Hs,ϕ(Γ) ≤ c ‖wk‖L2(Γ) (2.38)

for all k ∈ N. By virtue of (2.33) and (2.34), we get

limk→∞

‖wk‖L2(Γ) = ‖h‖L2(Γ) =

( ∞∑j=1

ϕ2s,m(λj) |cj(f)|2

)1/2

. (2.39)


It remains to pass to the limit as k →∞ in inequality (2.38) and apply equalities(2.37) and (2.39). We immediately obtain the equivalence of norms (2.36).


Proof of Theorem 2.7. It follows from the asymptotic relation (2.27) andthe inclusion ϕ ∈M that

ϕs,m(λj) j s/nϕ(j 1/n) as functions of j ≥ 1. (2.40)

Indeed, according to relation (2.20), we get

ϕs,m(λj) = λs/mj ϕ(λ

1/mj ) for λj ≥ 1.

Since ϕ ∈ M ⊂ QSV, there exists a positive function ϕ1 ∈ SV such thatϕ(t) ϕ1(t) whenever t 1. Hence,

ϕs,m(λj) λs/mj ϕ1(λ1/mj ) provided that j 1. (2.41)

Applying Proposition 1.2 (Uniform Convergence Theorem) to the function ϕ1 ∈SV, in view of relation (2.27), we conclude that

limj→∞

ϕ1(λ1/mj )

ϕ1(j 1/n)= lim

j→∞

ϕ1((λ1/mj j −1/n) j 1/n)

ϕ1(j 1/n)= 1,

whence, by virtue of relations (2.41) and (2.27), we obtain

ϕs,m(λj) j s/nϕ1(j1/n) j s/nϕ(j 1/n) whenever j 1. (2.42)

Since the functions ϕ and 1/ϕ are bounded on each segment [1, b], 1 < b <∞,relation (2.42) implies (2.40).

It remains to note that Theorem 2.7 immediately follows from Lemmas 2.3and 2.4 and relation (2.40).


Example 2.1. Let Γ be a circle of radius 1 and let A := 1 − d2/dt2, where tspecifies a natural parametrization on Γ. The eigenfunctions

hj(t) := (2π)−1eijt, j ∈ Z,

of the operator A form an orthonormal basis in L2(Γ). They correspond to theeigenvalues λj = 1 + j 2. Assume that s ∈ R and ϕ ∈ M. By Theorem 2.5, weobtain

‖f‖2Hs,ϕ(Γ) ‖ϕs,2(A0)f‖2L2(Γ)=

∞∑j=−∞

(1 + j 2)s ϕ2((1 + j 2)1/2) |cj(f)|2.


Note that we can now use the basis formed by the real-valued eigenfunctions

h0(t) := (2π)−1, hj(t) := π−1 cos jt, and h−j(t) := π−1 sin jt

with j ∈ N. Then

‖f‖2Hs,ϕ(Γ) |a0(f)|2 +

∞∑j=1

j 2s ϕ2(j)(|aj(f)|2 + |bj(f)|2

),

where a0(f), aj(f), and bj(f) are the Fourier coefficients of f with respectto these eigenfunctions. In this case, the space Hs,ϕ(Γ) is closely connectedwith the spaces of periodic real functions considered by A. I. Stepanets [248,Chapt. 1, § 7], [249, Part 1, Chapt. 3, Subsection 7.1].

2.1.4 Embedding theorem

We now prove an important theorem on the embedding of the space Hs,ϕ(Γ)

in the space Ck(Γ).

Theorem 2.8. Assume that a function ϕ ∈ M and an integer number k ≥ 0

are given. Then condition (1.37) is equivalent to the embedding

Hk+n/2,ϕ(Γ) → Ck(Γ). (2.43)

This embedding is compact.

Proof. Assume that relation (1.37) holds. Thus, by Theorem 1.15(iii), weget the continuous embedding

Hk+n/2,ϕ(Rn) → Ckb (Rn).

This immediately yields the continuous embedding (2.43) if we use the localdefinition 2.1 of the space Hk+n/2,ϕ(Γ). Indeed, for any f ∈ Hk+n/2,ϕ(Γ), weget

(χjf) αj ∈ Hk+n/2,ϕ(Rn) → Ckb (Rn) for each j ∈ 1, . . . , r.

Hence,

f =r∑j=1

χjf ∈ Ck(Γ).

Moreover,

‖f‖Ck(Γ) ≤r∑j=1

‖χjf‖Ck(Γ) ≤ c1r∑j=1

‖(χjf) αj‖Ckb (Rn)

≤ c2r∑j=1

‖(χjf) αj‖Hk+n/2,ϕ(Rn) ≤ c3 ‖f‖Hk+n/2,ϕ(Γ).


Here, the positive numbers c1, c2, and c3 are independent of f. Embedding(2.43) and its continuity are proved.

We now show that this embedding is compact. In view of Theorem 1.12(i),without loss of generality, we can assume that the function ϕ ∈ M is con-tinuous. By virtue of (1.37), the function ψ1 := ϕ2 satisfies the condition ofLemma 1.4. Let ψ0 be the function appearing in this lemma. Then the func-tion ϕ0 =

√ψ0 ∈M satisfies both the condition ϕ0(t)/ϕ(t)→ 0 as t→∞ and

inequality (1.37) with ϕ0 instead of ϕ. Hence, by virtue of the result provedabove and Theorem 2.3(iv), we find

Hk+n/2,ϕ(Γ) → Hk+n/2,ϕ0(Γ) → Ck(Γ).

Here, the first embedding is compact and the second embedding is continuous.Hence, embedding (2.43) is compact.

It remains to show that the inclusion Hk+n/2,ϕ(Γ) ⊆ Ck(Γ) implies condition(1.37). We again apply the local definition 2.1 and use the reasoning from theproof of assertion (iv) in Theorem 2.3. For any distribution u ∈ Hk+n/2,ϕ(Rn)with suppu ⊂ α−11 (U), we obtain

Θ(u α−11 ) ∈ Hk+n/2,ϕ(Γ) ⊂ Ck(Γ).

Thus, u ∈ Ck(Rn). Hence, by virtue of Proposition 1.5, we arrive at the inclu-sion

〈ξ〉k µ−1(ξ) ∈ L2(Rnξ ),

where

µ(ξ) := 〈ξ〉k+n/2 ϕ(〈ξ〉).

In view of (1.41), this inclusion is equivalent to (1.37).Theorem 2.8 is proved.

2.2 Elliptic operators on closed manifolds

In this section, we study elliptic PsDOs given on infinitely smooth and closed(i.e., compact and without boundary) manifolds. We prove that these PsDOsare bounded and Fredholm in appropriate pairs of spaces from the refinedSobolev scale. We also study the global and local smoothness of solutions to theelliptic equations. Moreover, we consider a class of parameter-elliptic PsDOsspecifying isomorphisms on the refined scale provided that the complex-valuedparameter is sufficiently large in the absolute value.

Section 2.2 Elliptic operators on closed manifolds 79

2.2.1 Pseudodifferential operators on closed manifolds

For the sake of convenience, we now recall the definition of PsDO on a closedmanifold Γ and the related notions required for our presentation. As above, weuse the terminology and notation taken from the survey [10, § 2].

Definition 2.5. Let m ∈ R. A linear operator A : C∞(Γ) → C∞(Γ) is calleda PsDO (on Γ) from the class Ψm(Γ) if the following conditions are satisfied:

(i) For all functions ϕ,ψ ∈ C∞(Γ) whose supports are disjoint, the mappingu 7→ ϕA(ψu), u ∈ C∞(Γ), is extended by continuity to an operator oforder −∞ on the Sobolev scale, i.e., to a bounded operator from Hs(Γ)

into Hs+r(Γ) for all s ∈ R and r > 0;

(ii) We now arbitrarily choose a local C∞-chart α : Rn → Γα on Γ and an openset Ω whose closure is contained in the coordinate neighborhood Γα ⊂ Γ.

Then there exists a PsDO AΩ ∈ Ψm(Rn) such that

(ϕA(ψu)) α = (ϕ α)AΩ((ψu) α) for any u ∈ C∞(Γ)

and all functions ϕ,ψ ∈ C∞(Γ) whose supports lie in Ω.

In connection with item (i) of this definition, we note that the condition“to be an operator of order −∞ on the Sobolev scale” used in (i) is equivalentto the condition “to be an integral operator

u(y) 7→∫Γ

K(x, y)u(y) dy, u ∈ C∞(Γ), (2.44)

on Γ with infinitely smooth kernel K(x, y).” Hence, the presented notion ofPsDO on the manifold Γ is not, in fact, related to any scale of Hilbert spaces.

We setΨ−∞(Γ) :=

⋂m∈R

Ψm(Γ), Ψ

∞(Γ) :=⋃m∈R

Ψm(Γ).

The class Ψ−∞(Γ) coincides with the class of all integral operators (2.44) withinfinitely smooth kernels. Each PsDO A ∈ Ψ∞(Γ) is continuous on C∞(Γ)

and can be uniquely extended to a linear continuous operator on D′(Γ). Thisoperator is also denoted by A.

Definition 2.6. Let m ∈ R. A PsDO A ∈ Ψm(Γ) is called polyhomogeneous(or classical) of order m on Γ if each AΩ in Definition 2.5 belongs to Ψm

ph(Rn).

By Ψmph(Γ) we denote the class of all polyhomogeneous PsDOs of order m

on Γ.


Definition 2.7. Let a PsDO A ∈ Ψmph(Γ) be given. Its principal symbol a0(x, ξ)

is defined as a function of the arguments x ∈ Γ and ξ ∈ T ∗xΓ, where ξ 6= 0,

which coincides (locally in x) with the principal symbol of the correspondingPsDO AΩ ∈ Ψm

ph(Rn).

Here, as usual, T ∗xΓ stands for the cotangent space to the manifold Γ ata point x ∈ Γ. It is important that the principal symbol of a polyhomogeneousPsDO on Γ does not depend on the choice of local charts on Γ and is an infinitelysmooth function of x and ξ positively homogeneous in ξ of order m.

Definition 2.8. A linear operator A+ : C∞(Γ)→ C∞(Γ) is said to be formallyadjoint to a PsDO A ∈ Ψ∞(Γ) if

(Au, v)Γ = (u,A+v)Γ for all u, v ∈ C∞(Γ).

If A = A+, then the PsDO A is called formally self-adjoint.

We note that the notion of formally adjoint PsDO is introduced with respectto the density dx given on Γ because (·, ·)Γ is the inner product in L2(Γ, dx).

If A ∈ Ψm(Γ) for some m ∈ R, then A+ ∈ Ψm(Γ). If, in addition, A is a poly-homogeneous PsDO with the principal symbol a0, then A+ is also a polyhomo-geneous PsDO with the complex conjugate principal symbol a0.

Definition 2.9. A PsDO A ∈ Ψmph(Γ) and its principal symbol a0(x, ξ) are

called elliptic on Γ if a0(x, ξ) 6= 0 for any point x ∈ Γ and each covectorξ ∈ T ∗xΓ \ 0.

If a PsDO A ∈ Ψmph(Γ) is elliptic on Γ, then A has a parametrix B ∈ Ψ

−mph (Γ),

which is also elliptic on Γ; i.e., the analog of Proposition 1.6 is true . However,we do not use this analog.

As an important example of elliptic PsDO on Γ,we can mention the Beltrami–Laplace operator ∆Γ. We now recall its definition. Let a Riemannian met-ric be introduced on the manifold Γ; i.e., let an infinitely smooth covariantreal tensor field g(x), x ∈ Γ, be given, where g(x) = (gj,k(x))

nj,k=1 is a sym-

metric positive definite matrix. The Riemannian metric defines a densitydx := (det g(x))1/2dx1 . . . dx1 on the local coordinates x = (x1, . . . , xn) in Γ.By definition, the action of the Beltrami–Laplace operator upon the functionu ∈ C2(Γ) is given by the formula

(∆Γu)(x) := (det g(x))−1/2n∑

j,k=1

∂xj((det g(x))1/2 gj,k(x) ∂xku(x)

),

where g−1(x) = (gj,k(x))nj,k=1 is the matrix inverse to g(x). The function ∆Γu

is independent of the choice of local coordinates on Γ. The principal symbol of


the differential operator ∆Γ is equal to

n∑j,k=1

gj,k(x) ξj ξk,

and, therefore, the ellipticity of ∆Γ follows from the positive definiteness ofthe matrix g−1(x). The operator ∆Γ is formally self-adjoint with respect to thedensity dx.

2.2.2 Elliptic operators on the refined scale

Consider a given PsDO A ∈ Ψmph(Γ) of order m ∈ R. In Subsections 2.2.2 and

2.2.3, we assume that A is elliptic on Γ.

We set

N := u ∈ C∞(Γ) : Au = 0 on Γ , (2.45)

N+ := v ∈ C∞(Γ) : A+v = 0 on Γ . (2.46)

Since both PsDOs A and A+ are elliptic on Γ, the spaces N and N+ are finite-dimensional [10, Theorem 2.3.3].

We now investigate the properties of the PsDO A on the refined Sobolevscale over Γ. First, we prove the following assertion about the action of eachPsDO on this scale (an analog of Lemma 1.6).

Lemma 2.5. Let G ∈ Ψr(Γ) for some r ∈ R. Then the restriction of themapping u 7→ Gu, u ∈ D′(Γ), to the space Hσ,ϕ(Γ) is a linear bounded operator

G : Hσ,ϕ(Γ)→ Hσ−r, ϕ(Γ) (2.47)

for all parameters σ ∈ R and ϕ ∈M.

Proof. In the Sobolev case where ϕ ≡ 1, this lemma is well known [10, The-orem 2.1.2]. We arbitrarily choose σ ∈ R and ϕ ∈ M. Consider the followingbounded linear operators:

G : Hσ∓1(Γ)→ Hσ∓1−r(Γ).

Further, we apply the interpolation with the same functional parameter ψ asin Theorem 2.2 for ε = δ = 1. We get the following bounded operator:

G :[Hσ−1(Γ), Hσ+1(Γ)

]ψ→[Hσ−r−1(Γ), Hσ−r+1(Γ)

]ψ.

This and Theorem 2.2 imply that operator (2.47) is well defined and bounded.Lemma 2.5 is proved.


In view of Lemma 1.6, we have the bounded operator

A : Hs+m,ϕ(Γ)→ Hs,ϕ(Γ) (2.48)

for any parameters s ∈ R and ϕ ∈ M. We now study the properties of thisoperator.

Theorem 2.9. The bounded operator (2.48) is a Fredholm operator for alls ∈ R and ϕ ∈M. Its kernel coincides with N . Its domain has the form

f ∈ Hs,ϕ(Γ) : (f, w)Γ = 0 for all w ∈ N+. (2.49)

The index of operator (2.48) is equal to dimN − dimN+ and does not dependon s and ϕ.

Proof. For ϕ ≡ 1 (the Sobolev scale), this theorem is known (see, e.g., [10,Theorems 2.2.6, 2.3.3, and 2.3.12] or [86, Theorem 19.2.1]). Thus, the generalcase ϕ ∈ M can be obtained with the help of interpolation with a functionalparameter. Namely, let s ∈ R. We have the following bounded Fredholm oper-ators:

A : Hs∓1+m(Γ)→ Hs∓1(Γ) (2.50)

with common kernel N and identical indices κ := dimN − dimN+. In thiscase,

A(Hs∓1+m(Γ)

)=f ∈ Hs∓1(Γ) : (f, w)Γ = 0 for all w ∈ N+

. (2.51)

We now apply interpolation with the functional parameter ψ from Theorem 2.2to (2.50), where ε = δ = 1. This yields the bounded operator

A :[Hs−1+m(Γ), Hs+1+m(Γ)

]ψ→[Hs−1(Γ), Hs+1(Γ)

]ψ,

which coincides, by virtue of Theorem 2.2, with operator (2.48). Hence, byTheorem 1.7, operator (2.48) is a Fredholm operator with kernel N and indexκ = dimN − dimN+. The domain of this operator is

Hs,ϕ(Γ) ∩A(Hs−1+m(Γ)

).

Hence, in view of (2.51), we conclude that it is equal to (2.49).Theorem 2.9 is proved.

By virtue of this theorem, N+ is the defect subspace of operator (2.48). Notethat, in view of Theorem 2.3(v), the operator

A+ : H−s,1/ϕ(Γ)→ H−s−m,1/ϕ(Γ) (2.52)


is adjoint to operator (2.48). Since PsDO A+ is elliptic on Γ, the boundedoperator (2.52) is Fredholm. By virtue of Theorem 2.9, it has the kernel N+

and the defect subspace N . We also note that the indices of operators (2.48)and (2.52) are equal to 0 provided that dim Γ ≥ 2 (see [18], [10, Sec. 2.3 f]).

If the spaces N and N+ are trivial, then Theorem 2.9 and the Banach theo-rem on inverse operator imply that operator (2.48) is an isomorphism. In thegeneral case, it is convenient to specify the isomorphism with the help of thefollowing projectors.

We represent the spaces of action of operator (2.48) in the form of the directsums of closed subspaces:

Hs+m,ϕ(Γ) = N uu ∈ Hs+m,ϕ(Γ) : (u, v)Γ = 0 for all v ∈ N

,

Hs,ϕ(Γ) = N+ uf ∈ Hs,ϕ(Γ) : (f, w)Γ = 0 for all w ∈ N+

.

The indicated decompositions in direct sums exist because the intersection oftheir terms is trivial and the finite dimension of the first term is equal to thecodimension of the second term. (Indeed, e.g., in the first sum, the quotientspace of the space Hs+m,ϕ(Γ) by the second term is the space dual to thesubspace N of the space H−s−m,1/ϕ(Γ)). By P and P+, we denote the obliqueprojectors of the spaces Hs+m,ϕ(Γ) and Hs,ϕ(Γ) onto the second terms of theindicated sums parallel to the first terms, respectively. These projectors areindependent of s and ϕ.

Theorem 2.10. For any s ∈ R and ϕ ∈M, the restriction of operator (2.48)to the subspace P(Hs+m,ϕ(Γ)) is an isomorphism

A : P(Hs+m,ϕ(Γ))↔ P+(Hs,ϕ(Γ)). (2.53)

Proof. By Theorem 2.9, N is the kernel and P+(Hs,ϕ(Γ)) is the domain ofoperator (2.48). Hence, operator (2.53) is a bijection. In addition, this operatoris bounded. Therefore, it is an isomorphism by the Banach theorem on inverseoperator. Theorem 2.10 is proved.

Theorem 2.10 yields the following a priori estimate for the solution of theelliptic equation Au = f on Γ (an analog of Theorem 1.16 for a closed manifold).

Theorem 2.11. Let s ∈ R, σ > 0, and ϕ ∈ M. There exists a number c =

c(s, σ, ϕ)>0 such that, for any distribution u∈Hs+m,ϕ(Γ), the a priori estimate

‖u‖Hs+m,ϕ(Γ) ≤ c(‖Au‖Hs,ϕ(Γ) + ‖u‖Hs+m−σ,ϕ(Γ)

)(2.54)

is true.


Proof. Since N is a finite-dimensional subspace of the spacesHs+m,ϕ(Γ) andHs+m−σ,ϕ(Γ), their norms are equivalent on N . In particular, the distributionu− Pu ∈ N satisfies the relation

‖u− Pu‖Hs+m,ϕ(Γ) ≤ c1 ‖u− Pu‖Hs+m−σ,ϕ(Γ)

with a constant c1 > 0 independent of u ∈ Hs+m,ϕ(Γ). This yields

‖u‖Hs+m,ϕ(Γ) ≤ ‖u− Pu‖Hs+m,ϕ(Γ) + ‖Pu‖Hs+m,ϕ(Γ)

≤ c1 ‖u− Pu‖Hs+m−σ,ϕ(Γ) + ‖Pu‖Hs+m,ϕ(Γ)

≤ c1 c2 ‖u‖Hs+m−σ,ϕ(Γ) + ‖Pu‖Hs+m,ϕ(Γ),

where c2 is the norm of the projector 1− P acting in the space Hs+m−σ,ϕ(Γ).

Thus,‖u‖Hs+m,ϕ(Γ) ≤ c1 c2 ‖u‖Hs+m−σ,ϕ(Γ) + ‖Pu‖Hs+m,ϕ(Γ). (2.55)

We now use the condition Au = f. Since N is the kernel of operator (2.48), andu−Pu ∈ N , we have APu = f. Hence, Pu is the preimage of the distributionf under isomorphism (2.53). Therefore,

‖Pu‖Hs+m,ϕ(Γ) ≤ c3 ‖f‖Hs,ϕ(Γ),

where c3 is the norm of the operator inverse to (2.53). This result and inequality(2.55) immediately yield estimate (2.54).


Note that if N = 0, i.e., the equation Au = f has at most one solution,then the norm ‖u‖Hs+m−σ(Γ) on the right-hand side of estimate (2.54) can beomitted. However, if N 6= 0, then this quantity can be made arbitrarily smallfor each distribution u by choosing a sufficiently large number σ.

2.2.3 Smoothness of solutions to the elliptic equation

Let Γ0 be a nonempty open subset of the manifold Γ. We study the localsmoothness of solutions to the elliptic equation Au = f on Γ0 in the refinedscale. First, we consider the case where Γ0 = Γ.

Theorem 2.12. Assume that the distribution u ∈ D′(Γ) is a solution of theequation Au = f on Γ, where f ∈ Hs,ϕ(Γ) for some parameters s ∈ R andϕ ∈M. Then u ∈ Hs+m,ϕ(Γ).

Proof. Since the manifold Γ is compact, the space D′(Γ) is the union ofthe Sobolev spaces Hσ(Γ), where σ ∈ R. Hence, for the distribution u ∈ D′(Γ),


there exists a number σ < s such that u ∈ Hσ+m(Γ). By virtue of Theorems 2.9and 2.3(iii), the equality

(Hs,ϕ(Γ)) ∩A(Hσ+m(Γ)) = A(Hs+m,ϕ(Γ))

holds. Hence, the condition f ∈ Hs,ϕ(Γ) implies that

f = Au ∈ A(Hs+m,ϕ(Γ)).

Thus, on the manifold Γ, the equalities Au = f and Av = f are true for somedistribution v ∈ Hs+m,ϕ(Γ). Therefore, A(u−v) = 0 on Γ and, by Theorem 2.9,we arrive at the inclusion

w := u− v ∈ N ⊂ C∞(Γ) ⊂ Hs+m,ϕ(Γ).

Hence, u = v + w ∈ Hs+m,ϕ(Γ).


We now consider the case Γ0 6= Γ. Denote

Hσ,ϕloc (Γ0) := h ∈ D′(Γ) : χh ∈ Hσ,ϕ(Γ) for all χ ∈ C∞(Γ), suppχ ⊂ Γ0.

Here, σ ∈ R, ϕ ∈M.

Theorem 2.13. Assume that the distribution u ∈ D′(Γ) is a solution of theequation Au = f on Γ0, where f ∈ Hs,ϕ

loc (Γ0) for some parameters s ∈ R andϕ ∈M. Then u ∈ Hs+m,ϕ

loc (Γ0).

Theorem 2.13 is proved by analogy with Theorem 1.18. In this case, it isnecessary to apply Theorem 2.12 instead of Theorem 1.17.

As applied to the scale of spaces Hs,ϕ(Γ), Theorem 2.13 improves the well-known assertions on the increase in the local smoothness of solutions to theelliptic equation on a manifold in the Sobolev scale [52, Chap. 2, Lemma 3.2],[232, Theorem 7.2]. It is easy to see that the local refined smoothness ϕ of theright-hand side of the elliptic equation is inherited by its solution.

Theorems 2.13 and 2.8 immediately yield the following sufficient conditionfor the continuity of derivatives of the solution u to the equation Au = f.

Corollary 2.3. Let an integer r ≥ 0 and a function ϕ ∈M satisfying condition(1.37) be given. Assume that the distribution u ∈ D′(Γ) is a solution of theequation Au = f on the set Γ0, where f ∈ Hr−m+n/2,ϕ

loc (Γ0). Then u ∈ Cr(Γ0).

Note that condition (1.37) is not only sufficient for the validity of the inclusionu ∈ Cr(Γ0) but also necessary in the class of all analyzed solutions of theequation Au = f ; see Remark 1.9.


2.2.4 Parameter-elliptic operators

Various classes of parameter-elliptic operators were studied by S. Agmon andL. Nirenberg [6], M. S. Agranovich and M. I. Vishik [13], A. N. Kozhevnikov [99]and their followers (see the survey [10] and the references therein). It was es-tablished that if the modulus of the complex-valued parameter is sufficientlylarge, then the elliptic operator specifies an isomorphism between suitableSobolev spaces. Moreover, the norm of the operator admits a two-sided es-timate with constants independent of the parameter. We consider a broadclass of parameter-elliptic PsDOs on the manifold Γ and extend the indicatedresult to the refined Sobolev scale.

The definition of parameter-elliptic PsDO is taken from the survey [10,Sec. 4.1]. We now fix arbitrary numbers q ∈ N and m > 0. Consider a PsDOA(λ) ∈ Ψmq(Γ) depending on a complex-valued parameter λ as follows:

A(λ) :=q∑

r= 0

λq−r A(r). (2.56)

Here, A(r), r = 0, . . . , q, are polyhomogeneous PsDOs on Γ of orders ordA(r) ≤mr and A0 is the operator of multiplication by a function a0 ∈ C∞(Γ).

Let K be a fixed closed angle in the complex plane with vertex at the origin(we do not exclude the case where K is degenerated into a ray).

Definition 2.10. A PsDO A(λ) is called parameter-elliptic in the angle K if

q∑r= 0

λq−r ar,0(x, ξ) 6= 0 (2.57)

for any x ∈ Γ, ξ ∈ T ∗xΓ, and λ ∈ K such that (ξ, λ) 6= 0. Here, ar,0(x, ξ)is the principal symbol of the PsDO A(r) in the case where ordA(r) = mr orar,0(x, ξ) ≡ 0 if ordA(r) < mr. Moreover, we assume that a0,0(x, ξ) ≡ a0(ξ)

and that the functions a1,0(x, ξ), a2,0(x, ξ), . . . are equal to 0 for ξ = 0 (the lastassumption is explained by the fact that the principal symbols are not initiallydefined for ξ = 0).

Example 2.2. Consider a PsDO A − λI, where A ∈ Ψmph(Γ), and I is the

identity operator. For this PsDO, the condition of parameter-ellipticity in theangle K simply means that a0(x, ξ) /∈ K for any ξ 6= 0. Here, as above, a0(x, ξ)is the principal symbol of A. This example is of importance for the spectraltheory of elliptic operators.

Further, in this subsection, we always assume that the PsDO A(λ) satisfiesDefinition 2.10.


It follows from this definition that the PsDO A(λ) is elliptic on Γ for anyfixed λ ∈ C. Indeed, the principal symbol of A(λ) is aq,0(x, ξ) for any λ. Asfollows from (2.57), for λ = 0, it satisfies the inequality aq,0(x, ξ) 6= 0 for anyx ∈ Γ and ξ ∈ T ∗xΓ \ 0. This fact means that the PsDO A(λ) is elliptic on Γ.

By Theorem 2.9, we get the following bounded Fredholm operator:

A(λ) : Hs+mq,ϕ(Γ)→ Hs,ϕ(Γ) (2.58)

for any s ∈ R, ϕ ∈M, and λ ∈ C. Moreover, since A(λ) is a parameter-ellipticPsDO in the angle K, it has important additional properties.

Theorem 2.14. The following assertions are true:

(i) There exists a number λ0 > 0 such that, for every value of the parameterλ ∈ K satisfying the condition |λ| ≥ λ0, the PsDO A(λ) establishes, forany s ∈ R and ϕ ∈M, the isomorphism

A(λ) : Hs+mq,ϕ(Γ)↔ Hs,ϕ(Γ). (2.59)

(ii) For arbitrary parameters s ∈ R and ϕ ∈ M, there exists a number c =

c(s, ϕ) ≥ 1 such that the two-sided estimate

c−1 ‖A(λ)u‖Hs,ϕ(Γ) ≤(‖u‖Hs+mq,ϕ(Γ) + |λ|q ‖u‖Hs,ϕ(Γ)

)≤ c

∥∥A(λ)u‖Hs,ϕ(Γ) (2.60)

holds for any λ ∈ K with |λ| ≥ λ0 and any u ∈ Hs+mq,ϕ(Γ).

For ϕ ≡ 1 (Sobolev spaces), this theorem is known [10, Theorem 4.1.2];in this case, the left inequality in the two-sided estimate (2.60) holds withoutthe assumption about the parameter-ellipticity of A(λ).

We now separately prove assertions (i) and (ii) of Theorem 2.14. The generalcase of ϕ ∈M is derived from the Sobolev case ϕ ≡ 1.

Proof of Theorem 2.14(i). Let s ∈ R and let ϕ ∈ M. Since, for ev-ery λ ∈ C, the PsDO A(λ) is elliptic on Γ, the bounded Fredholm operator(2.58) has the kernel N (λ) and the defect subspace N+(λ), which are finite-dimensional and independent of s and ϕ (by virtue of Theorem 2.9). Then weuse the fact that Theorem 2.14 holds for ϕ ≡ 1. Thus, there exists a numberλ0 > 0 such that, for any λ ∈ K satisfying the condition |λ| ≥ λ0, the PsDOA(λ) establishes an isomorphism

A(λ) : Hs+mq(Γ)↔ Hs+m(Γ).


Hence, for the indicated λ, the spaces N (λ) and N+(λ) are trivial, i.e., thelinear bounded operator (2.58) is a bijection. By the Banach theorem on in-verse operator, this yields isomorphism (2.59). Assertion (i) of Theorem 2.14is proved.

Assertion (ii) of Theorem 2.14 is proved with the help of interpolation withfunctional parameter. For this purpose, we need the following space:

Let a function ϕ ∈ M and numbers σ ∈ R, % > 0, θ > 0 be given. ByHσ,ϕ(Γ, %, θ) we denote the space Hσ,ϕ(Γ) endowed with the norm dependingon the number parameters % and θ in the following way:

‖h‖Hσ,ϕ(Γ,%,θ) :=(‖h‖2Hσ,ϕ(Γ) + %2 ‖h‖2Hσ−θ,ϕ(Γ)

)1/2.

This definition is correct in view of the continuous embedding Hσ,ϕ(Γ) →Hσ−θ,ϕ(Γ). This means that the norms in the spaces Hσ,ϕ(Γ, %, θ) and Hσ,ϕ(Γ)

are equivalent. The norm in the space Hσ,ϕ(Γ, %, θ) is generated by the innerproduct

(h1, h2)Hσ,ϕ(Γ,%,θ) := (h1, h2)Hσ,ϕ(Γ) + %2 (h1, h2)Hσ−θ,ϕ(Γ).

Hence, this is a Hilbert space. As above, we omit the index ϕ in the notationused in the Sobolev case ϕ ≡ 1. Returning to the formulation of assertion (ii)of Theorem 2.14, we note that

‖u‖Hs+mq,ϕ(Γ,|λ|q ,mq) ≤ ‖u‖Hs+mq,ϕ(Γ) + |λ|q ‖u‖Hs,ϕ(Γ)

≤√2 ‖u‖Hs+mq,ϕ(Γ,|λ|q ,mq).

By virtue of Theorem 2.2, the spaces[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ

and Hσ,ϕ(Γ, %, θ)

coincide up to equivalence of norms. Here, the numbers ε and δ are positive,and the functional parameter ψ is the same as in Theorem 2.2. It turns outthat the constants in the two-sided estimates for the norms in these spaces canbe made independent of the parameter %.

Lemma 2.6. Let σ ∈ R, let ϕ ∈ M, and let θ, ε, and δ be given positivenumbers. Then there exists a number c0 ≥ 1 such that the following two-sidedestimate of norms holds for any % > 0 and h ∈ Hσ,ϕ(Γ) :

c−10 ‖h‖Hσ,ϕ(Γ,%,θ) ≤ ‖h‖[Hσ−ε(Γ,%,θ), Hσ+δ(Γ,%,θ) ]ψ≤ c0 ‖h‖Hσ,ϕ(Γ,%,θ). (2.61)

Here, ψ is the interpolation parameter from Theorem 2.2 and the number c0 isindependent of % and h.


Proof. Assume that % > 0. First, we prove an analog of estimate (2.61)for the spaces of distributions in Rn. Then, by using the operators of “flat-tening” and “sewing,” we pass to the spaces of distributions on the manifold Γ

(cf. the proof of Theorem 2.2). By Hσ,ϕ(Rn, %, θ) we denote the space Hσ,ϕ(Rn)endowed with the Hilbert norm

‖w‖Hσ,ϕ(Rn,%,θ) :=(‖w‖2Hσ,ϕ(Rn) + %2 ‖w‖2Hσ−θ,ϕ(Rn)

)1/2=

( ∫Rn

〈ξ〉2σ(1 + %2 〈ξ〉−2θ

)ϕ2(〈ξ〉) |w(ξ)|2 dξ

)1/2

. (2.62)

For any fixed % > 0, this norm is equivalent to the norm in the space Hσ,ϕ(Rn).Hence, Hσ,ϕ(Rn, %, θ) is a Hilbert space. Similarly, we can define the spacesHσ−ε(Rn, %, θ) and Hσ+δ(Rn, %, θ). By virtue of Theorem 1.14, the spaces[

Hσ−ε(Rn, %, θ), Hσ+δ(Rn, %, θ)]ψ, (2.63)

and Hσ,ϕ(Rn, %, θ) are equal up to equivalence of norms for any fixed % > 0.

We now show that the equality of norms is indeed realized.We calculate the norm in space (2.63). By J we denote a pseudodifferential

operator in the space Rn with the symbol 〈ξ〉ε+δ, where the argument ξ ∈ Rn.It can be directly verified that J is a generating operator for the pair of spacesin relation (2.63). With the help of the isometric isomorphism

F : Hσ−ε(Rn, %, θ)↔ L2

(Rn, 〈ξ〉2(σ−ε)(1 + %2 〈ξ〉−2θ) dξ

),

where F is the Fourier transformation, the operators J and ψ(J) are reducedto the operators of multiplication by the functions 〈ξ〉ε+δ and

ψ(〈ξ〉ε+δ) = 〈ξ〉εϕ(〈ξ〉),

respectively. Hence,

‖w‖2[Hσ−ε(Rn,%,θ), Hσ+δ(Rn,%,θ)]ψ= ‖ψ(J)w‖2Hσ−ε(Rn,%,θ)

=

∫Rn

〈ξ〉2(σ−ε)(1 + %2 〈ξ〉−2θ

) ∣∣〈ξ〉εϕ(〈ξ〉) w(ξ)∣∣2 dξ= ‖w‖2Hσ,ϕ(Rn,%,θ).

Thus, we get

‖w‖[Hσ−ε(Rn,%,θ), Hσ+δ(Rn,%,θ)]ψ = ‖w‖Hσ,ϕ(Rn,%,θ) (2.64)


for any w ∈ Hσ,ϕ(Rn) and % > 0.We now deduce the two-sided inequality (2.61)from (2.64). To do this, we use the local Definition 2.1 of the spaces Hs,ϕ(Γ)

with s ∈ R and ϕ ∈ M for a fixed finite atlas and the resolution of identityon Γ. We consider the linear mapping of “flattening” of the manifold Γ :

T : h 7→((χ1h) α1, . . . , (χrh) αr

), h ∈ D′(Γ).

It can be directly verified that this mapping specifies the isometric operators

T : Hσ,ϕ(Γ, %, θ)→ (Hσ,ϕ(Rn, %, θ))r, (2.65)

T : Hs(Γ, %, θ)→ (Hs(Rn, %, θ))r, s ∈ σ − ε, σ + δ. (2.66)

Applying the interpolation with the parameter ψ to operators (2.66), we obtainthe bounded operator

T :[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ

→[(Hσ−ε(Rn, %, θ))r, (Hσ+δ(Rn, %, θ))r

]ψ. (2.67)

Since the presented pairs of spaces are normal, the norm of operator (2.67)does not exceed some number c1 := c(ψ, 1) independent of the parameter % byvirtue of Theorem 1.8. This result, Theorem 1.5, and equality (2.64) yield thebounded operator

T :[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ→ (Hσ,ϕ(Rn, %, θ))r (2.68)

whose norm ≤ c1.Parallel with the mapping T, we consider the linear mapping of “sewing”

K : (w1, . . . , wr) 7→r∑j=1

Θj

((ηjwj) α−1j

),

where w1, . . . , wr are distributions in Rn. Here, the function ηj ∈ C∞(Rn) isfinite and equal to 1 on the set α−1j (suppχj) and Θj is the operator of extensionby zero onto Γ. By virtue of (2.8), we get the bounded operators

K : (Hs(Rn))r → Hs(Γ), s ∈ R, (2.69)

K : (Hs,ϕ(Rn))r → Hs,ϕ(Γ), s ∈ R. (2.70)

Let c2 be the maximum of the norms of operators (2.69), where

s ∈ σ − ε, σ − ε− θ, σ + δ, σ + δ − θ,


and operators (2.70), where s ∈ σ, σ − θ. The number c2 is independent ofthe parameter %. It can be directly verified that the norms of the operators

K : (Hσ,ϕ(Rn, %, θ))r → Hσ,ϕ(Γ, %, θ), (2.71)

K : (Hs(Rn, %, θ))r → Hs(Γ, %, θ), s ∈ σ − ε, σ + δ, (2.72)

do not exceed the number c2. Applying the interpolation with the parameter ψto (2.72), we obtain the bounded operator

K :[(Hσ−ε(Rn, %, θ))r, (Hσ+δ(Rn, %, θ))r

]ψ

→[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ

whose norm does not exceed the number c1c2 by virtue of Theorem 1.8. Hence,in view of Theorem 1.5 and equality (2.64), we arrive at the bounded operator

K : (Hσ,ϕ(Rn, %, θ))r →[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ

(2.73)

whose norm ≤ c1c2.By virtue of (2.7), the product KT = I is the identity operator. Hence, in

view of of relations (2.65) (isometric operator) and (2.73), we get the boundedoperator

I = KT : Hσ,ϕ(Γ, %, θ)→[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ

whose norm ≤ c1c2.In addition, in view of (2.68) and (2.71) (the norm of the second operator

does not exceed the number c2), we arrive at one more bounded operator

I = KT :[Hσ−ε(Γ, %, θ), Hσ+δ(Γ, %, θ)

]ψ→ Hσ,ϕ(Γ, %, θ)

whose norm ≤ c1c2. This immediately yields the two-sided estimate (2.61),where the number c0 := max1, c1c2 is independent of the parameter %.


Proof of Theorem 2.14(ii). Let s ∈ R and let ϕ ∈ M. We recall thatTheorem 2.14 holds in the Sobolev case where ϕ ≡ 1. Therefore, there existsa number λ0 > 0 such that, for every value of the parameter λ ∈ K satisfyingthe condition |λ| ≥ λ0, the isomorphisms

A(λ) : Hs∓1+mq(Γ, |λ|q,mq)↔ Hs∓1(Γ) (2.74)

hold and the norm of operator (2.74), together with the norm of the inverseoperator, are bounded uniformly in the parameter λ. Let ψ be the interpo-lation parameter from Theorem 2.2, where we set ε = δ = 1. Applying the


interpolation with this parameter to (2.74), we obtain an isomorphism

A(λ) :[Hs−1+mq(Γ, |λ|q,mq), Hs+1+mq(Γ, |λ|q,mq)

]ψ

↔[Hs−1(Γ), Hs+1(Γ)

]ψ. (2.75)

In this case, by virtue of Theorem 1.8, the norm of operator (2.75) and thenorm of the inverse operator are bounded uniformly in the parameter λ. (Theadmissible pairs of spaces in relation (2.75) are normal.) It remains to useLemma 2.6, where we set

σ := s+mq, % := |λ|q, θ := mq, ε = δ = 1,

and Theorem 2.2. According to these assertions, relation (2.75) leads to theisomorphism

A(λ) : Hs+mq,ϕ(Γ, |λ|q,mq)↔ Hs,ϕ(Γ) (2.76)

such that the norm of operator (2.76), together with the norm of the inverseoperator, are bounded uniformly in the parameter λ. This yields the two-sidedestimate (2.60), where the number c is independent of the parameter λ and thedistribution u ∈ Hs+mq,ϕ(Γ). Assertion (ii) of Theorem 2.14 is proved.

Theorem 2.14(i) yields the following assertion about the indices of parameter-elliptic PsDOs (it should be compared with [13, § 6, Subsec. 4]).

Corollary 2.4. Assume that a PsDO A(λ) is parameter elliptic on a closedray K := λ ∈ C : argλ = const. Then the index of operator (2.58) is equal tozero for any λ ∈ C.

Proof. For any fixed λ ∈ C, the PsDO A(λ) is elliptic on Γ. Therefore,by virtue of Theorem 2.9, the index of operator (2.58) is finite and inde-pendent of s ∈ R and ϕ ∈ M. Moreover, this index is also independentof the parameter λ. Indeed, λ affects solely the lowest terms in sum (2.56):A(λ)−A(0) ∈ Ψm(q−1)(Γ). Hence, in view of Lemma 2.5, we have the boundedoperator

A(λ)−A(0) : Hs+mq,ϕ(Γ)→ Hs+m,ϕ(Γ).

At the same time, by virtue of Theorem 2.3(iii) and the conditionm > 0, we getthe compact embedding Hs+m,ϕ(Γ) → Hs,ϕ(Γ). This means that the operator

A(λ)−A(0) : Hs+mq,ϕ(Γ)→ Hs,ϕ(Γ)

is compact. Thus (see, e.g., [86, Corollary 19.1.8]), the Fredholm operatorsA(λ) and A(0) have the same index, i.e., this index is independent of the

Section 2.3 Convergence of spectral expansions 93

parameter λ. Then, by Theorem 2.14(i), isomorphism (2.59) is true for thevalues of the parameter λ ∈ K sufficiently large in the absolute value.

Hence, the index of the operator A(λ) is equal to zero for λ ∈ K, |λ| 1

and, therefore, for any λ ∈ C.Corollary 2.4 is proved.

2.3 Convergence of spectral expansions

In this subsection, we discuss the applications of the refined Sobolev scale tothe investigation of convergence of spectral expansions almost everywhere or inthe spaces Ck with integer k ≥ 0.

2.3.1 Convergence almost everywhere for generalorthogonal series

First, we present the classical results on convergence almost everywhere for ar-bitrary orthogonal series required in what follows. Let Γ be a measurable spacewith finite measure µ and let (hj)∞j=1 be an orthonormal system of functions inL2(Γ, µ). (The functions hj are, generally speaking, complex-valued.) We usethe symbol log to denote the logarithmic function for any fixed base a > 1.

Proposition 2.1 (Men’shov–Rademacher theorem). Let a sequence (aj)∞j=1 of

complex numbers be such that

L :=∞∑j=1

|aj |2 log2 (j + 1) <∞. (2.77)

Then the orthogonal series∞∑j=1

aj hj(x) (2.78)

converges µ-almost everywhere on Γ. In addition, if

S∗(x) := sup1≤k<∞

∣∣∣ k∑j=1

aj hj(x)∣∣∣,

is the majorant of the partial sums of series (2.78), then

‖S∗‖L2(Γ) ≤ C√L, (2.79)

where the number C > 0 depends only on Γ and µ.


This result was proved independently by D. E. Men’shov [134] and H. Ra-demacher [197] in the case where Γ is a finite interval on the axis R, µ isthe Lebesgue measure, and the functions hj are real-valued. The proof of theMen’shov–Rademacher theorem presented in the monograph [94, Chap. 8, § 1]remains valid in the analyzed general case.

Note that the Men’shov–Rademacher theorem is exact. Men’shov [134] con-structed an example of the orthonormal system (hj)

∞j=1 in L2((0, 1)) such that,

for any number sequence (ωj)∞j=1 satisfying the conditions

1 = ω1 ≤ ω2 ≤ ω3 ≤ . . . and limj→∞

ωj

log2 j= 0,

there exists a series of the form (2.78) divergent almost everywhere whose co-efficients satisfy the inequality

∞∑j=1

|aj |2 ωj <∞.

(The presentation of this result can be found, e.g., in [14, Sec. 2.4.1] or [94,Chap. 8, § 1]).

Note that (see [94, Chap. 8, § 2]) the convergence of series (2.78) µ-almosteverywhere does not, generally speaking, imply that this series converges un-conditionally µ-almost everywhere. We recall that series (2.78) is called uncon-ditionally convergent µ-almost everywhere on Γ if the series

∞∑j=1

aσ(j) hσ(j) (2.80)

converges µ-almost everywhere for any permutation of the series of positiveintegers σ = (σ(j))∞j=1. (In this case, the set of measure zero of all points atwhich series (2.80) diverges may depend on the permutation σ.)

Proposition 2.2 (Orlicz–Ul’yanov theorem). Let a sequence (aj)∞j=1 of com-

plex numbers and a (nonstrictly) increasing sequence (ωj)∞j=1 of positive num-

bers satisfy the conditions

∞∑j=2

|aj |2 (log2 j)ωj <∞, (2.81)

∞∑j=2

1

j (log j)ωj<∞. (2.82)

Then series (2.78) converges unconditionally µ-almost everywhere on Γ.


This is an equivalent formulation of the Orlicz theorem [178] proposed byP. L. Ul’yanov [261, § 4] (see also [262, § 9, Subsec. 1]). As shown by K. Tan-dori [251], the Orlicz theorem is the best possible assertion in a sense thatcondition (2.81) for the sequence (ωj)

∞j=1 cannot be weakened.

W. Orlicz and P. L. Ul’yanov restricted themselves to the case where Γ isa finite interval on the axis R, µ is the Lebesgue measure, and the functionshj are real valued. In the general case considered in our monograph, Propo-sition 2.2 remains true. This follows from the more general Tandori theorem[251] whose proof can be found in the monograph [94, Chap. 8, § 2, Theorem 5]and remains valid in the analyzed general situation.

2.3.2 Convergence almost everywhere for spectral expansions

We now study the convergence almost everywhere for the spectral expansionsin eigenfunctions of elliptic PsDOs. In this subsection, Γ is an infinitely smoothclosed (compact) manifold of dimension n ≥ 1 with a fixed density dx. Let Abe a classic elliptic PsDO of positive order on Γ. Assume that A is an (un-bounded) normal operator in the Hilbert space L2(Γ, dx). Let (hj)

∞j=1 be the

complete orthonormal system of the eigenfunctions of this operator. They areenumerated so that the absolute values of the corresponding eigenvalues forma monotonically nondecreasing sequence. For any function f ∈ L2(Γ, dx), wehave

f =∞∑j=1

cj(f)hj in L2(Γ, dx), (2.83)

where cj(f) := (f, hj)Γ are the Fourier coefficients of f in the basis (hj)∞j=1.

Series (2.83) is called convergent in the indicated sense on some functionalclass X(Γ) if, for any function f ∈ X(Γ), this series converges to f in a properway.

We now study the problem of convergence almost everywhere for series (2.83)in the Hörmander spaces. Note that if, for some function f ∈ L2(Γ, dx), se-ries (2.83) converges almost everywhere on Γ, then f is the sum of this seriesalmost everywhere on Γ. This follows from the fact that the convergence inL2(Γ, dx) and the convergence almost everywhere on Γ yield the convergencein the measure generated by the density dx (see, e.g., [23, Chap. I, Sec. 4;Chap. 6, Sec. 8.3]).

Consider the majorant of the partial sums of series (2.83) for the functionf ∈ L2(Γ, dx):

S∗(f, x) := sup1≤k<∞

∣∣∣ k∑j=1

cj(f)hj(x)∣∣∣.


For A = ∆Γ,Meaney [133] proved that series (2.83) converges almost everywhereon Γ for the functional class

H0+(Γ) :=⋃ε>0

Hε(Γ).

Moreover, for any ε > 0 and f ∈ Hε(Γ),

‖S∗(f, ·)‖L2(Γ,dx) ≤ Cε‖f‖Hε(Γ),

where the number Cε is independent of f.We generalize and improve this result by using Hörmander spaces. Denote

log∗ t := max1, log t.

Theorem 2.15. In the functional class H0,log∗(Γ), series (2.83) converges al-most everywhere on Γ. Moreover, the following estimate is true for any functionf ∈ H0,log∗(Γ) :

‖S∗(f, ·)‖L2(Γ,dx) ≤ C ‖f‖H0,log∗ (Γ), (2.84)

where the number C > 0 is independent of f.

Proof. Let f ∈ H0,log∗(Γ). If the operator A is self-adjoint and positivedefinite in L2(Γ, dx), then, by virtue of Theorem 2.7,

‖f‖2H0,log∗ (Γ)

∞∑j=1

(log∗(j 1/n))2 |cj(f)|2,

where log∗(j 1/n) log(j + 1) with j ≥ 1.

Hence,∞∑j=1

|cj(f)|2 log2(j + 1) ‖f‖2H0,log∗ (Γ) <∞.

Therefore, by the Men’shov–Rademacher theorem (Proposition 2.1), series (2.83)converges almost everywhere on Γ (to f) and estimate (2.84) is true.

The general case, where the operator A is normal, is reduced to the analyzedcase by the transition to the self-adjoint positive definite operator B := 1+A∗A.

In this case, one must take into account the fact that (hj)∞j=1 is the system ofeigenfunctions both of the normal operator A and the self-adjoint operator B.This system is complete in L2(Γ, dx) and the numbering of eigenfunctions isconsistent.


This result is refined by the following theorem on unconditional convergence:


Theorem 2.16. Assume that an increasing function ϕ ∈M satisfies the con-dition

∞∫2

dt

t (log t)ϕ2(t)<∞. (2.85)

Then, in the functional class H0,ϕ log∗(Γ), series (2.83) is unconditionally con-vergent almost everywhere on Γ.

Proof. Let f ∈ H0,ϕ log∗(Γ). If the operator A is self-adjoint and positivedefinite in L2(Γ, dx), then, by Theorem 2.7, we can write

‖f‖2H0,ϕ log∗ (Γ)

∞∑j=1

ϕ2(j 1/n)(log∗(j 1/n))2 |cj(f)|2. (2.86)

Consider an increasing sequence of numbers ωj := ϕ2(j 1/n), j ∈ N. By virtueof (2.86), we have

∞∑j=2

|cj(f)|2 (log2 j)ωj <∞. (2.87)

In addition, according to condition (2.85),

∞∑j=3

1

j (log j)ωj≤∞∫2

dτ

τ (log τ)ϕ2(τ1/n)=

∞∫21/n

n tn−1 dt

tn n (log t)ϕ2(t)<∞. (2.88)

By virtue of the Orlicz–Ul’yanov theorem (Proposition 2.2), inequalities (2.87)and (2.88) yield the unconditional convergence almost everywhere of series(2.83) in Γ.

As in the proof of Theorem 2.15, the general case, where the operator Ais normal, is reduced to the already analyzed case by the transition to theself-adjoint operator B := 1 +A∗A.


Theorems 2.15 and 2.16 belong to the theory of general orthogonal series.However, the conditions imposed on the function f are formulated in construc-tive terms of the smoothness of functions.

2.3.3 Convergence of spectral expansions in the metricof the space Ck

At the end of Section 2.3, we prove a criterion of convergence of series (2.83)in the spaces Ck(Γ) with integer k ≥ 0 on the classes Hs,ϕ(Γ).


Theorem 2.17. Let an integer k ≥ 0 and a function ϕ ∈ M be given. Series(2.83) converges in the space Ck(Γ) on the class Hk+n/2,ϕ(Γ) if and only if thefunction ϕ satisfies condition (1.37).

Proof. Sufficiency. Assume that ϕ satisfies condition (1.37). Let f ∈Hk+n/2,ϕ(Γ). As indicated in Remark 2.4, series (2.83) converges to f in thespace Hk+n/2,ϕ(Γ). By Theorem 2.8, inequality (1.37) yields the continuity ofthe embedding Hk+n/2,ϕ(Γ) → Ck(Γ). Hence, series (2.83) converges to f inthe space Ck(Γ).

Sufficiency is proved.

Necessity. Assume that, for any function f ∈ Hk+n/2,ϕ(Γ), series (2.83)converges (to f) in the space Ck(Γ). Then Hk+n/2,ϕ(Γ) ⊆ Ck(Γ), which yieldscondition (1.37) by virtue of Theorem 2.8.

Necessity is proved.Theorem 2.17 is proved.

2.4 RO-varying functions and Hörmander spaces

In the present section, we describe all (up to equivalence of norms) interpola-tion Hilbert spaces for the pairs of inner product Sobolev spaces Hs0(Rn) andHs1(Rn), where s0, s1 ∈ R and s0 < s1. It is shown that the class of these in-terpolation spaces is formed solely by isotropic Hörmander spaces with weightfunctions, RO-varying at infinity in a sense of Avakumović. We also considerthe indicated spaces over smooth closed manifolds and discuss their possibleapplications.

2.4.1 RO-varying functions in the sense of Avakumović

We now present a definition, which is of fundamental importance for our pre-sentation.

Definition 2.11. Let RO be the set of all Borel measurable functions

ϕ : [1,∞)→ (0,∞)

for which one can find numbers a > 1 and c ≥ 1 such that

c−1 ≤ ϕ(λt)

ϕ(t)≤ c for any t ≥ 1 and λ ∈ [1, a] (2.89)

(the constants a and cmay depend on ϕ). These functions are called RO-varyingat infinity.

Section 2.4 RO-varying functions and Hörmander spaces 99

The class of RO-varying (or OR-varying) functions was introduced byV. G. Avakumović [19] in 1936 and studied fairly comprehensively (see, e.g.,[26, Sec. 2.0–2.2] or [235, Appendix]).

We now recall some well-known properties of functions from the class RO.It is clear that the number a for the function ϕ ∈ RO in relation (2.89) can bemade arbitrarily large. This yields the following property:

Proposition 2.3. If ϕ ∈ RO, then the functions ϕ and 1/ϕ are bounded onevery segment [1, b], where 1 < b <∞.

Proposition 2.4. The following description of the class RO is true:

ϕ ∈ RO ⇔ ϕ(t) = exp

(β(t) +

t∫1

ε(τ)

τdτ

), t ≥ 1,

where β and ε are real-valued Borel measurable functions bounded on the semi-axis [1,∞).

Proposition 2.5. For any function ϕ : [1,∞) → (0,∞), condition (2.89) isequivalent to the following fact: there exist numbers s0, s1 ∈ R, s0 ≤ s1, andc ≥ 1 such that

t−s0ϕ(t) ≤ c τ−s0ϕ(τ) and τ−s1ϕ(τ) ≤ c t−s1ϕ(t) (2.90)

for all t ≥ 1 and τ ≥ t.

Condition (2.90) indicates that the function t−s0ϕ(t) is equivalent to an in-creasing function and the function t−s1ϕ(t) is equivalent to a decreasing func-tion on the semiaxis [1,∞). In this case, the property of equivalence is under-stood in a weak sense, whereas the properties of increasing and decreasing areunderstood in the nonstrict sense.

Setting λ := τ/t in condition (2.90), we rewrite it in the equivalent form

c−1λs0 ≤ ϕ(λt)

ϕ(t)≤ cλs1 for all t ≥ 1 and λ ≥ 1. (2.91)

For any function ϕ ∈ RO, we denote

σ0(ϕ) := sup s0 ∈ R : (2.91) is true,

σ1(ϕ) := inf s1 ∈ R : (2.91) is true.

It is clear that−∞ < σ0(ϕ) ≤ σ1(ϕ) <∞.


The numbers σ0(ϕ) and σ1(ϕ) are the lower and upper Matuszewska indicesof ϕ (see [130] and [26, Sec. 2.1.2]).

We now additionally define the function ϕ(t) := ϕ2(1)/ϕ(t−1) for 0 < t < 1.This gives a function ϕ positive on the semiaxis (0,∞) and such that (2.91)yields the condition

c−2λs0 ≤ ϕ(λt)

ϕ(t)≤ c2λs1 for all t > 0 and λ ≥ 1.

This means that the numbers σ0(ϕ) and σ1(ϕ) are equal to the lower and upperstretching indices of the function ϕ, respectively [109, Chap. II, § 1, Subsec. 2].They are given by the formulas

σ0(ϕ) = limλ→0+

logmϕ(λ)

logλ, σ1(ϕ) = lim

λ→∞

logmϕ(λ)

logλ, (2.92)

wheremϕ(λ) := sup

t>0

ϕ(λt)

ϕ(t), λ > 0,

is the stretching function for ϕ. Note that the right-hand sides of relations (2.92)are equal, by definition, to the lower and upper Boyd indices of the function mϕ

(see [27]).In an important case where

σ0(ϕ) = σ1(ϕ) =: σ,

the number σ is called the order of variation of the function ϕ. Note that everyBorel measurable function ϕ : [1,∞)→ (0,∞) belongs to the class RO and hasthe order of variation σ provided that it is a quasiregularly varying function oforder σ at∞ and that both ϕ and 1/ϕ are bounded on each segment [1, b], where1 < b < ∞. This follows from [235, Sec. 1.5, Subsec. 4] and Proposition 2.5.In particular,M⊂ RO.

2.4.2 Interpolation spaces for a pair of Sobolev spaces

In this subsection, we describe all interpolation Hilbert spaces for the pairs ofinner product Sobolev spaces. For this purpose, we first select the followingclass of inner product Hörmander spaces:

Definition 2.12. Let ϕ ∈ RO. By definition, the linear spaceHϕ(Rn) is formedby all distributions u ∈ S ′(Rn) such that their Fourier transform u is locallyLebesgue summable in Rn and satisfies the inequality∫

Rn

ϕ2(〈ξ〉) |u(ξ)|2 dξ <∞.


In the space Hϕ(Rn), the inner product of distributions u1 and u2 is definedby the formula

(u1, u2)Hϕ(Rn) :=∫Rn

ϕ2(〈ξ〉) u1(ξ) u2(ξ) dξ.

The inner product specifies the norm in a standard way.

For every functional parameter ϕ ∈ RO, the space Hϕ(Rn) is a special(isotropic) Hilbert case of Hörmander spaces: Hϕ(Rn) = B2,µ(Rn), whereµ(ξ) := ϕ(〈ξ〉), ξ ∈ Rn. In this case, we can show that µ is a weight func-tion.

Lemma 2.7. Let ϕ ∈ RO. Then the function µ(ξ) := ϕ(〈ξ〉) of the argumentξ ∈ Rn is a weight function in a sense of Definition 1.8.

Proof. Let ξ, η ∈ Rn. The inequality |〈ξ〉 − 〈η〉| ≤ | |ξ| − |η| | is checked byraising it to the square. Thus, for 〈ξ〉 ≥ 〈η〉, we find

〈ξ〉〈η〉

= 1 +〈ξ〉 − 〈η〉〈η〉

≤ 1 + |ξ| − |η| ≤ 1 + |ξ − η|.

Hence, by virtue of Proposition 2.5, we get

ϕ(〈ξ〉)ϕ(〈η〉)

≤ c(〈ξ〉〈η〉

)s1≤ c (1 + |ξ − η|)max0,s1.

At the same time, if 〈η〉 ≥ 〈ξ〉, then

ϕ(〈ξ〉)ϕ(〈η〉)

≤ c(〈ξ〉〈η〉

)s0= c

(〈η〉〈ξ〉

)−s0≤ c (1 + |ξ − η|)max0,−s0.

Thus, for any ξ, η ∈ Rn, we obtain

µ(ξ)

µ(η)=ϕ(〈ξ〉)ϕ(〈η〉)

≤ c (1 + |ξ − η|)l,

where the numbers c ≥ 1 and l := max0,−s0, s1 are independent of ξ and η.This means that µ is a weight function in a sense of Definition 1.8.


Note that if ϕ0 ∈ M and s ∈ R, then the function ϕs(t) := tsϕ0(t) belongsto the class RO. Hence, the class of spaces Hϕ(Rn) : ϕ ∈ RO contains therefined Sobolev scale.

We now indicate the properties of the spaces Hϕ(Rn), which follow from thecorresponding properties of the Hörmander spaces [81, Sec. 2.2].


Proposition 2.6. Let ϕ,ϕ1 ∈ RO. Then

(i) Hϕ(Rn) is a separable Hilbert space.

(ii) The set C∞0 (Rn) is dense in Hϕ(Rn).

(iii) The function ϕ(t)/ϕ1(t) is bounded in the vicinity of ∞ if and only ifHϕ1(Rn) → Hϕ(Rn). This embedding is continuous and dense.

(iv) For any real numbers s0 < σ0(ϕ) and s1 > σ1(ϕ), the following continuousand dense embeddings are true: Hs1(Rn) → Hϕ(Rn) → Hs0(Rn).

(v) The spaces Hϕ(Rn) and H1/ϕ(Rn) are mutually dual with respect to theextension by continuity of the inner product in the space L2(Rn).

(vi) For every fixed integer k ≥ 0, the condition

∞∫1

t2k+n−1 ϕ−2(t) dt <∞ (2.93)

is equivalent to the embedding Hϕ(Rn) → Ckb (Rn). This embedding is con-tinuous.

We now make some comments to Proposition 2.6. Assertion (iv) follows fromassertion(iii) and inequality (2.91) in which we set t := 1. Since

ϕ ∈ RO⇔ 1/ϕ ∈ RO,

the space H1/ϕ(Rn) from assertion (v) is well defined. Assertion (vi) followsfrom the Hörmander embedding theorem (Proposition 1.5). In this case, itshould be taken into account that (1.33) ⇔ (2.93) for the function µ(ξ) :=ϕ(〈ξ〉) of the argument ξ ∈ Rn if we pass to the spherical coordinates.

We now study the interpolation properties of the Hörmander spaces Hϕ(Rn),where the function parameter ϕ ∈ RO.

Theorem 2.18. For given functions ϕ0, ϕ1 ∈ RO and ψ ∈ B, the functionϕ0/ϕ1 is assumed to be bounded in the vicinity of ∞. Let ψ be an interpolationparameter and let

ϕ(t) := ϕ0(t)ψ(ϕ1(t)/ϕ0(t)) for t ≥ 1.

Then ϕ ∈ RO and

[Hϕ0(Rn), Hϕ1(Rn)]ψ = Hϕ(Rn) with equality of the norms. (2.94)


Proof. First, we show that ϕ ∈ RO. By definition, the function ϕ isBorel measurable on the semiaxis [1,∞). We now show that it satisfies con-dition (2.89). Since ϕ0, ϕ1 ∈ RO, there exist numbers a > 1 and c > 1 suchthat

c−1 ≤ ϕj(λt)

ϕj(t)≤ c for all t ≥ 1, λ ∈ [1, a], and j ∈ 0, 1. (2.95)

The boundedness of the function ϕ0/ϕ1 in the vicinity of∞ and Proposition 2.3imply that there exists a number ε > 0 such that

ϕ1(t)

ϕ0(t)> ε for any t ≥ 1. (2.96)

Further, since ψ is an interpolation parameter, the function ψ is pseudoconcavein the vicinity of∞ by Theorem 1.9. Thus, by virtue of Lemmas 1.1 and 1.2, thefunction ψ is weakly equivalent to a concave function on the semiaxis (ε,∞).This is equivalent to the following condition: there exists a number c0 > 1 suchthat

ψ(τ)

ψ(t)≤ c0 max

1,τ

t

for all τ > ε and t > ε. (2.97)

This yields the inequality

ψ(t)

ψ(τ)≥ c−10 min

1,t

τ

for all τ > ε and t > ε. (2.98)

Relations (2.95), (2.96), and (2.97) now imply that, for any t ≥ 1 and λ ∈ [1, a],we get

ϕ(λt)

ϕ(t)=ϕ0(λt)

ϕ0(t)· ψ(ϕ1(λt)/ϕ0(λt))

ψ(ϕ1(t)/ϕ0(t))≤ c · c0 max

1,ϕ1(λt)/ϕ0(λt)

ϕ1(t)/ϕ0(t)

≤ c3c0.

Similarly, relations (2.95), (2.96), and (2.98) yield

ϕ(λt)

ϕ(t)≥ c−1c−10 min

1,ϕ1(λt)/ϕ0(λt)

ϕ1(t)/ϕ0(t)

≥ c−3c−10 .

Thus, the function ϕ satisfies condition (2.89) and, therefore, ϕ ∈ RO.We now prove equality (2.94). By virtue of Proposition 2.6(iii), the pair

[Hϕ0(Rn), Hϕ1(Rn)] is admissible. The pseudodifferential operator with thesymbol ϕ1(〈ξ〉)/ϕ0(〈ξ〉), where ξ ∈ Rn, is the generating operator J for thispair. By using the Fourier transformation

F : Hϕ0(Rn)↔ L2(Rn, ϕ20(〈ξ〉) dξ),


the generating operator J is reduced to the form of multiplication by the func-tion ϕ1(〈ξ〉)/ϕ0(〈ξ〉). Hence, the operator ψ(J) is reduced to the form of mul-tiplication by the function

ψ

(ϕ1(〈ξ〉)ϕ0(〈ξ〉)

)=

ϕ(〈ξ〉)ϕ0(〈ξ〉)

.

Thus, for any function u ∈ C∞0 (Rn), we can write

‖u‖2[Hϕ0 (Rn),Hϕ1 (Rn)]ψ = ‖ψ(J)u‖2Hϕ0 (Rn) =

∫Rn

ϕ20(〈ξ〉) |(ψ(J)u)(ξ)|2 dξ

=

∫Rn

ϕ2(〈ξ〉) |u(ξ)|2 dξ = ‖u‖2Hϕ(Rn).

This yields the equality of spaces (2.94) because the set C∞0 (Rn) is dense ineach of these spaces. This fact follows from Proposition 2.6(ii) and Theorem 1.1according to which the space Hϕ1(Rn) is continuously and densely embeddedin [Hϕ0(Rn), Hϕ1(Rn)]ψ.


Theorem 2.19. Assume that a function ϕ ∈ RO and real numbers s0 and s1such that s0 < σ0(ϕ) and s1 > σ1(ϕ) are given. Also let

ψ(t) :=

t−s0/(s1−s0) ϕ(t1/(s1−s0)) for t ≥ 1,

ϕ(1) for 0 < t < 1.

Then the function ψ ∈ B is an interpolation parameter and

[Hs0(Rn), Hs1(Rn)]ψ = Hϕ(Rn) with equality of the norms. (2.99)

Proof. Sinceϕ(t) = ts0 ψ(ts1/ts0)

for t ≥ 1, Theorem 2.19 follows from Theorems 1.9 and 2.18 if we prove thatthe function ψ belongs to the set B and is pseudoconcave in the vicinity of ∞.By virtue of Proposition 2.5, the function ψ satisfies condition (2.97) for ε = 1.

Indeed, if t ≥ 1 and τ ≥ 1, then

ψ(τ)

ψ(t)=(τt

)−s0/(s1−s0) ϕ(τ1/(s1−s0))ϕ(t1/(s1−s0))

≤(τt

)−s0/(s1−s0)c max

(τt

)s1/(s1−s0),(τt

)s0/(s1−s0)

= c maxτt, 1.


In particular, this implies that the function 1/ψ is bounded on the semiaxis[1,∞) and, hence, ψ ∈ B in view of Proposition 2.3. It remains to useLemma 1.2 according to which condition (2.97) is equivalent to the require-ment that the function ψ is pseudoconcave on the semiaxis (ε,∞).


We can now prove the following fundamental property of the class of Hör-mander spaces:

Hϕ(Rn) : ϕ ∈ RO. (2.100)

Theorem 2.20. The class of spaces (2.100) coincides (up to equivalence ofnorms) with the set of all Hilbert interpolation spaces for the pairs of innerproduct Sobolev spaces

[Hs0(Rn), Hs1(Rn)], (2.101)

where s0, s1 ∈ R and s0 < s1.

Proof. By virtue of Theorem 2.19, each space Hϕ(Rn), where ϕ ∈ RO, is aninterpolation space for the pair of spaces (2.101) provided that s0 < σ0(ϕ) ands1 > σ1(ϕ). Conversely, if H is an interpolation Hilbert space for the pair ofspaces (2.101), where s0, s1 ∈ R and s0 < s1, then, by Proposition 1.1, we get

H = [Hs0(Rn), Hs1(Rn)]ψ with equivalence of the norms

for a function ψ ∈ B pseudoconcave in the vicinity of ∞. By Theorem 1.9, thisfunction is an interpolation parameter. According to Theorem 2.18, this yieldsH = Hϕ(Rn), where the function ϕ ∈ RO is given by the formula

ϕ(t) := ts0ψ(ts1−s0) for t ≥ 1.


In view of this theorem, the class of spaces (2.100) is called the extendedSobolev scale (by means of the Hilbert interpolation spaces).

Remark 2.5. By virtue of Theorems 2.18 and 1.9 and Proposition 1.1, theclass of spaces (2.100) is closed relative to the interpolation as a result of whichwe obtain a Hilbert space.

We now consider the extended Sobolev scale over an infinitely smooth closedmanifold Γ of dimension n ≥ 1. The following theorem gives equivalent defini-tions of the space Hϕ(Γ), ϕ ∈ RO (cf. Subsection 2.1.1). We use the notationfrom Subsection 2.1.1.

Theorem 2.21. Let ϕ ∈ RO. The following definitions give the same Hilbertspace Hϕ(Γ) up to equivalence of norms:


(i) The linear space Hϕ(Γ) consists, by the definition, of all distributions f ∈D′(Γ) such that (χjf) αj ∈ Hϕ(Rn) for any j ∈ 1, . . . , r. The spaceHϕ(Γ) is endowed with the inner product given by the formula

(f1, f2)Hϕ(Γ) :=r∑j=1

((χjf1) αj , (χjf2) αj)Hϕ(Rn)

and the corresponding Hilbert norm.

(ii) Let k0 and k1 be integers such that k0 < σ0(ϕ) and k1 > σ1(ϕ). By thedefinition,

Hϕ(Γ) := [Hk0(Γ), Hk1(Γ)]ψ,

where the interpolation parameter ψ is given by the formula

ψ(t) =

t−k0/(k1−k0) ϕ(t1/(k1−k0)) for t ≥ 1,

ϕ(1) for 0 < t < 1.

(iii) By the definition, the space Hϕ(Γ) is the completion of the linear manifoldC∞(Γ) with respect to the Hilbert norm

f 7→ ‖ϕ((1− ∆Γ)1/2) f‖L2(Γ), f ∈ C∞(Γ).

This theorem is a special case of the following two theorems. As the basicdefinition of the space Hϕ(Γ), we take assertion (i) of Theorem 2.21.

Theorem 2.22. The interpolation theorems 2.18 and 2.19 remain true if,in their formulations, Rn is replaced by Γ and the equality of norms is replacedby their equivalence.

Theorem 2.22 is proved by analogy with Theorem 2.2.Let A be an elliptic PsDO of order m > 0. Assume that A is an (unbounded)

self-adjoint positive-definite operator in the Hilbert space L2(Γ). We addition-ally define the function ϕ ∈ RO by the equality ϕ(t) := ϕ(1) for 0 < t < 1.

Theorem 2.23. For any ϕ ∈ RO, the norm in the space Hϕ(Γ) is equivalentto the norm

f 7→ ‖ϕ(A1/m) f‖L2(Γ) (2.102)

in the dense set C∞(Γ). Thus, the space Hϕ(Γ) coincides with the completionof the set C∞(Γ) with respect to norm (2.102).

Theorem 2.23 is proved by analogy with Theorem 2.5.At the end of this subsection, we present an analog of Theorem 2.20 for the

class of spacesHϕ(Γ) : ϕ ∈ RO. (2.103)


Theorem 2.24. The class of spaces (2.103) coincides (up to equivalence ofnorms) with the set of all Hilbert interpolation spaces for the pairs of innerproduct Sobolev spaces [Hs0(Γ), Hs1(Γ)], where s0, s1 ∈ R and s0 < s1.

This theorem is proved by analogy with Theorem 2.20.The class of spaces (2.103) is called the extended Sobolev scale over the

manifold Γ.

2.4.3 Applications to elliptic operators

We now discuss some applications of the extended Sobolev scale to ellipticPsDOs.

First, we consider the PsDOs in Rn. It is useful to compare the results pre-sented in what follows with the results from Section 1.4. We preliminarilyconsider the action of PsDOs on the extended Sobolev scale (2.100). We set%(t) := t for t ≥ 1.

Lemma 2.8. Let A be a PsDO from the class Ψm(Rn), where m ∈ R. Thenthe restriction of the mapping u 7→ Au, u ∈ S ′(Rn), to the space Hϕ%m(Rn) isa linear bounded operator

A : Hϕ%m(Rn)→ Hϕ(Rn) for any ϕ ∈ RO. (2.104)

This lemma is proved with the help of the interpolation theorem 2.19 by analogywith Lemma 1.6.

Assume that the PsDO A ∈ Ψmph(Rn) is uniformly elliptic in Rn. Then map-

ping (2.104) has the following properties:

Theorem 2.25. Assume that a function ϕ ∈ RO and a number σ > 0 aregiven. There exists a number c = c(ϕ, σ) > 0 such that, for any distributionu ∈ Hϕ%m(Rn), the following a priori estimate is true:

‖u‖Hϕ%m (Rn) ≤ c(‖Au‖Hϕ(Rn) + ‖u‖Hϕ%m−σ (Rn)

).

This theorem is proved by analogy with Theorem 1.16.

Let V be an arbitrary nonempty open subset of the space Rn. For ϕ ∈ RO,we set

Hϕint(V ) :=

w ∈ H−∞(Rn) : χw ∈ Hϕ(Rn)

for all χ ∈ C∞b (Rn), suppχ ⊂ V, dist(suppχ, ∂V ) > 0.

Theorem 2.26. Assume that u ∈ H−∞(Rn) is a solution of the equationAu = f on the set V, where f ∈ Hϕ

int(V ) for a certain parameter ϕ ∈ RO.Then u ∈ Hϕ%m

int (V ).


This theorem is proved by analogy with Theorem 1.18.

Theorem 2.26 and Proposition 2.6(vi) yield the following sufficient condition(sharper than Theorem 1.19) for the existence of continuous derivatives of thesolutions to the equation Au = f.

Theorem 2.27. Let an integer r ≥ 0 and a function ϕ ∈ M satisfying thecondition

∞∫1

t2(r−m)+n−1ϕ−2(t) dt <∞,

be given. Assume that a distribution u ∈ H−∞(Rn) is a solution of the equationAu = f on the open set V ⊆ Rn and that f ∈ Hϕ

int(V ). Then the assertion ofTheorem 1.19 is true.

We now briefly consider the application of the spaces Hϕ(Γ), ϕ ∈ RO, toelliptic PsDOs on the infinitely smooth closed manifold Γ.

Let a PsDO A ∈ Ψmph(Γ), where m ∈ R, be elliptic on Γ. For the operator A,

the finite-dimensional spacesN andN+ are given by relations (2.45) and (2.46).

Theorem 2.28. For any parameter ϕ ∈ RO, the restriction of the mappingu 7→ Au, u ∈ D′(Γ), to the space Hϕ%m(Γ) is a bounded operator

A : Hϕ%m(Γ)→ Hϕ(Γ). (2.105)

This operator is Fredholm. Its kernel coincides with N and the domain is givenby the formula

f ∈ Hϕ(Γ) : (f, w)Γ = 0 for all w ∈ N+.

The index of operator (2.105) is equal to dimN − dimN+. It is independentof ϕ.

This theorem is proved by analogy with Theorem 2.9. Theorem 2.28 yieldsanalogs of Theorems 2.10–2.13 for the extended Sobolev scale over Γ. This canbe established by using the same reasoning as in Subsections 2.2.2 and 2.2.3.We omit the formulations of these analogs.


Section 2.1. The proposed equivalent definitions of the Hörmander spaces onclosed (compact) smooth manifolds are similar to the definitions used for theSobolev spaces; see, e.g., the monographs by J.-L. Lions and E. Magenes [121,Sec. 7.3] and M. Taylor [253, Chap. I, § 5].


G. Shlenzak [231] used a certain class of locally introduced Hörmander spacesover the boundary of an Euclidean domain. However, the independence of thesespaces and their topologies of the choice of local charts is not proved in [231].

The real Hörmander spaces over a circle, where the functions are given bytrigonometric Fourier series, were used by J. Pöschel [194], P. Djakov andB. S. Mityagin [44, 45], and V. A. Mikhailets and V. Molyboga [140, 141, 142].These spaces are closely related to the spaces of periodic real functions intro-duced by A. I. Stepanets [248, 249]. They are extensively used in the approxi-mation theory.

All results presented in Section 2.1 (except Theorem 2.7) were obtained bythe authors in [153, Sec. 3.3 and 3.4]. The proof of Theorem 2.7 is presentedhere for this first time. These results were partially announced in [159, Sec. 7.2].Numerous results, including Theorems 2.2, 2.3, and 2.8, remain valid for thecompact smooth manifolds with boundary [145, Sec. 3].Section 2.2. The analysis of classical PsDOs on smooth manifolds withoutboundary was developed by L. Hörmander [82]. The systematic presentationof the theory of elliptic PsDOs on these manifolds can be found, e.g., in hismonograph [86, Chap. 19] and in the survey by M. S. Agranovich [10, § 2].The classical results presented in these works concerning the Fredholm prop-erty of elliptic PsDOs and the regularity of solutions on the Sobolev scale havefound various applications to the theory of elliptic boundary-value problems fordifferential equations, spectral theory, the theory of function spaces, etc. (seealso the monographs by M. Taylor [253], F. Treves [254, 255], and M. A. Shu-bin [232].

The elliptic PsDOs realizing isomorphisms on the Sobolev scale are of sub-stantial independent interest. A broad class of operators of this kind, namely,the elliptic operators with parameter, was selected and studied by S. Agmonand L. Nirenberg [2, 6] and M. S. Agranovich and M. I. Vishik [13]; see alsothe survey by M. S. Agranovich [10, § 4]. These operators found importantapplications to the spectral theory and to the theory of parabolic equations.

All theorems in Section 2.2 were proved in [165].Section 2.3. For any orthogonal series, the classical theorem on convergencealmost everywhere was proved independently by D. E. Men’shov [134] andH. Rademacher [197]. The theorems on unconditional convergence were es-tablished by W. Orlicz [178] and K. Tandori [251]. We use the equivalentformulation of the Orlicz theorem proposed by P. L. Ul’yanov [261, § 4] (seealso [262, § 9, Subsec. 1]). All these theorems were proved by the mentionedauthors for real-valued series given on a finite interval of the axis R and for theLebesgue measure. The presentation of these results can be found, e.g., in themonographs by G. Alexits [14] and B. S. Kashin and A. A. Saakyan [94].

Apparently, the most general versions of the Men’shov–Rademacher, Orlicz–Ul’yanov, and Tandori theorems are considered in [157] (see also [158]). The


complete characterization of the coefficients of arbitrary orthogonal series con-vergent almost everywhere was given by A. Paszkiewicz [185].

Theorems 2.15 and 2.16 were announced in [156, Sec. 5]. They generalizeand significantly improve the result obtained by C. Meaney [133] who used theSobolev scale and considered the expansions in eigenfunctions of the Beltrami–Laplace operator. Theorem 2.17 was presented in [159]

Section 2.4. The class of RO-varying functions was introduced by V. G. Ava-kumović [19] in 1936 and studied fairly comprehensively; see, e.g., the mono-graphs by N. H. Bingham, C. M. Goldie, and J. L. Teugels [26, Sec. 2.0–2.2]and E. Seneta [235, Appendix].

The Hörmander spaces parametrized by RO-varying functions, the extendedSobolev scale formed by these spaces, and their applications to scalar ellipticoperators were studied in [152, 156, 160]. All theorems presented in Section 2.4were proved in the cited works. The parameter-elliptic operators were investi-gated on this scale by A. A. Murach and T. N. Zinchenko in [174].

Chapter 3

Semihomogeneous elliptic boundary-valueproblems

3.1 Regular elliptic boundary-value problems

In the present section, we give a definition of the regular elliptic boundary-valueproblem in a bounded Euclidean domain and formulate some notions relatedto this class of problems.

3.1.1 Definition of the problem

In Chapters 3 and 4, we assume that Ω is an arbitrary bounded domain inthe Euclidean space Rn (n ≥ 2) with a boundary Γ, which is supposed to bean infinitely smooth manifold of dimension n − 1 that has no boundary. Thedomain Ω is locally located on one side of Γ.We set Ω := Ω∪Γ and Ω := Rn\Ω.For a point x ∈ Γ, let ν(x) denote the unit vector of the inner normal to theboundary Γ at x. Also let ν denote the infinitely smooth vector field of theseunit vectors.

In the domain Ω, we consider the boundary-value problem

Lu ≡∑|µ|≤ 2q

lµ(x)Dµu = f in Ω, (3.1)

Bj u ≡∑|µ|≤mj

bj,µ(x)Dµu = gj on Γ, j = 1, . . . , q, (3.2)

where L = L(x,D) is a linear differential expression of even order 2q ≥ 2 on Ω

andBj = Bj(x,D), j = 1, . . . , q,

are boundary linear differential expressions of orders ordBj = mj ≤ 2q − 1on Γ. All coefficients of differential expressions L and Bj are supposed to becomplex-valued functions that are infinitely smooth:

lµ ∈ C∞(Ω ) and bj,µ ∈ C∞(Γ).

We set B := (B1, . . . , Bq).In relations (3.1) and (3.2) (and in what follows), we use the standard nota-

tion:µ := (µ1, . . . , µn) is a multiindex,

|µ| := µ1 + . . .+ µn,

112 Chapter 3 Semihomogeneous elliptic boundary-value problems

Dµ := Dµ11 . . . Dµn

n ,

Dk := i∂/∂xk for k = 1, . . . , n,

i is the imaginary unit, and x = (x1, . . . , xn) is a point in the space Rn.In addition, we assume that (3.1), (3.2) is a regular elliptic boundary-value

poblem in the domain Ω.Let us recall the corresponding definition (see, e.g., [121, Chap. 2, Sec. 1] or

[258, Sec. 5.2.1]).The above-mentioned differential expressions L(x,D) for fixed x ∈ Ω and

Bj(x,D) for fixed x ∈ Γ are associated with the following homogeneous char-acteristic polynomials, respectively:

L(0)(x, ξ) :=∑|µ|=2q

lµ(x) ξµ and B

(0)j (x, ξ) :=

∑|µ|=mj

bj,µ(x) ξµ.

These polynomials are also called principal symbols of expressions L and Bj .Here, the variable ξ = (ξ1, . . . , ξn) ∈ Cn and ξµ := ξµ11 . . . ξµnn .

Definition 3.1. The boundary-value problem (3.1), (3.2) is called regular el-liptic in the domain Ω if the following conditions are satisfied:

(i) The differential expression L is properly elliptic on Ω; i.e., for any pointx ∈ Ω and any linearly independent vectors ξ′, ξ′′ ∈ Rn, the polynomialL(0)(x, ξ′ + τξ′′) in the variable τ has exactly q roots τ+j (x; ξ′, ξ′′), j =1, . . . , q, with positive imaginary parts and exactly q roots with negativeimaginary parts (with regard for multiplicities of the roots).

(ii) The system of boundary expressions B1, . . . , Bq satisfies the complement-ing condition with respect to L on Γ, i.e., for any point x ∈ Γ and any vectorξ 6= 0 tangent to Γ at x, the polynomials B(0)

j (x, ξ + τν(x)), j = 1, . . . , q,in the variable τ are linearly independent modulo the polynomial

q∏j=1

(τ − τ+j (x; ξ, ν(x))).

(iii) The system of boundary expressions B1, . . . , Bq is normal, i.e., theirorders mj , j = 1, . . . , q, are mutually distinct and B

(0)j (x, ν(x)) 6= 0 for

any x ∈ Γ.

In what follows, we always assume (except for Subsection 4.1.3) that theboundary-value problem (3.1), (3.2) is regular elliptic in its domain Ω.

Remark 3.1. Condition (i) of Definition 3.1 yields

L(0)(x, ξ) 6= 0 for all x ∈ Ω and ξ ∈ Rn \ 0, (3.3)

Section 3.1 Regular elliptic boundary-value problems 113

i.e., the differential expression L is elliptic on Ω. If n ≥ 3, then conditions (i)and (3.3) are equivalent (see, e.g., [121, Chap. 2, Sec. 1.1] or [258, Sec. 5.2.1]).If all coefficients of the expression L are real, then this equivalence holds alsofor n = 2.

Remark 3.2. The complementing condition (Definition 3.1(ii)) was firstformulated by Ya. B. Lopatinskii [123], [124] and, in particular cases, byZ. Ya. Shapiro [230]. Other equivalent forms of this condition are also known[11, Sec. 1.3]. The condition of normality for the system of boundary expres-sions (Definition 3.1(iii)) was independently introduced by N. Aronszajn andA. N. Milgram [16] and M. Schechter [222].

Example 3.1. Let k ∈ Z and 0 ≤ k ≤ q. The system of boundary expressions

Bju := ∂k+j−1u/∂νk+j−1, j = 1, . . . , q,

is normal and satisfies the complementing condition on Γ with respect to any dif-ferential expression L, which is properly elliptic on Ω (see, e.g., [258, Sec. 5.2.1,Remark 4]). If k = 0, then we have the Dirichlet boundary-value problem forthe equation Lu = f.

3.1.2 Formally adjoint problem

Along with problem (3.1), (3.2), we consider the boundary-value problem

L+v ≡∑|µ|≤ 2q

Dµ(lµ(x) v) = ω in Ω, (3.4)

B+j v = hj on Γ, j = 1, . . . , q. (3.5)

It is formally adjoint to the problem (3.1), (3.2) with respect to Green’s formula

(Lu, v)Ω +

q∑j=1

(Bju, C+j v)Γ = (u, L+v)Ω +

q∑j=1

(Cju, B+j v)Γ, (3.6)

which is valid for all functions u, v ∈ C∞(Ω ) (see, e.g., [121, Chap. 2, Sec. 2.2]or [258, Sec. 5.4.2]). Here, B+

j , Cj, and C+j are normal systems of bound-

ary linear differential expressions with coefficients from C∞(Γ). Their orderssatisfy the condition

ordBj + ordC+j = ordCj + ordB+

j = 2q − 1. (3.7)

Additionally, we should note that here and hereinafter (·, ·)Ω and (·, ·)Γ are usedto indicate as inner products in the spaces L2(Ω) and L2(Γ), respectively, aswell as natural extensions of these inner products by continuity.


The differential expression L+ is called formally adjoint to the expression L,and the system of boundary expressions B+

1 , . . . , B+q is called adjoint to the

system B1, . . . , Bq with respect to L. The adjoint system is not uniquely de-fined but all adjoint systems are equivalent in the following sense [227, Sec. 10-4]:

If we have a distinct system B+1 , . . . , B

+q , which is adjoint to B1, . . . , Bq

with respect to L, then

v ∈ C∞(Ω ) : B+j v = 0 on Γ for every j = 1, . . . , q

= v ∈ C∞(Ω ) : B+j v = 0 on Γ for every j = 1, . . . , q. (3.8)

It is known that a boundary-value problem is regular elliptic if and only ifthe problem, which is formally adjoint to it, is regular elliptic (see, e.g., [121,Chap. 2, Sec. 2.5] or [227, Sec. 10-3]).

We set

N := u ∈ C∞(Ω ) : Lu = 0 in Ω, Bju = 0 on Γ, j = 1, . . . , q,

N+ := v ∈ C∞(Ω ) : L+v = 0 in Ω, B+j v = 0 on Γ, j = 1, . . . , q.

Due to equality (3.8), the set N+ does not depend on the choice of the adjointsystem of boundary expressions B+

1 , . . . , B+q . Since problems (3.1), (3.2) and

(3.4), (3.5) are regular elliptic, spaces N and N+ are finite-dimensional (see,e.g., [121, Chap. 2, Sec. 5.4])

Example 3.2. Consider the Dirichlet problem for the differential equationLu = f, where the expression L is properly elliptic on Ω. For this boundary-value problem, its adjoint system is the Dirichlet problem for the equationL+v = ω. In this case, dimN = dimN+ [121, Chap. 2, Sec. 2.5, 8.5].

In the present chapter, we study semihomogeneous regular elliptic boundary-value problems; i.e., for (3.1) and (3.2) we suppose that either f = 0 in thedomain Ω or all gj = 0 on the boundary Γ. These two important subclasses ofproblems are studied separately in Sections 3.3 and 3.4.

3.2 Hörmander spaces for Euclidean domains

In this section, we study the classes of Hörmander spaces separately for openand closed Euclidean domains. In the first case, the space consists of distri-butions defined in an open domain, whereas in the second case, it is formedby distributions in Rn, which are supported on a closed domain [269, § 3, Sub-cec. 1 and 2]. For open and closed domains, we consider Ω and Ω, respectively.We use the indicated spaces in order to investigate the boundary-value prob-lem (3.1), (3.2). We discover a relation between these spaces and the spaces over

Section 3.2 Hörmander spaces for Euclidean domains 115

the boundary Γ. As usual, D′(Ω) is a topological linear space of all distributionsdefined in the domain Ω.

3.2.1 Spaces for open domains

Let s ∈ R and ϕ ∈M.

Definition 3.2. The linear space Hs,ϕ(Ω) consists of all distributions w ∈Hs,ϕ(Rn) restricted to the domain Ω. In the space Hs,ϕ(Ω), the norm of anydistribution u ∈ Hs,ϕ(Ω) is given by the formula

‖u‖Hs,ϕ(Ω) := inf‖w‖Hs,ϕ(Rn) : w ∈ Hs,ϕ(Rn), w = u on Ω

. (3.9)

Theorem 3.1. The space Hs,ϕ(Ω) is a separable Hilbert space with respect tonorm (3.9).

Proof. In accordance with the definition, Hs,ϕ(Ω) is the factor space of theHilbert space Hs,ϕ(Rn) by its subspace

w ∈ Hs,ϕ(Rn) : suppw ⊆ Ω. (3.10)

(Linear manifold (3.10) is closed in the topology of the space Hs,ϕ(Rn) dueto the continuous embedding Hs,ϕ(Rn) → D′(Rn) (see Theorem 3.6 below)).Hence, Hs,ϕ(Ω) is a Hilbert space with respect to the inner product

(u1, u2)Hs,ϕ(Ω) := (w1 −Πw1, w2 −Πw2)Hs,ϕ(Rn). (3.11)

Here, uj ∈ Hs,ϕ(Ω), wj ∈ Hs,ϕ(Rn), uj = wj in Ω for j ∈ 1, 2, and Π isthe orthoprojector of Hs,ϕ(Rn) onto subspace (3.10). (The right-hand side ofequality (3.11) is independent of the choice of w1 and w2.) The norm definedby formula (3.9) is generated by this inner product. Due to Definition 3.2, theseparability property of the space Hs,ϕ(Rn) implies the separability propertyfor Hs,ϕ(Ω).


In the particular case where ϕ ≡ 1, the space Hs,ϕ(Ω) is also denoted byHs(Ω). This is the Sobolev space of order s ∈ R over the domain Ω [258,Sec. 4.2.].

The class of spaces

Hs,ϕ(Ω) : s ∈ R, ϕ ∈M (3.12)

is called the refined Sobolev scale over the domain Ω or, shortly, the refinedscale. Let us study the properties of this scale.


Theorem 3.2. Let a function ϕ ∈ M and positive numbers ε and δ be given.Then, for any s ∈ R,

[Hs−ε(Ω), Hs+δ(Ω)]ψ = Hs,ϕ(Ω) (3.13)

with equivalence of norms. Here, ψ is the interpolation parameter from Theo-rem 1.14.

Proof. The pair of Sobolev spaces on the left-hand side of (3.13) is admissi-ble. Consider the operator RΩ of the restriction of distributions u ∈ D′(Rn) tothe domain Ω. We have the following bounded and surjective linear operators:

RΩ : Hs−ε(Rn)→ Hs−ε(Ω), RΩ : Hs+δ(Rn)→ Hs+δ(Ω), (3.14)

RΩ : Hs,ϕ(Rn)→ Hs,ϕ(Ω). (3.15)

By considering the interpolation with parameter ψ applied to (3.14), we obtainthe bounded operator

RΩ : [Hs−ε(Rn), Hs+δ(Rn)]ψ → [Hs−ε(Ω), Hs+δ(Ω)]ψ.

In view of Theorem 1.14, this implies the boundedness of the operator

RΩ : Hs,ϕ(Rn)→ [Hs−ε(Ω), Hs+δ(Ω)]ψ.

Since operator (3.15) is surjective, this yields the inclusion

Hs,ϕ(Ω) ⊆ [Hs−ε(Ω), Hs+δ(Ω)]ψ. (3.16)

Let us prove that the inverse inclusion is also true and continuous. It is shown inmonograph [258, Theorem 4.2.2] that for each k ∈ N one can construct a linearmapping Tk, which extends each distribution u ∈ H−k(Ω) onto the space Rn,such that

Tk : Hσ(Ω)→ Hσ(Rn) for |σ| < k (3.17)

is a bounded operator. Let us pick k ∈ N such that |s− ε| < k and |s+ δ| < kand consider bounded operators (3.17) for σ = s− ε and σ = s+ δ. Since ψ isan interpolation parameter, we have the bounded operator

Tk : [Hs−ε(Ω), Hs+δ(Ω)]ψ → [Hs−ε(Rn), Hs+δ(Rn)]ψ.

Hence, in view of Theorem 1.14, we obtain the bounded operator

Tk : [Hs−ε(Ω), Hs+δ(Ω)]ψ → Hs,ϕ(Rn). (3.18)

The product of bounded operators (3.15) and (3.18) generates the boundedidentity operator

I = RΩTk : [Hs−ε(Ω), Hs+δ(Ω)]ψ → Hs,ϕ(Ω).


Thus, along with inclusion (3.16), its inverse continuous embedding holds too.Hence, the equality of spaces (3.13) takes place. Moreover, by the Banachinverse operator theorem, norms in these spaces are equivalent.

Theorem 3.2 is proved.Let us recall that

C∞0 (Ω) := u ∈ C∞(Ω) : suppu ⊂ Ω.

We identify functions u ∈ C∞0 (Ω) with their extensions by zero onto Rn. In whatfollows, it should be clear from context in which one of the domains (Ω or Rn)the function u ∈ C∞0 (Ω) is defined.

Theorem 3.3. Let s ∈ R and ϕ,ϕ1 ∈M. Then the following assertions hold:

(i) The set C∞(Ω ) is dense in the space Hs,ϕ(Ω).

(ii) If s < 1/2, then the set C∞0 (Ω) is dense in the space Hs,ϕ(Ω).

(iii) For any number ε > 0, the dense compact embedding Hs+ε,ϕ1(Ω) →Hs,ϕ(Ω) takes place.

(iv) The function ϕ/ϕ1 is bounded in a neighborhood of ∞ if and only ifHs,ϕ1(Ω) → Hs,ϕ(Ω). This embedding is dense and continuous. It is com-pact if and only if ϕ(t)/ϕ1(t)→ 0 as t→∞.

Proof. Assertion (i) follows immediately from the fact that the set C∞0 (Rn)is dense in the space Hs,ϕ(Rn) [269, Lemma 3.1].

Assertion (ii) holds in the Sobolev case where ϕ ≡ 1 (see, e.g., [258, Theo-rem 4.7.1 (d)]). From this result, the assertion can be proved for any ϕ ∈ Mby using Theorems 3.2 and 1.1.

Let s < 1/2. Due to the above-mentioned theorems, the dense continuousembedding Hs+δ(Ω) → Hs,ϕ(Ω) takes place for any δ > 0. Moreover, if s+ δ <1/2, then the set C∞0 (Ω) is dense in the Sobolev space Hs+δ(Ω). Hence, thisset is also dense in Hs,ϕ(Ω). Assertion (ii) is proved.

Assertions (iii) and (iv) are contained in Theorems 7.4 and 8.1 from [269].The density of these embeddings is implied by assertion (i).


Theorem 3.4. Let a function ϕ ∈ M and an integer k ≥ 0 be given. Thencondition (1.37) is equivalent to the embedding

Hk+n/2,ϕ(Ω) → Ck(Ω ). (3.19)

This embedding is compact.


Proof. Assume that the function ϕ satisfies condition (1.37). Then, byTheorem 1.15(iii), the continuous embedding

Hk+n/2,ϕ(Rn) → Ckb (Rn) (3.20)

takes place. This yields continuous embedding (3.19). Indeed, for arbitraryfunctions u ∈ Hk+n/2,ϕ(Ω) and w ∈ Hk+n/2,ϕ(Rn) such that u = w in Ω, wehave w ∈ Ckb (Rn), u ∈ Ck(Ω ), and

‖u‖Ck(Ω ) ≤ ‖w‖Ckb (Rn) ≤ c ‖w‖Hk+n/2,ϕ(Rn),

where c is the norm of the embedding operator (3.20). Passing to the infimumover the functions w in this inequality, we obtain the estimate

‖u‖Ck(Ω ) ≤ c ‖u‖Hk+n/2,ϕ(Ω).

The continuity of embedding (3.19) is proved. Its compactness can be provedsimilarly to the proof of compactness for embedding (2.43) in Theorem 2.8.It remains to note that inclusion (2.43) leads to property (1.37) due to Propo-sition 1.5 (for p = q = 2, V := Ω) and equivalence (1.41).


Remark 3.3. For k = 0, the statement of the theorem can be found in Theo-rem 7.5 in the paper [269] by L. R. Volevich and B. P. Paneah.

Let us consider the problem of existence of traces on the boundary Γ forarbitrary distributions u ∈ Hs,ϕ(Ω) and properties of these traces. Since Γ isa closed compact infinitely smooth manifold of dimension n − 1, the refinedSobolev scale over Γ is defined.

Theorem 3.5. For arbitrary parameters s > 1/2 and ϕ ∈M, the linear map-ping

u 7→ u Γ, u ∈ C∞(Ω ), (3.21)

(u Γ is the trace of a function u on Γ) can be uniquely extended (by continuity)to the bounded surjective operator

RΓ : Hs,ϕ(Ω)→ Hs−1/2,ϕ(Γ). (3.22)

This operator has a right inverse operator

SΓ : Hs−1/2,ϕ(Γ)→ Hs,ϕ(Ω), (3.23)

which is linear and bounded and such that the mapping SΓ does not depend ons and ϕ.


Proof. In the case where ϕ ≡ 1 (Sobolev spaces), this theorem is known(see, e.g., the monograph by H. Triebel [258, Lemma 5.4.4]). By using aninterpolation, we extend this case onto the general case of any ϕ ∈M.

Let us choose a number ε > 0 such that s− ε > 1/2. We have the followingbounded linear operators:

RΓ : Hs∓ε(Ω)→ Hs∓ε−1/2,ϕ(Γ),

SΓ : Hs∓ε−1/2,ϕ(Γ)→ Hs∓ε,ϕ(Ω).

Applying the interpolation with the parameter ψ from Theorem 1.14, wherewe set δ := ε, in view of Theorems 2.2 and 3.2 we obtain bounded operators(3.22) and (3.23). Since RΓSΓh = h for any h ∈ Hs−ε−1/2(Γ), operator (3.23)is a right inverse operator to operator (3.22), and the latter one is surjective.


Remark 3.4. Theorems on traces for hyperplanes and flat pieces of a boundarywere proved for general Hörmander spaces in [81, Theorem 2.2.8] and [269,Theorems 6.1, 6.2, and 7.6].

Corollary 3.1. Let σ > 0 and ϕ ∈M. Then

Hσ,ϕ(Γ) = RΓf : f ∈ Hσ+1/2,ϕ(Ω), (3.24)

‖h‖Hσ,ϕ(Γ) inf‖f‖Hσ+1/2,ϕ(Ω) : f ∈ Hσ+1/2,ϕ(Ω), RΓf = h

. (3.25)

Proof. We set s := σ + 1/2 in Theorem 3.5. Equality (3.24) holds becauseoperator (3.22) is surjective. Let us prove the equivalence of norms (3.25). Forarbitrary functions h ∈ Hσ,ϕ(Γ) and f ∈ Hσ+1/2,ϕ(Ω) that satisfy the equalityRΓf = h, we have

‖h‖Hσ,ϕ(Γ) ≤ c1 ‖f‖Hσ+1/2,ϕ(Ω),

where c1 is the norm of operator (3.22). Passing to the infimum over thefunctions f in this inequality, we obtain the following estimate:

‖h‖Hσ,ϕ(Γ) ≤ c1 inf‖f‖Hσ+1/2,ϕ(Ω) : f ∈ Hσ+1/2,ϕ(Ω), RΓf = h

.

The inverse estimate is implied by the fact that

‖f‖Hσ+1/2,ϕ(Ω) ≤ c2 ‖h‖Hσ,ϕ(Γ)

for the function f := SΓh, where c2 is the norm of operator (3.23).Corollary 3.1 is proved.


Remark 3.5. If s < 1/2, then mapping (3.21) cannot be extended to a con-tinuous operator RΓ : Hs,ϕ(Ω) → D′(Γ). Indeed, if it were possible, then byTheorem 3.3 (ii) we would have RΓf = 0 for any distribution f ∈ Hs,ϕ(Ω).However, it is not true, e.g., for the function f := 1 on Ω. Since the set C∞0 (Ω)is dense in H1/2(Ω), this remark remains true for Sobolev spaces in the limitcase s = 1/2 [258, Theorem 4.7.1 (d)].

3.2.2 Spaces for closed domains

Let s ∈ R and ϕ ∈ M. Also let Q be an arbitrary nonempty closed subset ofthe space Rn.

Definition 3.3. The linear space Hs,ϕQ (Rn) consists of all distributions w ∈

Hs,ϕ(Rn) such that their supports suppw ⊆ Q. The space Hs,ϕQ (Rn) is endowed

with the inner product and norm from the space Hs,ϕ(Rn).

Theorem 3.6. The space Hs,ϕQ (Rn) is a separable Hilbert space.

Proof. Let a sequence (wj) be fundamental in Hs,ϕQ (Rn). Since the space

Hs,ϕ(Rn) is complete, the sequence has a limit w in this space. In view ofcontinuity of the embeddingHs,ϕ(Rn) → D′(Rn), we can conclude that wj → win D′(Rn) as j → ∞. This result along with inclusions suppwj ⊆ Q yieldsuppw ⊆ Q. Thus, the sequence (wj) has a limit w in the space Hs,ϕ

Q (Rn).This proves the completeness of the space. The space is separable because it isa subspace of the separable space Hs,ϕ(Rn).


In the Sobolev case (ϕ ≡ 1), we omit the index ϕ in the notation of the spaceHs,ϕQ (Rn) and other spaces considered in this chapter.It is important for us to consider the case where Q := Ω. We are going

to study the properties of the space Hs,ϕ

Ω(Rn) and its relation to the refined

Sobolev scale over Ω.

Theorem 3.7. Let a function ϕ ∈ M and positive numbers ε and δ be given.Then, for any s ∈ R,

[Hs−εΩ

(Rn), Hs+δΩ

(Rn)]ψ = Hs,ϕ

Ω(Rn) (3.26)


Proof. Since Ω is a bounded domain with infinitely smooth boundary, theset C∞0 (Ω) is dense in the space Hσ

Ω(Rn) for any σ ∈ R (see, e.g., [258, The-

orem 4.3.2/1 (b)]). Hence, the continuous embedding Hs+δΩ

(Rn) → Hs−εΩ

(Rn)


is dense, the pair of spaces Hs−εΩ

(Rn) and Hs+δΩ

(Rn) is admissible, and theleft-hand side of formula (3.26) is defined.

We will deduce this formula from Theorem 1.14 by using Theorem 1.6 (in-terpolation of subspaces). To this end, we have to construct a mapping whichis a projector for any space Hσ(Rn) with s− ε ≤ σ ≤ s+ δ onto the subspaceHσ

Ω(Rn). We construct this mapping as follows. Let us pick a number r > 0

such that |x| < r for all x ∈ Ω and set

G = x ∈ Rn : |x| < 4r \Ω.

Let R denote the mapping that corresponds each distribution in Rn to itsrestriction in the domain G. In such a way, we obtain the linear boundedoperator

R : Hσ(Rn)→ Hσ(G) for every σ ∈ R. (3.27)

Note that G is a bounded open domain with infinitely smooth boundary. Then(see, e.g., [258, Theorem 4.2.2]), for any compact set K ⊂ R, there existsa linear mapping T, which is a bounded operator

T : Hσ(G)→ Hσ(Rn) for every σ ∈ K (3.28)

that extends the distribution from the domain G onto Rn. This means thatoperator (3.28) is right inverse to operator (3.27). We take K = [s − ε, s + δ]and consider the mapping

P0 : u 7→ u− TRu, u ∈ Hs−ε(Rn).

Due to the boundedness of operators (3.27) and (3.28), we have the boundedlinear operator

P0 : Hσ(Rn)→ Hσ(Rn) for any σ ∈ [s− ε, s+ δ].

It projects the space Hσ(Rn) onto the subspace HσG(Rn) for any σ ∈ [s − ε,

s+ δ]. Indeed,

u ∈ Hσ(Rn)⇒ RP0 u = R(u− TRu) = Ru−RTRu = Ru−Ru = 0

⇒ P0 u ∈ HσG(Rn).

In addition,

u ∈ HσG(Rn) ⇒ Ru = 0 ⇒ P0u = u− TRu = u.

Now we choose a function χ ∈ C∞(Rn) such that χ(x) = 1 for |x| ≤ 2r andχ(x) = 0 for |x| ≥ 3r and consider the mapping

P : u 7→ χ · P0 u, u ∈ Hs−ε(Rn).


Since the operator of multiplication by the function χ is bounded in any Sobolevspace over Rn, we conclude that P is a projector of the space Hσ(Rn) onto thesubspace Hσ

Ω(Rn) for any σ ∈ [s − ε, s + δ]. Hence, in view of Theorems 1.6

and 1.14, we have the following equalities of the spaces up to equivalence oftheir norms:

[Hs−εΩ

(Rn), Hs+δΩ

(Rn)]ψ = [Hs−ε(Rn), Hs+δ(Rn)]ψ ∩ Hs−εΩ

(Rn)

= Hs,ϕ(Rn) ∩ Hs−εΩ

(Rn)

= Hs,ϕ

Ω(Rn).


Theorem 3.8. Let s ∈ R and ϕ ∈M. Then the following assertions hold:

(i) The set C∞0 (Ω) is dense in Hs,ϕ

Ω(Rn).

(ii) If |s| < 1/2, then spaces Hs,ϕ(Ω) and Hs,ϕ

Ω(Rn) are equal to each other as

completions of C∞0 (Ω) with respect to equivalent norms.

(iii) Spaces Hs,ϕ

Ω(Rn) and H−s,1/ϕ(Ω) are mutually dual with respect to the

extension of the inner product in L2(Ω) by continuity.

(iv) Assertions (iii) and (iv) of Theorem 3.3 remain true if one replace spacesH ·,·(Ω) by spaces H ·,·

Ω(Rn) with the same upper indices.

Proof. Assertions (i), (iii), and (iv) are contained, respectively, in Lemma 3.3,Section 3.4, and Theorems 7.1 and 8.1 in [269].

Let us prove assertion (ii). Assume that |s| < 1/2. Assertion (ii) for theSobolev case (ϕ ≡ 1) was proved, e.g., in [258, Theorem 4.3.2/1 (a) and (c)].From this result, we can prove the assertion for any ϕ ∈ M with the help ofinterpolation. Let us pick a number ε > 0 such that |s∓ ε| < 1/2. The identitymapping on the set C∞0 (Ω) can be extended by continuity to isomorphisms

I : Hs∓εΩ

(Rn)↔ Hs∓ε(Ω).

By applying the interpolation with the parameter ψ from Theorem 1.14 underthe choice of δ := ε, we obtain, in view of Theorems 3.2 and 3.7, an extraisomorphism

I : Hs,ϕ

Ω(Rn)↔ Hs,ϕ(Ω).

This proves assertion (ii).Theorem 3.8 is proved.


3.2.3 Rigging of L2(Ω) with Hörmander spaces

In the study of the elliptic problem with homogeneous boundary conditions,the following scale of Hörmander spaces proves to be useful.

Definition 3.4. Let s ∈ R and ϕ ∈ M. If s ≥ 0, then Hs,ϕ,(0)(Ω) denotes theHilbert space Hs,ϕ(Ω). If s < 0, then Hs,ϕ,(0)(Ω) denotes the Hilbert spaceHs,ϕ

Ω(Rn), which is dual to the space H−s,1/ϕ(Ω) with respect to the extension

of the inner product in L2(Ω) by continuity (due to Theorem 3.8(iii)).

The scale of spaces

Hs,ϕ,(0)(Ω) : s ∈ R, ϕ ∈M (3.29)

is two-sided in the parameter s and is refined in the parameter ϕ. For s ≥ 0, thepositive part of the scale consists of spaces Hs,ϕ,(0)(Ω) = Hs,ϕ(Ω) of distribu-tions defined in the domain Ω. For s < 0, the negative part of the scale consistsof spaces Hs,ϕ,(0)(Ω) = Hs,ϕ

Ω(Rn) of distributions supported on the closure of

the domain Ω. Thus, scale (3.29) is formed by the spaces of distributions ofvarious nature.

We identify (and this is natural) the functions from the space L2(Ω) = H0(Ω)with their extensions by zero over Rn. In this sense, L2(Ω) = H0

Ω(Rn). In view

of Theorems 3.3(i), (iii), and 3.8(i), we have dense continuous embeddings

Hs,ϕ,(0)(Ω)← L2(Ω)← H−s,1/ϕ,(0)(Ω) for all s < 0, ϕ ∈M. (3.30)

Here, the flanked spaces are mutually dual relative to the extension of the innerproduct in L2(Ω) by continuity. Thus, we obtain a rigging of the Hilbert spaceL2(Ω) with spaces of scale (3.29). (For the definition of Hilbert rigging andrelated concepts, see [21, Chap. 1, Sec. 1.1] and [23, Chap. 14, Sec. 1.1].)

In the Sobolev case where ϕ ≡ 1, rigging (3.30) was introduced and studiedby Yu. M. Berezansky (see [21, Chap. 1, Sec. 3] or [23, Chap. 14, Sec. 3 and 4]).In this case, we denote the space Hs,ϕ,(0)(Ω) also by Hs,(0)(Ω).

The fact that continuous embeddings (3.30) are dense implies that functionsof the class C∞(Ω ) (extended by zero over Rn) form a dense subset in anynegative space Hs,ϕ,(0)(Ω), s < 0. They are also dense in any positive spaceHs,ϕ,(0)(Ω), s > 0. This allows us to consider a continuous dense embedding ofthe form Hr,χ,(0)(Ω) → Hs,ϕ,(0)(Ω), where s, r ∈ R and ϕ, χ ∈ M. Naturally,this means that

‖u‖Hs,ϕ,(0)(Ω) ≤ const ‖u‖Hr,χ,(0)(Ω) for any u ∈ C∞(Ω ),

and moreover, the identity mapping on the set C∞(Ω ) can be extended bycontinuity to a bounded injective operator I : Hr,χ,(0)(Ω) → Hs,ϕ,(0)(Ω).


It is called the operator of embedding of the space Hr,χ,(0)(Ω) into the spaceHs,ϕ,(0)(Ω) (on this subject, see [23, Chap. 15, Sec. 7]).

Let us study the properties of scale (3.29).

Theorem 3.9. Let s ∈ R and ϕ,ϕ1 ∈M. Then the following assertions hold:

(i) Up to equivalence of norms,

Hs,ϕ,(0)(Ω) = Hs,ϕ

Ω(Rn) for s < 1/2 (3.31)

Hs,ϕ,(0)(Ω) = Hs,ϕ(Ω) for s > −1/2. (3.32)

(ii) If s < 1/2, then the set C∞0 (Ω) is dense in Hs,ϕ,(0)(Ω).

(iii) Spaces Hs,ϕ,(0)(Ω) and H−s, 1/ϕ,(0)(Ω) are mutually dual (with equal normsfor s 6= 0 and equivalent norms for s = 0) relative to the extension of theinner product in L2(Ω) by continuity.

(iv) For any ε > 0, the compact and dense embedding

Hs+ε,ϕ1,(0)(Ω) → Hs,ϕ,(0)(Ω)

takes place.

Proof. (i) Equalities (3.31) and (3.32) are implied immediately by Theo-rem 3.8(ii) and the definition of the space Hs,ϕ,(0)(Ω).

(ii) This assertion is a consequence of equality (3.31) and Theorem 3.8(i).

(iii) For s 6= 0, assertion (iii) is justified in Definition 3.4. For s = 0, itfollows from Theorem 3.8(iii) and formula (3.31). Namely,

(H0,ϕ,(0)(Ω))′ = (H0,ϕ(Ω))′ = H0,1/ϕ

Ω(Rn) = H0,1/ϕ,(0)(Ω)

(the latter equality holds up to equivalence of norms).

(iv) For s ≥ 0, assertion (iv) coincides with Theorem 3.3(iii). In the casewhere s+ε ≤ 0, it is contained in Theorem 3.8(iv); and if s+ε = 0 in this case,then we use (3.31). In the remaining case where s < 0 < s + ε, the requiredresult is implied by the following dense compact embeddings

Hs+ε,ϕ1,(0)(Ω) → H0,ϕ1,(0)(Ω) → Hs,ϕ,(0)(Ω).


Theorem 3.10. Let a function ϕ ∈ M and positive numbers ε, δ be given.Then, for any s ∈ R,

[Hs−ε,(0)(Ω), Hs+δ,(0)(Ω)]ψ = Hs,ϕ,(0)(Ω) (3.33)



Proof. In the case where s − ε ≥ 0 or s + δ ≤ 0, this theorem followsimmediately from interpolation theorems 3.2 and 3.7. Consider the remainingcase where s − ε < 0 < s + δ. We set λ := minε/2, δ/2, 1/4. Since we haveeither s∓ λ < 1/2 or s∓ λ > −1/2, we conclude that

[Hs−λ,(0)(Ω), Hs+λ,(0)(Ω)]η = Hs,ϕ,(0)(Ω) (3.34)

in view of Theorem 3.9(i) and the mentioned interpolation theorems. Here, theinterpolation parameter η is given by the formula

η(t) :=

t1/2 ϕ(t1/(2λ)) for t ≥ 1,

ϕ(1) for 0 < t < 1.

Additionally, since s− ε < s∓ λ < s+ δ, we can use the result [121, Chap. 1,Theorem 12.5] by J.-L. Lions and E. Magenes in order to prove that[

Hs−ε,(0)(Ω), Hs+δ,(0)(Ω)]ψ∓

=[(H−s+ε(Ω))′, Hs+δ(Ω)

]ψ∓

= Hs∓λ,(0)(Ω). (3.35)

Here, ψ∓(t) := tθ∓ and the number θ∓ ∈ (0, 1) is determined by the conditions∓λ = (1−θ∓)(s−ε)+θ∓(s+δ). This implies the equality θ∓ = (ε∓λ)/(ε+δ).

In view of the reiteration theorem 1.3, equalities (3.34) and (3.35) (withequivalence of norms) yield the equality

Hs,ϕ,(0)(Ω) = [Hs−λ,(0)(Ω), Hs+λ,(0)(Ω)]η

=[[Hs−ε,(0)(Ω), Hs+δ,(0)(Ω)]ψ− , [H

s−ε,(0)(Ω), Hs+δ,(0)(Ω)]ψ+

]η

= [Hs−ε,(0)(Ω), Hs+δ,(0)(Ω)]ω.

Here,

ω(t) := ψ−(t) η(ψ+(t)

ψ−(t)

)= tθ− η(tθ+−θ−) = t(ε−λ)/(ε+δ) η(t2λ/(ε+δ))

= t(ε−λ)/(ε+δ) tλ/(ε+δ) ϕ(t1/(ε+δ))

= ψ(t) for t ≥ 1.

Hence (see Remark 1.1), equality (3.34) holds up to equivalence of norms.Theorem 3.10 is proved.


3.3 Boundary-value problems for homogeneouselliptic equations

Consider the regular elliptic boundary-value problem (3.1), (3.2) in the casewhere the elliptic equation (3.1) is homogeneous:

Lu = 0 in Ω, Bj u = gj on Γ for j = 1, . . . , q. (3.36)

Let us study the operator B = (B1, . . . , Bq), which corresponds to this problem,on the refined Sobolev scale.

3.3.1 Main result: boundedness and Fredholm propertyof the operator

We setK∞L (Ω) := u ∈ C∞(Ω ) : Lu = 0 in Ω (3.37)

and associate the linear mapping

u 7→ Bu = (B1 u, . . . , Bq u), u ∈ K∞L (Ω), (3.38)

with problem (3.36).Let s ∈ R and ϕ ∈M. We denote

Ks,ϕL (Ω) := u ∈ Hs,ϕ(Ω) : Lu = 0 in Ω . (3.39)

Since the embedding Hs,ϕ(Ω) → D′(Ω) is continuous, Ks,ϕL (Ω) is a closed

subspace of Hs,ϕ(Ω).Indeed, let u ∈ Hs,ϕ(Ω), and let a sequence (uj) ⊂ Ks,ϕ

L (Ω) be such thatuj → u inHs,ϕ(Ω) as j →∞. Then uj → u in D′(Ω), and this yields 0 = Luj →Lu in D′(Ω) as j →∞. Hence, Lu = 0 in the domain Ω, i.e., u ∈ Ks,ϕ

L (Ω).We treat Ks,ϕ

L (Ω) as a Hilbert space with respect to the inner product inHs,ϕ(Ω).

Let us formulate the main result of Section 3.3.

Theorem 3.11. For arbitrary parameters s ∈ R and ϕ ∈ M, the set K∞L (Ω)is dense in Ks,ϕ

L (Ω) and the mapping (3.38) can be uniquely extended (by con-tinuity) to the bounded linear operator

B : Ks,ϕL (Ω)→

q⊕j=1

Hs−mj−1/2, ϕ(Γ) =: Hs,ϕ(Γ). (3.40)

It is a Fredholm operator with the kernel N and the domain(g1, . . . , gq) ∈ Hs,ϕ(Γ) :

q∑j=1

(gj , C+j v)Γ = 0 for all v ∈ N+

. (3.41)

The index of operator (3.40) does not depend on s and ϕ.

Section 3.3 Boundary-value problems for homogeneous elliptic equations 127

Note that the term (gj , C+j v)Γ in formula (3.41) is the value of the antilinear

functional gj at the function C+j v ∈ C∞(Ω ). This implies that set (3.41) is

closed in the space Hs,ϕ(Γ). Then, in accordance with Theorem 3.11, the set

G := (C+1 v, . . . , C

+q v)

: v ∈ N+

(3.42)

is a defect subspace of operator (3.40): this set is orthogonal to the domainof the operator with respect to the extension of the inner product in (L2(Γ))

q

by continuity. The index of operator (3.40) is equal to dimN − dimG. It isclear that dimG ≤ dimN+, where the strict inequality is also possible. This isimplied by [85, Theorem 13.6.15].

In the case where ϕ ≡ 1 and s /∈ 1/2 − k : k ∈ N, the statement of The-orem 3.11 is contained in the Lions–Magenes theorem [121, Chap. 2, Sec. 7.3]on solvability of the inhomogeneous problem (3.1), (3.2) in the two-sided scaleof Sobolev spaces. The general case, where ϕ ∈M and s ∈ R, can be deducedfrom the Lions–Magenes theorem by using the interpolation with a suitablefunctional parameter and subsequent restriction of the operator of the problemto the space Ks,ϕ

L (Ω). The required steps are represented in Subsection 3.3.4.In Subsections 3.3.2 and 3.3.3, we present necessary results from the theory ofinterpolations and elliptic boundary-value problems.

Note that Theorem 3.11 provides new results even in the Sobolev case ϕ ≡ 1if the number s < 0 is half-integer (see Remark 3.6 in Subsection 3.3.3). Thistheorem can be regarded as an analog of the Harnack theorem on convergenceof sequences of harmonic functions (see, e.g., [161, Cap. 11, Sec. 9]). In thiscase, we have to use the metric in Hs,ϕ(Ω) instead of the uniform metric.

With respect to Theorem 3.11, we also mention the work [233] by Seeley onCauchy data for solutions of the homogeneous elliptic equation in the two-sidedscale of Sobolev spaces (additionally, see [11, Subsec. 5.4 b]).

3.3.2 A theorem on interpolation of subspaces

In this subsection, we formulate and prove an assertion (which is somewhatawkward) that concerns the interpolation of subspaces related to a linear op-erator. This assertion plays an important role in the proof of the main result.In the case of (complex) holomorphic interpolations, the assertion was provedby J.-L. Lions and E. Magenes [121, Theorem 14.3]. We show that this resultis also true for interpolations of Hilbert spaces with a functional parameter.In this case, unlike the cited monograph, our proof does not use the interpola-tion functor construction.

First of all, note that we accept the following notation. Let H, Φ, and Ψ beHilbert spaces and Φ → Ψ be continuous. Also let a linear bounded operatorT : H → Ψ be given. We denote

(H)T,Φ = u ∈ H : Tu ∈ Φ.


The space (H)T,Φ is a Hilbert space with respect to the graphic inner product

(u1, u2)(H)T,Φ = (u1, u2)H + (Tu1, Tu2)Φ

and it does not depend on Ψ.

Theorem 3.12. Let six Hilbert spaces X0, Y0, Z0, X1, Y1, and Z1 and threelinear mappings T, R, and S be given and the following conditions be satisfied:

(i) Pairs X = [X0, X1] and Y = [Y0, Y1] are admissible;

(ii) Z0 and Z1 are subspaces of a linear space E;

(iii) For any j ∈ 0, 1, the continuous embedding Yj → Zj takes place;

(iv) The mapping T is defined on X0 and, for any j ∈ 0, 1, specifies thebounded operator T : Xj → Zj;

(v) The mapping R is defined on E and, for any j ∈ 0, 1, specifies thebounded operator R : Zj → Xj for each j ∈ 0, 1;

(vi) The mapping S is defined on E and, for any j ∈ 0, 1, specifies thebounded operator S : Zj → Yj for each j ∈ 0, 1;

(vii) For any ω ∈ E, the equality TRω = ω + Sω is valid.

Then the pair of spaces [ (X0)T,Y0 , (X1)T,Y1 ] is admissible, and, for any inter-polation parameter ψ ∈ B, the equality of spaces

[ (X0)T,Y0 , (X1)T,Y1 ]ψ = (Xψ)T,Yψ . (3.43)

takes place up to equivalence of norms.

Proof. In view of conditions (iii) and (iv), spaces (Xj)T,Yj , j ∈ 0, 1, arewell defined. We prove that the space on the right-hand side of equality (3.43)is well defined too. Due to condition (i), spacesXψ and Yψ are defined; for thesespaces, continuous embeddings Xψ → X0 and Yψ → Y0 take place. The firstembedding and condition (iv) for j = 0 yield the boundedness of the operatorT : Xψ → Z0. In addition, the second embedding and condition (iii) yield thecontinuity of the embedding Yψ → Z0. Thus, the space on the right-hand side ofequality (3.43) is well defined and it is a Hilbert space likewise spaces (Xj)T,Yjfor j ∈ 0, 1 are.

In our proof, we consider the mapping

Pu = −RTu+ u, u ∈ X0. (3.44)

For any j ∈ 0, 1, the operator P : Xj → Xj is bounded due to conditions (iv)and (v). Moreover, for any u ∈ Xj , conditions (vi) and (vii) imply that

TPu = −TRTu+ Tu = −(Tu+ STu) + Tu = −STu ∈ Yj ,


i.e., Pu ∈ (Xj)T,Yj . In addition, the boundedness of the operator P : Xj → Xj

and conditions (iv), (vi) yield the estimate

‖Pu‖2(Xj)T,Yj = ‖Pu‖2Xj + ‖TPu‖

2Yj = ‖Pu‖

2Xj + ‖ − STu‖

2Yj ≤ c1‖u‖

2Xj ,

where the number c1 > 0 does not depend on u. Thus, mapping (3.44) setsbounded operators

P : Xj → (Xj)T,Yj for each j ∈ 0, 1. (3.45)

Besides, consider restrictions of the mapping R to Yj for j ∈ 0, 1. Due to con-ditions (iii) and (v), we have the bounded operator R : Yj → Xj . Moreover, forany ω ∈ Yj , conditions (vi) and (vii) imply the equality TRω = ω + Sω ∈ Yj ,i.e., Rω ∈ (Xj)T,Yj . In addition, conditions (iii), (vi), and (vii) and the bound-edness of the operator R : Yj → Xj yield the estimate

‖Rω‖2(Xj)T,Yj = ‖Rω‖2Xj + ‖TRω‖

2Yj = ‖Rω‖

2Xj + ‖ω + Sω‖2Yj

≤ ‖Rω‖2Xj +(‖ω∥∥Yj

+∥∥Sω‖Yj)2

≤ c2 ‖ω‖2Yj +(‖ω‖Yj + c3 ‖ω‖Zj

)2≤ c4 ‖ω‖2Yj ,

where constants c2, c3, and c4 are independent of ω. Thus, operators

R : Yj → (Xj)T,Yj for j ∈ 0, 1 (3.46)

are bounded.We use operators (3.45) and (3.46) in order to prove that the pair of spaces

[ (X 0)T,Y0 , (X1)T,Y1 ] (3.47)

is admissible. First we prove that the space (Xj)T,Yj is separable for eachj ∈ 0, 1. Due to condition (i), spaces Xj and Yj are separable. Considerarbitrary countable sets X0

j and Y 0j that belong to and are dense in Xj and Yj ,

respectively. Using these sets, we construct the countable set

Q = Pu0 +Rv0 : u0 ∈ X0j , v0 ∈ Y 0

j

and approximate any u ∈ (Xj)T,Yj by elements of this set. Since u ∈ Xj andTu ∈ Yj , one can find sequences of elements uk ∈ X0

j and vk ∈ Y 0j such that

uk → u in Xj and vk → Tu in Yj as k →∞. Due to bounded operators (3.45)and (3.46) and equality (3.44), we obtain

wk := Puk +Rvk → Pu+RTu = u in (Xj)T,Yj as k →∞, (3.48)


with wk ∈ Q. Hence, the countable set Q is dense in the space (Xj)T,Yj , i.e.,the space is separable.

In order to prove that pair (3.47) is admissible, it remains to prove the densityof the continuous embedding (X1)T,Y1 → (X0)T,Y0 . Fix any u ∈ (X0)T,Y0 . Thenu ∈ X0 and Tu ∈ Y0. Due to condition (i), the space X1 is dense in X0 andthe space Y1 is dense in Y0. Hence, there exist sequences of elements uk ∈ X1

and vk ∈ Y1 such that uk → u in X 0 and vk → Tu in Y0 as k →∞. From here,by using operators (3.45) and (3.46) and equality (3.44), we obtain (3.48) forj = 0 and prove that wk ∈ (X1)T,Y1 . Thus, (X1)T,Y1 is dense in (X0)T,Y0 .

Further, let us prove formula (3.43). First we show that the continuousembedding

[ (X0)T,Y0 , (X1)T,Y1 ]ψ → (Xψ)T,Yψ (3.49)

takes place. In view of definition of the space (Xj)T,Yj , we have boundedoperators

I : (Xj)T,Yj → Xj and T : (Xj)T,Yj → Yj for each j ∈ 0, 1,

where, as usual, I stands for the identity mapping. For the interpolation pa-rameter ψ, this implies the boundedness of operators

I : [ (X0)T,Y0 , (X1)T,Y1 ]ψ → Xψ,

T : [ (X0)T,Y0 , (X1)T,Y1 ]ψ → Yψ.

Hence, if u ∈ [ (X0)T,Y0 , (X1)T,Y1 ]ψ, then u ∈ Xψ, Tu ∈ Yψ and

‖u‖2Xψ + ‖Tu‖2Yψ ≤ c∥∥u∥∥2

[(X0)T,Y0 , (X1)T,Y1 ]ψ

for a constant c that does not depend on u. In other words, the continuousembedding (3.49) takes place.

In view of the Banach inverse operator theorem, it remains to prove theinclusion, which is inverse to (3.49). To this end, we apply the interpolationwith the parameter ψ to (3.45) and (3.46). We obtain bounded operators

P : Xψ → [ (X0)T,Y0 , (X1)T,Y1 ]ψ,

R : Yψ → [ (X0)T,Y0 , (X1)T,Y1 ]ψ.

Hence, if u ∈ (Xψ)T,Yψ (i.e., u ∈ Xψ and Tu ∈ Yψ ), then, in view of (3.44), weconclude that

u = Pu+RTu ∈ [ (X 0)T,Y0 , (X1)T,Y1 ]ψ.

Thus, we have the inclusion inverse to (3.49).Theorem 3.12 is proved.


3.3.3 Elliptic boundary-value problem in Sobolev spaces

To prove the main result, we use the classical theorem on solvability of theinhomogeneous boundary-value problem (3.1), (3.2) in positive Sobolev spaces(see, e.g., [11, Theorems 2.4.1 and 4.3.1] or [121, Chap. 2, Theorem 5.4]).

Proposition 3.1. The mapping

u 7→ (Lu,B1 u, . . . , Bq u), u ∈ C∞(Ω ), (3.50)

can be uniquely extended (by continuity) to the bounded Fredholm operator

(L,B) : Hs(Ω)→ Hs−2q(Ω)⊕Hs(Γ) (3.51)

for any real s ≥ 2q. The kernel of the operator coincides with N, and the domainconsists of the vectors

(f, g1, . . . , gq) ∈ Hs−2q(Ω)⊕Hs(Γ)

that satisfy the condition

(f, v)Ω +

q∑j=1

(gj , C+j v)Γ = 0 for all v ∈ N+. (3.52)

The index of operator (3.51) is equal to dimN − dimN+ and does not dependon s.

In the theorem and in what follows, we use the notation

Hs(Γ) :=q⊕j=1

Hs−mj−1/2(Γ), s ∈ R,

(cf. formula (3.40))Proposition 3.1 was extended to the case of any real s by J.-L. Lions and

E. Magenes [119, 120, 121] and Ya. A. Roitberg [202, 203, 209] (see also thepresentation of Roitberg’s results in [21, Chapt. III, § 6]). In that case, the op-erator (L,B) was studied in spaces that were constructed in different wayson the base of Sobolev spaces of relevant orders. In our proof, we use theconstruction suggested by J.-L. Lions and E. Magenes in the monograph [121,Chap. 2, Sections 6 and 7] because, unlike the cited papers by Ya. A. Roitberg.the construction by J.-L. Lions and E. Magenes remains applicable within theframework of spaces of distributions in the domain Ω. For the sake of simplicity,we consider here only integer s because it is enough to meet our purposes. Thegeneral case is considered in Subsection 4.4.1.


Let a function %1 ∈ C∞(Ω ) be positive in Ω and equal to the distance tothe boundary Γ in a certain neighborhood of the boundary.

For any integer σ ≥ 0, we set

Ξσ(Ω) :=

u ∈ D′(Ω) : %|µ|1 Dµu ∈ L2(Ω), |µ| ≤ σ

, (3.53)

where µ is an n-dimensional multiindex. The space Ξσ(Ω) is a Hilbert spacewith respect to the inner product

(u1, u2)Ξσ(Ω) :=∑|µ|≤σ

(%|µ|1 Dµu1, %

|µ|1 Dµu2

)L2(Ω)

.

We have the following dense continuous embeddings:

Hσ0 (Ω) → Ξ

σ(Ω) → L2(Ω), (3.54)

where Hσ0 (Ω) is the closure of the set C∞0 (Ω) in the topology of the space

Hσ(Ω).Let Ξ−σ(Ω) denote the Hilbert space dual to Ξσ(Ω) with respect to the

inner product in L2(Ω). Since spaces Hσ0 (Ω) and H−σ(Ω) are mutually dual

with respect to the same inner product [258, Theorem 4.8.2(a)], relation (3.54)yields the continuity of dense embeddings

L2(Ω) → Ξ−σ(Ω) → H−σ(Ω) (3.55)

for any integer σ > 0. The embedding on the right-hand side implies that thespace Ξ−σ(Ω) consists of distributions in the domain Ω.

For any integer s < 2q, we define the linear space

DsL(Ω) := u ∈ Hs(Ω) : Lu ∈ Ξ

s−2q(Ω)

with the graphic inner product

(u1, u2)DsL(Ω) = (u1, u2)Hs(Ω) + (Lu1, Lu2)Ξs−2q(Ω).

The space DsL(Ω) is complete with respect to this inner product.

The set C∞(Ω ) is dense in the space DsL(Ω). Due to (3.54) and the bounded-

ness property of the operator L : H2q(Ω) → L2(Ω), we have dense continuousembeddings

H2q(Ω) → DsL(Ω) → Hs(Ω) for any integer s < 2q. (3.56)

Remark 3.6. In the cited monograph by J.-L. Lions and E. Magenes [121],Hs(Ω) for s < 0 denotes the space dual to H−s0 (Ω) with respect to the innerproduct in L2(Ω). For all s < 0 that are not half-integers, this dual spacecoincides with our space Hs(Ω) [258, Theorem 4.8.2(a)]. For half-integer s < 0,this dual space does not coincide with Hs(Ω) (see also Subsection 4.4.1).


The following result was proved by J.-L. Lions and E. Magenes [121, Theo-rems 6.7 and 7.4].

Proposition 3.2. Mapping (3.50) can be uniquely extended (by continuity) tothe bounded Fredholm operator

(L,B) : DsL(Ω)→ Ξ

s−2q(Ω)⊕Hs(Γ) (3.57)

for any integer s < 2q. The kernel of the operator coincides with N and itsdomain consists of the vectors

(f, g1, . . . , gq) ∈ Ξs−2q(Ω)⊕Hs(Γ)

that satisfy condition (3.52). The index of operator (3.51) is equal to dimN −dimN+ and does not depend on s.

For our study, we have to establish an assertion on the isomorphism relatedto the operator that corresponds to a homogeneous boundary-value Dirichletproblem. Fix an integer r ≥ 1 and consider the r th iteration Lr of the expres-sion L. Let Lr+ denote the expression, which is formally adjoint to Lr. Look atthe linear differential expression LrLr+ + 1 of order 4qr with coefficients fromC∞(Ω ). For any integer σ ≥ 2qr, we set

HσD(Ω) := u ∈ Hσ(Ω) : γju = 0 on Γ, j = 0, . . . , 2qr − 1.

Here and in what follows, γju := (∂ju/∂jν) Γ is the trace operator for thenormal derivative of order j on Γ. In this case, the trace is understood inthe sense of Theorem 3.5 that states the existence of the bounded operatorγj : Hσ(Ω) → Hσ−j−1/2(Γ). Therefore, the linear space Hσ

D(Ω) is completewith respect to the inner product in the space Hσ(Ω).

Lemma 3.1. Let an integer r ≥ 1. The restriction of the mapping u 7→LrLr+u+ u, where u ∈ D′(Ω), defines the isomorphism

LrLr+ + 1 : HσD(Ω)↔ Hσ−4qr(Ω) (3.58)

for any integer σ ≥ 2qr.

Proof. The differential expression LrLr++1 is properly elliptic in Ω since theexpression L is properly elliptic. Consider the inhomogeneous boundary-valueDirichlet problem

LrLr+ u+ u = f in Ω,

γj u = gj on Γ for j = 0, . . . , 2qr − 1.


This is a regular elliptic boundary-value problem, and it was proved in [121,Chap. 2, Theorem 8.3 and Remark 8.5] that the operator of this problem isa bounded Fredholm operator with zero index in the pair of spaces

(LrLr+ + 1; γ0, . . . , γ2qr−1) : Hσ(Ω)

→ Hσ−4qr(Ω)⊕2qr−1⊕j=0

Hσ−j−1/2(Γ) (3.59)

for any integer σ ≥ 2qr. The kernel ND of operator (3.59) belongs to C∞(Ω).Using integration by parts, it is easy to verify that this kernel is trivial:

u ∈ ND ⇒ (u, u)Ω = −(LrLr+u, u)Ω = −(Lr+u, Lr+u)Ω ≤ 0

⇒ u = 0.

Note that by applying the method of integration by parts and transferringthe differential expression Lr of order 2qr, we obtain expressions of the form( · , γju)Γ, where j = 0, . . . , 2qr − 1. These expressions are equal to zero foru ∈ ND. Hence, we conclude that operator (3.59) is an isomorphism and itsrestriction to the subspace Hσ

D(Ω) defines isomorphism (3.58).Lemma 3.1 is proved.

Remark 3.7. An assertion similar to Lemma 3.1 can be found in Triebel’smonograph [258, Sec. 5.7.1, Remark 1].

3.3.4 Proof of the main result

We now prove the main result of Section 3.3, namely, we prove Theorem 3.11.

Proof of Theorem 3.11. Let s ∈ R and ϕ ∈ M. Choose an integer r ≥ 1such that

2q(1− r) < s < 2qr (3.60)

and use Proposition 3.2 for the integer s = 2q(1−r) ≤ 0 and Proposition 3.1 fors = 2qr ≥ 2q. We conclude that mapping (3.50) can be extended by continuityto bounded Fredholm operators

(L,B) : D2q(1−r)L (Ω)→ Ξ

−2qr(Ω)⊕H2q(1−r)(Γ), (3.61)

(L,B) : H2qr(Ω)→ H2q(r−1)(Ω)⊕H2qr(Γ) (3.62)

with the same index and common kernel N .Note that pairs of spaces

[D2q(1−r)L (Ω), H2qr(Ω)] and [Ξ−2qr(Ω), H2q(r−1)(Ω)] (3.63)


are admissible. Indeed, in view of (3.55) and (3.56), the following dense con-tinuous embeddings take place:

H2qr(Ω) → H2q(Ω) → D2q(1−r)L (Ω),

H2q(r−1)(Ω) → L2(Ω) → Ξ−2qr(Ω).

Hence, the spaces on the right-hand sides in pairs (3.63) are embedded contin-uously and densely into their respective spaces on the left-hand sides. This im-plies that the spaces on the left-hand sides are separable because the (Sobolev)spaces on the right-hand sides are separable. Thus, both pairs in (3.63) areadmissible.

In view of (3.60), we set

ε := s− 2q(1− r) > 0 and δ := 2qr − s > 0. (3.64)

Let ψ be the interpolation parameter from Theorems 3.2 and 2.2 relative tochosen parameters ϕ, ε, and δ. If we apply the interpolation with the param-eter ψ to the action spaces of bounded Fredholm operators (3.61) and (3.62),then by Theorems 1.7 and 1.5 we obtain the bounded Fredholm operator

(L,B) : [D2q(1−r)L (Ω), H2qr(Ω)]ψ

→ [Ξ−2qr(Ω), H2q(r−1)(Ω)]ψ ⊕ [H2q(1−r)(Γ),H2qr(Γ)]ψ. (3.65)

By virtue of Theorems 2.2, 1.5 and in view of (3.64), we have

[H2q(1−r)(Γ),H2qr(Γ)]ψ =

[ q⊕j=1

H2q(1−r)−mj−1/2(Γ),

q⊕j=1

H2qr−mj−1/2(Γ)

]ψ

=

q⊕j=1

[H2q(1−r)−mj−1/2(Γ), H2qr−mj−1/2(Γ)]ψ

=

q⊕j=1

[Hs−mj−1/2−ε(Γ), Hs−mj−1/2+δ(Γ)]ψ

=

q⊕j=1

Hs−mj−1/2,ϕ(Γ)

= Hs,ϕ(Γ)

up to equivalence of norms. Denoting

Z(Ω) := [Ξ−2qr(Ω), H2q(r−1)(Ω)]ψ, (3.66)


we can see that operator (3.65) is a bounded Fredholm operator in the pair ofspaces

(L,B) :[D

2q(1−r)L (Ω), H2qr(Ω)

]ψ→ Z(Ω)⊕Hs,ϕ(Γ). (3.67)

This operator has the same kernel N and the same index as operators (3.61)and (3.62) have.

By considering Z(Ω), let us describe the domain of operator (3.67). We aregoing to apply Theorem 3.12. To this end, we set

X0 = H2q(1−r)(Ω), Y0 = Ξ−2qr(Ω), Z0 = E = H−2qr(Ω),

X1 = H2qr(Ω), Y1 = Z1 = H2q(r−1)(Ω),

T = L

in the conditions of the theorem. Since the second pair in formula (3.63) isadmissible, the embedding on the right-hand side of (3.55) implies that con-ditions (i), (ii), and (iii) of Theorem 3.12 are satisfied. Condition (iv) of thistheorem is also valid because the operator L : Hσ(Ω)→ Hσ−2q(Ω) is boundedfor any σ ∈ R.

Additionally, we have to define linear mappings R and S that satisfy condi-tions (v), (vi), and (vii). We define them as follows. By applying Lemma 3.1 andconsidering the mapping (LrLr++1)−1, which is inverse to isomorphism (3.58),we obtain the bounded linear operator

(LrLr+ + 1)−1 : Hσ−4qr(Ω)→ Hσ(Ω) (3.68)

for any integer σ ≥ 2qr. Then we set

R = Lr−1Lr+(LrLr+ + 1)−1 and S = −(LrLr+ + 1)−1.

In view of (3.68) for σ = 2qr and σ = 2q(3r− 1), we obtain bounded operators

R : Z0 = H−2qr(Ω)→ H2qr−2q(2r−1)(Ω) = X0,

R : Z1 = H2q(r−1)(Ω)→ H2q(3r−1)−2q(2r−1)(Ω) = X1,

S : Z0 = H−2qr(Ω)→ H2qr(Ω) → H0(Ω) → Ξ−2qr(Ω) = Y0,

S : Z1 = H2q(r−1)(Ω)→ H2q(3r−1)(Ω) → H2qr(Ω) = X1.

In addition, on the set E = H−2qr(Ω), the following equalities hold:

TR = LLr−1Lr+(LrLr+ + 1)−1

= (LrLr+ + 1− 1)(LrLr+ + 1)−1

= 1− S.


Thus, all conditions of Theorem 3.12 are satisfied. By this theorem, for theinterpolation parameter ψ, we have the equality of spaces up to equivalence ofnorms: [

(X0)L,Y0 , (X1)L,Y1]ψ= (Xψ)L,Yψ . (3.69)

Here,

(X0)L,Y0 =u ∈ H2q(1−r)(Ω) : Lu ∈ Ξ

−2qr(Ω)= D

2q(1−r)L (Ω),

and the norms of spaces, which are located at the edges, are equal. Furthermore,due to the boundedness of the operator L : H2qr(Ω) → H2q(r−1)(Ω), we havethe equality

(X1)L,Y1 =u ∈ H2qr(Ω) : Lu ∈ H2q(r−1)(Ω)

= H2qr(Ω)

with equivalence of norms in spaces, which are located at the edges. In addition,by Theorem 3.2 and in view of (3.64), we have

Xψ = [H2q(1−r)(Ω), H2qr(Ω)]ψ = [Hs−ε(Ω), Hs+δ(Ω)]ψ = Hs,ϕ(Ω).

Thus, relation (3.69) takes the form

[D2q(1−r)L (Ω), H2qr(Ω)]ψ = u ∈ Hs,ϕ(Ω) : Lu ∈ Z(Ω). (3.70)

In the latter space, the graphic inner product is defined and the space is com-plete with respect to this inner product (we used the equality Yψ = Z(Ω) herein accordance with notation (3.66)).

By substituting equality (3.70) into (3.67), we obtain the bounded operator

(L,B) : u ∈ Hs,ϕ(Ω) : Lu ∈ Z(Ω) → Z(Ω)⊕Hs,ϕ(Γ). (3.71)

In accordance with the results proved above, it is a Fredholm operator thathas the kernel N. Moreover, since operator (3.71) is obtained by using theinterpolation procedure applied to Fredholm operators (3.61) and (3.62), byTheorem 1.7 and Proposition 3.2 we conclude that the range of operator (3.71)is equal to

Z(Ω)⊕Hs,ϕ(Γ) ∩ (L,B)(D

2q(1−r)L (Ω)

).

In other words, the range consists of the vectors

(f, g1, . . . , gq) ∈ Z(Ω)⊕Hs,ϕ(Γ)

that satisfy condition (3.52). The restriction of operator (3.71) to the subspace

Ks,ϕL (Ω) = u ∈ Hs,ϕ(Ω) : Lu = 0 in Ω


defines the bounded operator

B : Ks,ϕL (Ω)→ Hs,ϕ(Γ). (3.72)

Its kernel is equal to N ∩ Ks,ϕL (Ω) = N (and hence, it is finite-dimensional)

and its range coincides with subspace (3.41). Therefore, the range is closed,its codimension is finite and equal to the dimension of the space G definedby (3.42). Thus, operator (3.72) is a Fredholm operator that has the kernel N,range (3.41), and the finite index dimN−dimG, which does not depend on s, ϕ.

It remains to show that the set K∞L (Ω) is dense in Ks,ϕL (Ω) and opera-

tor (3.72) is an extension of mapping (3.38) by continuity. In this connection,let us note the following. Since operator (3.71) is an extension of (3.50), op-erator (3.72) is an extension of (3.38) in accordance with its definition (3.72).Therefore, to complete the proof, we have to verify that the set K∞L (Ω) is densein Ks,ϕ

L (Ω). To this end, we consider the isomorphism

B : Ks,ϕL (Ω)/N ↔ Rs,ϕ(Γ) (3.73)

generated by the Fredholm operator (3.72). Here, Rs,ϕ(Γ) denotes the do-main (3.41) of operator (3.72). Consider the isomorphism B−1, which is inverseto (3.73). It puts each vector g = (g1, . . . , gq) ∈ Rs,ϕ(Γ) in a correspondencewith the coset

B−1g = [u ] = u+ w : w ∈ N

of the element u ∈ Ks,ϕL (Ω) such that Bu = g.

As a preliminary result, we prove that mapping (3.73) possesses the followingproperty of increase in smoothness:

g ∈ Rs,ϕ(Γ) ∩ (C∞(Γ))q

⇒(B−1g = [u ] for some u ∈ K∞L (Ω)

). (3.74)

Letg = (g1, . . . , gq) ∈ Rs,ϕ(Γ) ∩ (C∞(Γ))q.

Since Rs,ϕ(Γ) is equal to set (3.41), by virtue of Proposition 3.1 the ellipticboundary-value problem (3.36) has a solution u ∈ H2q(Ω). The right-handsides of equations in this problem are infinitely smooth. Hence [121, Chap. 2,Sec. 5.4], the inclusion u ∈ C∞(Ω ) takes place. Thus, for operator (3.72), wehave u ∈ K∞L (Ω) and Bu = g, and this proves (3.74).

Now it is easy to prove the required density. Let us pick any distributionu ∈ Ks,ϕ

L (Ω) and consider the vector

g = Bu ∈ Rs,ϕ(Γ) ⊂ Hs,ϕ(Γ). (3.75)


Since the set C∞(Γ) is dense in Hσ,ϕ(Γ) for every σ ∈ R, there exists a sequenceof vectors g(k) such that

g(k) ∈ (C∞(Γ))q and g(k) → g in Hs,ϕ(Γ) as k →∞. (3.76)

Further, observe thatRs,ϕ(Γ) andG are (closed) subspaces inHs,ϕ(Γ) satisfyingthe conditions Rs,ϕ(Γ)∩G = 0 and codimRs,ϕ(Γ) = dimG. This implies thatthe space Hs,ϕ(Γ) is the direct sum of these subspaces. Due to this sum, wecan write g = g + 0 and g(k) = h(k) + ω(k), where h(k) ∈ Rs,ϕ(Γ) and ω(k) ∈ G.Using this result along with (3.76), we obtain the following two assertions:

h(k) = g(k) − ω(k) ∈ Rs,ϕ(Γ) ∩ (C∞(Γ))q,

h(k) → g in Rs,ϕ(Γ) (i.e., in Hs,ϕ(Γ)) as k →∞.

In view of (3.74), the first assertion implies B−1h(k) = [uk ] for some uk ∈K∞L (Ω). In view of (3.73) and (3.75), the second assertion yields

[uk ] = B−1h(k) → B−1g = [u ],

i.e.,[uk − u ]→ 0 in Ks,ϕ

L (Ω)/N as k →∞.This means that

uk − u+ wk → 0 in Ks,ϕL (Ω) as k →∞

for a sequence of functions wk ∈ N ⊂ K∞L (Ω). Thus, any distribution u ∈Ks,ϕL (Ω) can be approximated in the space Ks,ϕ

L (Ω) by a sequence of functionsuk + wk ∈ K∞L (Ω). Hence, the set K∞L (Ω) is dense in Ks,ϕ

L (Ω).Theorem 3.11 is proved.

3.3.5 Properties of solutions to the homogeneouselliptic equation

Theorem 3.11 implies that the operator (3.40), which corresponds to problem(3.36), is an isomorphism in the case of trivial kernel N and trivial defectsubspace G. In the general case, this operator defines an isomorphism

B : Ks,ϕL (Ω)/N ↔ Rs,ϕ(Γ) for all s ∈ R and ϕ ∈M. (3.77)

Recall that Rs,ϕ(Γ) is a subspace (3.41). (Note that the operator inverse to(3.77) is bounded by the Banach inverse operator theorem.) The collection ofisomorphisms (3.77) gives a solution to problem (3.36) for any distributionsg1, . . . , gq ∈ D′(Γ) that satisfy the condition

(g1, C+1 v)Γ + . . .+ (gq, C

+q v)Γ = 0 for any v ∈ N+.

In this case, the following a priori estimate holds for the solution u.


Theorem 3.13. Let parameters s ∈ R, ϕ ∈ M and ε > 0 be given. Thereexists a number c = c(s, ϕ, ε) > 0 such that, for any u ∈ Ks,ϕ

L (Ω), the followingestimate holds:

‖u‖Hs,ϕ(Ω) ≤ c(‖Bu‖Hs,ϕ(Γ) + ‖u‖Hs−ε(Ω)

). (3.78)

Proof. Due to isomorphism (3.77), we have

inf‖u+ w‖Hs,ϕ(Ω) : w ∈ N

≤ c0 ‖Bu‖Hs,ϕ(Γ) (3.79)

for any distribution u ∈ Ks,ϕL (Ω), where c0 is the norm of the operator, which

is inverse to (3.77). Since N is a finite-dimensional subspace for both Hs,ϕ(Ω)and Hs−ε(Ω), the norms in these two spaces are equivalent on N. In particular,for any w ∈ N, we have

‖w‖Hs,ϕ(Ω) ≤ c1 ‖w‖Hs−ε(Ω),

with a certain number c1 that does not depend on u and w. In addition, wehave

‖w‖Hs−ε(Ω) ≤ ‖u+ w‖Hs−ε(Ω) + ‖u‖Hs−ε(Ω)

≤ c2 ‖u+ w‖Hs,ϕ(Ω) + ‖u‖Hs−ε(Ω),

where c2 is the norm of the embedding operator Hs,ϕ(Ω) → Hs−ε(Ω). Hence,

‖u‖Hs,ϕ(Ω) ≤ ‖u+ w‖Hs,ϕ(Ω) + ‖w‖Hs,ϕ(Ω)

≤ ‖u+ w‖Hs,ϕ(Ω) + c1 ‖w‖Hs−ε(Ω)

≤ (1 + c1c2) ‖u+ w‖Hs,ϕ(Ω) + c1 ‖u‖Hs−ε(Ω).

By taking the infimum over all w ∈ N and applying inequality (3.79), we obtain

‖u‖Hs,ϕ(Ω) ≤ (1 + c1c2) c0 ‖Bu‖Hs,ϕ(Γ) + c1 ‖u‖Hs−ε(Ω);

i.e., estimate (3.78) with c := max(1 + c1c2)c0, c1 is deduced.Theorem 3.13 is proved.

If the right-hand side of inequality (3.78) is finite, the left-hand side is finiteas well.

Theorem 3.14. Let s ∈ R, ϕ ∈ M, and ε > 0. Assume that a distributionu ∈ Hs−ε(Ω) is a solution to problem (3.36) where

Bju = gj ∈ Hs−mj−1/2, ϕ(Γ) for each j ∈ 1, . . . , q. (3.80)

Then u ∈ Hs,ϕ(Ω).


Proof. By condition of the theorem, we have u ∈ Ks−ε, 1L (Ω) and Bu = g

where g = (g1, . . . , gq). Hence, due to (3.80) and Theorem 3.11 (the descriptionof the domain of operator (3.40)), we also have

g ∈ B(Ks−ε, 1L (Ω)) ∩Hs,ϕ(Γ) = B(Ks,ϕ

L (Ω)).

Therefore, there exists a distribution u0 ∈ Ks,ϕL (Ω) such that Bu0 = g. From

this by using Theorem 3.11 (the description of the kernel of operator (3.40)),we obtain successively:

B(u− u0) = 0,

w := u− u0 ∈ N ⊂ C∞(Ω ),

u = u0 + w ∈ Hs,ϕ(Ω).


Theorem 3.14 states an increase in smoothness of the solution u to prob-lem (3.36) up to the boundary Γ. In this case, we can see that the refinedsmoothness ϕ of the right-hand sides of equalities in the problem is inheritedby the solution. Note that any solution of the homogeneous elliptic equationLu = 0 in the domain Ω possesses the property u ∈ C∞(Ω) (see, e.g., [81, The-orem 7.4.1]). Therefore, it is essential in Theorem 3.14 that the smoothness ofthe solution u increases until it reaches the boundary of the domain Ω.

Corollary 3.2. Let σ ∈ R. Assume that a distribution u ∈ Hσ(Ω) is a solutionto problem (3.36) with

gj ∈ Hm−mj+(n−1)/2, ϕ(Γ) for each j ∈ 1, . . . , q (3.81)

where m := maxm1, . . . ,mq and the function ϕ∈M satisfies condition (1.37).Then u ∈ Cm(Ω ), and moreover, since u ∈ C∞(Ω) as well, the distribution uis a classical solution to problem (3.36).

Proof. Condition (3.81) coincides with (3.80) if we set s = m+n/2. Hence,in view of Theorems 3.14 and 3.4, we have

u ∈ Hm+n/2 ,ϕ(Ω) → Cm(Ω ),

Q.E.D.

Note that the left-hand sides of equalities in problem (3.36) for the classicalsolution u are computed by using classical derivatives. In this case, we haveBju ∈ C(Γ).


3.4 Elliptic problems with homogeneous boundaryconditions

Let us consider the regular elliptic boundary-value problem (3.1), (3.2) in thecase where boundary conditions (3.2) are homogeneous:

Lu = f in Ω, (3.82)

Bj u = 0 on Γ for j = 1, . . . , q. (3.83)

We study properties of the mapping u 7→ Lu, where u satisfies equalities (3.83),on the refined Sobolev scale.

3.4.1 Theorem on isomorphisms for elliptic operators

We now introduce the required spaces of distributions that satisfy homogeneousboundary conditions. Let s ∈ R and ϕ ∈ M. In this section, for the sake ofbrevity, we denote the Hilbert space Hs,ϕ,(0)(Ω) by Hs,ϕ.

Let the abbreviation (b.c.) denote homogeneous boundary conditions (3.83).We set

C∞(b.c.) := u ∈ C∞(Ω ) : Bju = 0 on Γ, j = 1, . . . , q

and use Hs,ϕ(b.c.) in order to denote the closure of the set C∞(b.c.) in thetopology of the Hilbert space Hs,ϕ.

Along with problem (3.82), (3.83), we consider the following formally adjointproblem with homogeneous boundary conditions:

L+ v = g in Ω, (3.84)

B+j v = 0 on Γ for j = 1, . . . , q. (3.85)

Let (b.c.)+ stand for the homogeneous boundary conditions (3.85). Then weset

C∞(b.c.)+ := v ∈ C∞(Ω ) : B+j v = 0 on Γ, j = 1, . . . , q

and use Hs,ϕ(b.c.)+ in order to denote the closure of the set C∞(b.c.)+ in thetopology of the Hilbert space Hs,ϕ.

Linear spaces Hs,ϕ(b.c.) and Hs,ϕ(b.c.)+ are Hilbert spaces with respect tothe inner product in Hs,ϕ.

In view of (3.8), the set C∞(b.c.)+ and, consequently, the space Hs,ϕ(b.c.)+

are independent of the choice of the system of boundary expressions

B+1 , . . . , B

+q ,

Section 3.4 Elliptic problems with homogeneous boundary conditions 143

which is adjoint to the system B1, . . . , Bq with respect to the differentialexpression L.

Due to Green formula (3.6), we have

(Lu, v)Ω = (u, L+v)Ω (3.86)

for all u ∈ C∞(b.c.) and v ∈ C∞(b.c.)+.

Therefore, the image Lu of an arbitrary function u ∈ C∞(b.c.) can be naturallytreated as the bounded antilinear functional (Lu, ·)Ω on the space

H2q−s,1/ϕ(b.c.)+.

Moreover, since spaces H2q−s, 1/ϕ and Hs−2q, ϕ are mutually dual with re-spect to the form (·, ·)Ω, the subspace H2q−s, 1/ϕ(b.c.)+ and the factor spaceHs−2q, ϕ/Ms−2q, ϕ , where

Ms−2q, ϕ :=h ∈ Hs−2q, ϕ : (h,w)Ω = 0 for all w ∈ C∞(b.c.)+

,

are also mutually dual with respect to this form. Hence, Lu can be treated asthe coset

Lu+ h : h ∈Ms−2q, ϕ,

which belongs to the factor space Hs−2q, ϕ/Ms−2q, ϕ.We treat the linear mapping u 7→ Lu as an operator that acts from the

space Hs,ϕ(b.c.) into the space Hs−2q, ϕ/Ms−2q, ϕ, which is identified with thedual space (H2q−s, 1/ϕ(b.c.)+)′ (it consists of antilinear functionals). In order toformulate the theorem on isomorphisms for the operator L, we have to defineprojectors of the space Hs,ϕ onto its subspaces, which are orthogonal, respec-tively, to N and N+ regarding to the sesquilinear form ( · , · )Ω. Such projectorsdo exist since spaces N and N+ are finite-dimensional.

Lemma 3.2. Let s ∈ R, and let ϕ ∈ M. Every element u ∈ Hs,ϕ admits theunique representation in the form u = u0 + u1, where u0 ∈ N, and u1 ∈ Hs,ϕ

satisfies the condition (u1, w)Ω = 0 for any w ∈ N. In this case, the mappingP : u 7→ u1 is a projector of the space Hs,ϕ onto the subspace

u1 ∈ Hs,ϕ : (u1, w)Ω = 0 for all w ∈ N, (3.87)

and the image Pu does not depend on s and ϕ. The restriction of the mappingP on Hs,ϕ(b.c.) is a projector of the space Hs,ϕ(b.c.) onto the subspace

u1 ∈ Hs,ϕ(b.c.) : (u1, w)Ω = 0 for all w ∈ N. (3.88)

Furthermore, the statement of the lemma remains true if we replace N by N+,P by P+, and (b.c.) by (b.c.)+. In such a case, Ms,ϕ is a subspace of P+(Hs,ϕ).


Proof. As it was mentioned above, N is a finite-dimensional subspaceof Hs,ϕ. It is clear that dimN is equal to the codimension of subspace (3.87)and, moreover, N and (3.87) have the trivial intersection. Hence, the spaceHs,ϕ can be decomposed into the direct sum of subspaces N and (3.87) withthe projector P onto subspace (3.87) such that the projector does not dependon s and ϕ. In view of the inclusion N ⊂ Hs,ϕ(b.c.), this implies that thespace Hs,ϕ(b.c.) can be decomposed into the direct sum of subspaces N and(3.88) with the same projector P onto subspace (3.88). Thus, as we can see,the existence of the projector P is implied by two conditions: (1) the space Nis finite-dimensional and (2) N ⊂ Hs,ϕ(b.c.). So, since the space N+ is finite-dimensional and N+ ⊂ Hs,ϕ(b.c.)+, the lemma remains true for the projectorP+ with the replacements mentioned above in the formulation of the lemma.Finally, we note that the last assertion of the lemma is implied by the inclusionN+ ⊂ C∞(b.c.)+.


We now formulate the main result of Section 3.4, namely, the theorem onisomorphisms, which are generated by the elliptic operator L on the refinedSobolev scale.

Theorem 3.15. Let s ∈ R, ϕ ∈M, and

s 6= j + 1/2 for each j ∈ 0, 1, . . . , 2q − 1. (3.89)

The mapping u 7→ Lu, where u ∈ C∞(b.c.) and Lu is treated either as the cosetLu+h : h ∈Ms−2q, ϕ or as the functional (Lu, · )Ω, can be uniquely extended(by continuity) to the bounded linear operator

L : Hs,ϕ(b.c.)→ Hs−2q, ϕ/Ms−2q, ϕ =(H2q−s, 1/ϕ(b.c.)+

)′. (3.90)

The restriction of operator (3.90) onto subspace (3.87) generates the isomor-phism

L : P (Hs,ϕ(b.c.))↔ P+(Hs−2q, ϕ)/Ms−2q, ϕ. (3.91)

Remark 3.8. In the Sobolev case (ϕ ≡ 1), Theorem 3.15 was proved byYu. M. Berezansky, S. G. Krein, and Ya. A. Roitberg for integer s (see [22,Theorem 2] and [21, Chap. 3, Theorem 6.12]). For all real s, it was proved inthe monograph by Ya. A. Roitberg [209, Theorem 5.5.2] (see also survey [11,Sec. 7.9 c]). For half-integer s ∈ j + 1/2 : j = 0, 1, . . . , 2q − 1, the spaces ofaction of operator (3.90) were defined with the help of interpolation.

Note that if spaces N and N+ are trivial, then operator (3.90) becomes anisomorphism

Theorem 3.15 yields the Fredholm property of operator (3.90) and an a prioriestimate for the solutions of the equation Lu = f.


Theorem 3.16. Let s ∈ R and ϕ ∈ M. Additionally, let condition (3.89) besatisfied. Then the following assertions hold:

(i) The bounded operator (3.90) is a Fredholm operator with the kernel N andrange

[f ] ∈ Hs−2q, ϕ/Ms−2q, ϕ : (f, w)Ω = 0 for all w ∈ N+

(3.92)

where [f ] = f+h : h ∈Ms−2q, ϕ is the coset of the element f ∈ Hs−2q, ϕ.The index of this operator is equal to dimN −dimN+ and does not dependon s and ϕ.

(ii) For any solution u ∈ Hs,ϕ(b.c.) of the equation Lu = [f ], the followinga priori estimate takes place:

For any ε > 0, there exists a number c = c(s, ϕ, ε) > 0, which does notdepend on u, such that

‖u‖Hs,ϕ ≤ c(‖f‖Hs−2q,ϕ + ‖u‖Hs−ε

). (3.93)

We prove Theorems 3.15 and 3.16 later in Subsection 3.4.3.

Theorem 3.16 implies that N+ is a defect subspace of operator (3.90). IfN = 0, then we can omit the term ‖u‖Hs−ε in estimate (3.93).

Let us note the following. Since the formally adjoint boundary-value problem(3.84), (3.85) is regular elliptic, Theorems 3.15 and 3.16 remain true if wereplace the operator L by the operator L+ (with other obvious changes in theformulation). Namely, the linear mapping v 7→ L+v, where v ∈ C∞(b.c.)+, canbe uniquely extended (by continuity) to the bounded Fredholm operator

L+ : H2q−s, 1/ϕ(b.c.)+ → H−s, 1/ϕ/M+−s, 1/ϕ = (Hs,ϕ(b.c.))′ (3.94)

with the kernel N+ and the defect subspace N. Here, we denote

M+−s, 1/ϕ =

h ∈ H−s, 1/ϕ : (h,w)Ω = 0 for all w ∈ C∞(b.c.)

,

and use the same assumptions on parameters s and ϕ as in Theorems 3.15and 3.16. The restriction of operator (3.94) to the subspace

P+(H2q−s, 1/ϕ(b.c.)+)

=v ∈ H2q−s, 1/ϕ(b.c.)+ : (v, w)Ω = 0 for all w ∈ N+

defines the isomorphism

L+ : P+(H2q−s, 1/ϕ(b.c.)+)↔ P (H−s, 1/ϕ)/M+−s, 1/ϕ.


Since operators (3.90) and (3.94) are bounded, equality (3.86) can be extendedby continuity to the relation

(Lu, v)Ω = (u, L+v)Ω

for all u ∈ Hs,ϕ(b.c.) and v ∈ H2q−s, 1/ϕ(b.c.)+.

This means that operators (3.90) and (3.94) are mutually adjoint with respectto the sesquilinear form ( · , · )Ω that plays the role of extension by continuityfor the inner product in L2(Ω).

3.4.2 Interpolation and homogeneous boundary conditions

Let us study interpolation properties of spaces, which are the action spacesfor operator (3.90). These properties play an important role in the proof ofTheorems 3.15 and 3.16. Recall that we omit the index ϕ in the notationof spaces introduced in the previous subsection (Subsection 3.4.1) when weconsider the Sobolev case ϕ ≡ 1.

Theorem 3.17. Let s ∈ R and ϕ ∈M. Additionally, let

s 6= mj + 1/2 for each j ∈ 1, . . . , q. (3.95)

Then the following assertions hold:

(i) For s given in such a way, there exists a number % = %(s) > 0 (that doesnot depend on ϕ) such that, for any ε ∈ (0, %), one has

[Hs−ε(b.c.), Hs+ε(b.c.)]ψ = Hs,ϕ(b.c.)

up to equivalence of norms, where ψ does not depend on s and representsthe interpolation parameter from Theorem 1.14 taken for ε = δ.

(ii) If the number s, which satisfies condition (3.95), is positive, then Hs,ϕ(b.c.)consists of all distributions u ∈ Hs,ϕ(Ω) that meet the condition Bj u = 0on Γ for every j ∈ 1, . . . , q with s > mj + 1/2.

(iii) If s < 1/2, then Hs,ϕ(b.c.) = Hs,ϕ.

(iv) Assertions (i), (ii), and (iii) remain true even if we replace mj by m+j ,

(b.c.) by (b.c.)+, and Bj by B+j in their statements.

Remark 3.9. Regarding assertion (i) of Theorem 3.17, we note that the in-terpolation with the power parameter of Sobolev spaces under homogeneousboundary conditions was studied by P. Grisvard [66] and R. T. Seeley [234](see also [258, Sec. 4.3.3]). Their results imply that condition (3.95) cannot be


neglected. In assertion (ii) of Theorem 3.17, we treat the expression Bj u inthe sense of Theorem 3.5 on traces. The presence of condition (3.95) in thisassertion is justified by arguments represented in Remark 3.5.

Proof of Theorem 3.17. In this proof, we consider cases s > 0 and s < 1/2separately.

Case s > 0. By changing (if it is necessary) the numbering of operators inthe system Bj : j = 1, . . . , q, we obtain

0 ≤ m1 < m2 < . . . < mq ≤ 2q − 1.

Additionally, we set m0 := −1/2 and mq+1 := +∞. Due to condition (3.95),one can find a number r ∈ 0, 1, . . . , q such that mr + 1/2 < s < mr+1 + 1/2.Let % = %(s) denote the distance from a point s to the set

j + 1/2 : j = −1/2, 0, 1, . . . , 2q − 1 \ s.

Pick any ε ∈ (0, %). Then

s∓ ε 6= j + 1/2 for each j ∈ 0, 1, . . . , 2q − 1, (3.96)

0 ≤ mr + 1/2 < s− ε < s < s+ ε < mr+1 + 1/2. (3.97)

Let ψ be the interpolation parameter from Theorem 1.14 taken for ε = δ. Thisparameter is independent of s. We interpolate the pair of spaces Hs∓ε(b.c.)with the parameter ψ by using Theorems 3.10 and 1.6 (the interpolation ofsubspaces). For this purpose, we have to define a projector P of each spaceHs∓ε(Ω) = Hs∓ε onto the subspace Hs∓ε(b.c.). Such a projector does exist dueto the following arguments.

At first, we assume that r 6= 0 and consider the collection Bj : j = 1, . . . , r.Since the system Bj : j = 1, . . . , q is normal, this collection, as its part, isa normal system of boundary expressions. We now refer to the monograph byH. Triebel [258, Lemma 5.4.4] where it was shown how to construct a linearmapping P, which is a projector of each of the spaces Hσ(Ω) = Hσ withσ > mr + 1/2 onto the subspace

u ∈ Hσ(Ω) : Bj u = 0 on Γ, j = 1, . . . , r . (3.98)

Let σ = s ∓ ε in this case. Then due to (3.97) we can assert that subspace(3.98) admits the following description:

u ∈ Hσ(Ω) : Bj u = 0 on Γ

for all j ∈ 1, . . . , q such that σ > mj + 1/2. (3.99)


At the same time, as it was shown in the monograph by Ya. A. Roitberg [209,Sec. 5.5.2, p. 167], due to condition (3.96) the regular ellipticity of the problem(3.82), (3.83) implies the density of the set C∞(b.c.) in subspace (3.99) of thespace Hσ(Ω). Hence, the set C∞(b.c.) is dense in space (3.98), i.e., this spacecoincides with Hσ(b.c.). Thus, the mapping P represents the required projectorof the space Hs∓ε(Ω) onto the subspace Hs∓ε(b.c.).

If r = 0, then 0 < s ∓ ε < mj + 1/2 for each j ∈ 1, . . . , q. Hence [209,Sec. 5.5.2, p. 167], the set C∞(b.c.) is dense in the space Hs∓ε(Ω) = Hs∓ε and,in consequence, Hs∓ε(b.c.) = Hs∓ε(Ω). This means that for r = 0 we can usethe identity mapping in order to represent P. Thus, the required projector Pis constructed.

Due to Theorems 1.6 (the interpolation of subspaces) and 3.10, this allowsus to write down the following equalities for spaces, up to equivalence of theirnorms:

[Hs−ε(b.c.), Hs+ε(b.c.)]ψ

= [Hs−ε(Ω), Hs+ε(Ω)]ψ ∩ Hs−ε(b.c.)

= Hs,ϕ(Ω) ∩ Hs−ε(b.c.)

= Hs,ϕ(Ω) ∩ u ∈ Hs−ε(Ω) : Bj u = 0 on Γ, j = 1, . . . , r

=u ∈ Hs,ϕ(Ω) : Bj u = 0 on Γ

for all j ∈ 1, . . . , q such that s > mj + 1/2. (3.100)

Note that the last equality in (3.100) is implied by condition (3.97). Thus, theinterpolation space

[Hs−ε(b.c.), Hs+ε(b.c.)]ψ

coincides (up to equivalence of norms) with subspace (3.100) of the spaceHs,ϕ(Ω) = Hs,ϕ. Hence, in view of Theorem 1.1, the space Hs+ε(b.c.) is em-bedded in (3.100) continuously and densely. This implies that the set C∞(b.c.)is dense in (3.100), i.e., space (3.100) coincides with Hs,ϕ(b.c.). Thus, asser-tions (i) (for s > 0) and (ii) are proved.

Case s < 1/2. We set % = 1/2 − s > 0 and pick any number ε ∈ (0, %).Since s − ε < s < s + ε < 1/2, in view of 3.9(ii) the set C∞0 (Ω) is dense inspacesHs,ϕ andHs∓ε, and then, a fortiori, is the wider set C∞(b.c.). Therefore,Hs,ϕ(b.c.) = Hs,ϕ and Hs∓ε(b.c.) = Hs∓ε. In view of Theorem 3.10, this yieldsassertion (i) of the theorem for s < 1/2 immediately.

Assertion (iii) of the theorem is a consequence of the fact that the setC∞(b.c.) is dense in the space Hs,ϕ.


The proof of Theorem 3.17 under the condition (b.c.) is completed. Notethat it is based only on the property of regular ellipticity of the boundary-valueproblem (3.82), (3.83). Since the formally adjoint problem (3.84), (3.85) is alsoregular elliptic, assertion (iv) is true.


Let σ∈R and ϕ∈M.We study the properties of the space (H−σ, 1/ϕ(b.c.)+)′,which is antidual to the space H−σ, 1/ϕ(b.c.)+.

Recall that

Mσ,ϕ = h ∈ Hσ,ϕ : (h,w)Ω = 0 for all w ∈ C∞(b.c.)+.

We claim that the set Mσ,ϕ is closed in the space Hσ,ϕ. Indeed, due to The-orem 3.9(iii), the function w ∈ C∞(b.c.)+ ⊂ H−σ, 1/ϕ generates the linearcontinuous functional ( · , w)Ω on the space Hσ,ϕ. Hence, if for a sequence ofdistributions hj ∈Mσ,ϕ we have hj → h in Hσ,ϕ as j →∞, then

(h,w)Ω = limj→∞

(hj , w)Ω = 0 for every w ∈ C∞(b.c.)+,

i.e., h ∈ Mσ,ϕ. Thus, Mσ,ϕ is a subspace of the Hilbert space Hσ,ϕ. Therefore,the factor space Hσ,ϕ/Mσ,ϕ is also a Hilbert space.

Theorem 3.18. Let σ ∈ R and ϕ ∈M. Then the following assertions hold:

(i) If σ > −1/2, then Mσ,ϕ = 0.

(ii) The factor space Hσ,ϕ/Mσ,ϕ and the subspace H−σ, 1/ϕ(b.c.)+ are mutuallydual (with equality of norms for s 6= 0 and equivalence of norms for s = 0)with respect to extension by continuity for the inner product in L2(Ω).More precisely, with respect to the bilinear form

([u] , v

)Ω

:=(u, v)

Ω, where

[u] = u + h : h ∈ Mσ,ϕ is the coset of the element u ∈ Hσ,ϕ, andv ∈ H−σ, 1/ϕ(b.c.)+.

(iii) Compact dense embeddings

Hσ+ε/Mσ+ε → Hσ,ϕ/Mσ,ϕ → Hσ−ε/Mσ−ε

take place for any ε > 0;

(iv) If the number σ satisfies the condition

−σ 6= m+j + 1/2 for all j ∈ 1, . . . , q, (3.101)

then there exists a number % = %(σ) > 0 (which does not depend on ϕ)such that, for any ε ∈ (0, %), the following equality of spaces is valid (up toequivalence of norms):

[Hσ−ε/Mσ−ε, Hσ+ε/Mσ+ε ]ψ = Hσ,ϕ/Mσ,ϕ.


Here, ψ does not depend on σ and represents the interpolation parameterfrom Theorem 3.10 taken for ε = δ.

Proof. (i) Assume that σ > −1/2 and h ∈ Mσ,ϕ. Due to Theorem 3.9(iii),the functional (h, · )Ω is bounded on the space H−σ, 1/ϕ. By assumption, it isequal to zero on the set C∞(b.c.)+, which is dense in this space by Theo-rem 3.9(ii). Hence, this functional is equal to zero as an element of the dualspace (H−σ, 1/ϕ)′. Due to Theorem 3.9(iii), this implies the equality h = 0.Assertion (i) is proved.

(ii) Since the set C∞(b.c.)+ is dense in the space H−σ, 1/ϕ(b.c.)+, we have

Mσ,ϕ =h ∈ Hσ,ϕ : (h,w)Ω = 0 for all w ∈ H−σ, 1/ϕ(b.c.)+

and, hence,

H−σ, 1/ϕ(b.c.)+ =w ∈ H−σ, 1/ϕ : (h,w)Ω = 0 for all h ∈Mσ,ϕ

.

Now we can see that assertion (ii) is a consequence of Theorem 3.9(iii) and thewell-known theorem on duals of a subspace and a factor space of a given space(see, e.g., [107, Chap. 1, § 4, Sec. 5]).

(iii) By Theorem 3.9(iv), the following dense compact embeddings take place:

Hσ+ε → Hσ,ϕ → Hσ−ε for any ε > 0.

This implies that mappings

u+ h : h ∈Mσ+ε 7→ u+ h : h ∈Mσ,ϕ 7→ u+ h : h ∈Mσ−ε,

where u ∈ Hσ+ε for the first mapping and u ∈ Hσ,ϕ for the second one, definedense compact embeddings indicated in assertion (iii).

(iv) Assume that the number s = −σ satisfies condition (3.101). Considerthe case s > 0 first. Let us reuse the proof of Theorem 3.17 where, insteadof problem (3.82), (3.83), we consider formally adjoint problem (3.84), (3.85).For our case, in this proof we have to replace mj , Bj , (b.c.), and P by m+

j , B+j ,

(b.c.)+, and P+ respectively. Due to the above-proved result, P+ is a projectorof every space Hs∓ε onto the subspace Hs∓ε(b.c.)+. Here we have ε ∈ (0, %)for a sufficiently small positive number %, and s ∓ ε > 0. Let Π+ be the op-erator adjoint to P+ relative to the sesquilinear form ( · , · )Ω. By virtue ofTheorem 3.9(iii) and proved assertion (ii) of the present theorem, we have thelinear bounded operator

Π+ : Hσ±ε/Mσ±ε → Hσ±ε. (3.102)

It has the following property:

u ∈ Hσ±ε ⇒ u−Π+[u] ∈Mσ±ε. (3.103)


Here, [u] = u + h : h ∈ Mσ±ε is the coset of u. The property is implied bythe fact that for all u ∈ Hσ±ε and w ∈ C∞(b.c.)+ we have

(Π+[u], w)Ω = ( [u], P+w)Ω = ( [u], w)Ω = (u,w)Ω,

i.e., (u−Π+[u], w)Ω = 0.Properties (3.102) and (3.103) imply that the mapping

u 7→ u−Π+[u], u ∈ Hσ±ε,

is a projector of the space Hσ±ε = Hσ±εΩ

(Rn) onto the subspace Mσ±ε (theequality follows from the condition σ ± ε = −(s∓ ε) < 0). This allows us, dueto Theorems 1.6 (the interpolation of factor spaces) and 3.10, to write downthe following equalities for spaces up to equivalence of norms:[

Hσ−ε/Mσ−ε , Hσ+ε/Mσ+ε

]ψ

=[Hσ−ε

Ω(Rn)/Mσ−ε , H

σ+εΩ

(Rn)/Mσ+ε

]ψ

=[Hσ−ε

Ω(Rn), Hσ+ε

Ω(Rn)

]ψ

/([Hσ−ε

Ω(Rn), Hσ+ε

Ω(Rn) ]ψ ∩ Mσ−ε

)= Hσ,ϕ

Ω(Rn)

/(Hσ,ϕ

Ω(Rn) ∩ Mσ−ε

)= Hσ,ϕ

/(Hσ,ϕ ∩ Mσ−ε

)= Hσ,ϕ /Mσ,ϕ.

Here, ψ is the interpolation parameter from Theorem 3.10 where we set ε = δ.Thus, assertion (iv) is proved for s = −σ > 0.

In the opposite case where σ ≥ 0, the proof is trivial due to already provedassertion (i). Indeed, if we set % := 1/2, then by virtue of Theorems 3.9(i) and3.10, for any ε ∈ (0, 1/2) we can write down the following equalities:

[Hσ−ε/Mσ−ε , Hσ+ε/Mσ+ε ]ψ = [Hσ−ε, Hσ+ε ]ψ

= [Hσ−ε(Ω), Hσ+ε(Ω) ]ψ

= Hσ,ϕ(Ω) = Hσ,ϕ

= Hσ,ϕ /Mσ,ϕ.

As above, the equalities for spaces are valid up to equivalence of norms. Asser-tion (iv) is proved.



As it was already mentioned, we identify the factor space Hσ,ϕ/Mσ,ϕ and thedual space (H−σ, 1/ϕ(b.c.)+)′ in the sense of Theorem 3.18(ii). In this relation,for each distribution u ∈ Hσ,ϕ, its coset [u] = u+ h : h ∈ Mσ,ϕ is identifiedwith the antilinear bounded functional (u, · )Ω on the space H−σ, 1/ϕ(b.c.)+.The relevant norms of the coset and the functional are equal for s 6= 0 andequivalent for s = 0. For the sake of brevity, we represent this identification ofspaces (in a somewhat conditional manner) in the form of their equality,

Hσ,ϕ/Mσ,ϕ =(H−σ, 1/ϕ(b.c.)+

)′, (3.104)

for any σ ∈ R and ϕ ∈M.

3.4.3 Proofs of theorems on isomorphisms andthe Fredholm property

Let us prove Theorems 3.15 and 3.16 formulated in Subsection 3.4.1.

Proof of Theorem 3.15. For ϕ ≡ 1 (Sobolev spaces), this theorem wasproved in the monograph by Ya. A. Roitberg [209, Theorem 5.5.2]. The casewhere ϕ ∈ M is arbitrary can be deduced from that result by using the inter-polation with function parameter.

Due to condition (3.89), we have

s 6= mj + 1/2, 2q − s 6= m+j + 1/2 for each j ∈ 1, . . . , q. (3.105)

Hence, by Theorems 3.17(i) and 3.18(iv) (we set σ = s − 2q in the latter one)there exists a sufficiently small number ε > 0 such that

[Hs−ε(b.c.), Hs+ε(b.c.) ]ψ = Hs,ϕ(b.c.), (3.106)

[Hs−2q−ε/Ms−2q−ε , Hs−2q+ε/Ms−2q+ε ]ψ = Hs−2q, ϕ/Ms−2q, ϕ, (3.107)

where s ∓ ε 6= j + 1/2 for each j ∈ 0, 1, . . . , 2q − 1, and ψ is a certaininterpolation parameter. We now refer to Theorem 5.5.2 from the monographby Ya. A. Roitberg [209, p. 168]. In accordance with that theorem, the linearmapping u 7→ Lu for u ∈ C∞(b.c.) can be continuously extended to boundedoperators

L : Hs∓ε(b.c.)→ Hs∓ε−2q/Ms∓ε−2q

and related isomorphisms

L : P (Hs∓ε(b.c.) )↔ P+(Hs∓ε−2q)/Ms∓ε−2q.

Applying the interpolation with the parameter ψ to these extensions, we obtainthe bounded operator

L :[Hs−ε(b.c.), Hs+ε(b.c.)

]ψ

→[Hs−ε−2q/Ms−ε−2q , H

s+ε−2q/Ms+ε−2q]ψ

(3.108)


and the isomorphism

L :[P (Hs−ε(b.c.)), P (Hs+ε(b.c.))

]ψ

↔[P+(Hs−ε−2q)/Ms−ε−2q , P

+(Hs+ε−2q)/Ms+ε−2q]ψ. (3.109)

(The pairs of spaces in (3.109) are admissible; this is implied by Theorem 1.6,see the reasoning below.) Due to interpolation formulas (3.106) and (3.107)(and also equality (3.104)), operator (3.108) turns into the bounded operator(3.90) from the theorem under consideration. It remains to show that (3.109)is equal to isomorphism (3.91).

First we prove that the domain of isomorphism (3.109) coincides with thesubspace P (Hs,ϕ(b.c.)). By Lemma 3.2 the mapping P is a projector of thespace Hs∓ε(b.c.) onto the subspace

P (Hs∓ε(b.c.)) = u ∈ Hs∓ε(b.c.) : (u,w)Ω = 0 for all w ∈ N.

Due to Theorem 1.6 (the interpolation of subspaces) and formula (3.106), thisallows us to write down the following equalities for spaces up to equivalence ofnorms:

[P (Hs−ε(b.c.)), P (Hs+ε(b.c.)) ]ψ

= [Hs−ε(b.c.), Hs+ε(b.c.) ]ψ ∩ P (Hs−ε(b.c.))

= Hs,ϕ(b.c.) ∩ P (Hs−ε(b.c.))

= P (Hs,ϕ(b.c.)).

Thus,[P (Hs−ε(b.c.)), P (Hs+ε(b.c.)) ]ψ = P (Hs,ϕ(b.c.)). (3.110)

Let us show that the range of isomorphism (3.109) is equal to the space

P+(Hs−2q, ϕ)/Ms−2q, ϕ.

By Lemma 3.2 the mapping P+ is a projector of the space Hs∓ε−2q onto thesubspace

P+(Hs∓ε−2q) = f ∈ Hs∓ε−2q : (f, w)Ω = 0 for all w ∈ N+.

Consider the linear mapping

[f ] = f + h : h ∈Ms∓ε−2q

7→ [P+f ] = P+f + h : h ∈Ms∓ε−2q, (3.111)

where f ∈ Hs∓ε−2q. This mapping is well defined. Indeed, since Ms∓ε−2q ⊂P+(Hs∓ε−2q), we have P+h = h for any h ∈ Ms∓ε−2q. Hence, the coset [P+f ]


does not depend on the choice of a representative f from the coset [f ]. Prop-erties of the mapping P+ imply that mapping (3.111) is a projector of thespaceHs∓ε−2q/Ms∓ε−2q onto the subspace P+(Hs∓ε−2q)/Ms∓ε−2q. Due to The-orem 1.6 (the interpolation of subspaces) and formula (3.107), this allows usto write down the following equalities for spaces up to equivalence of theirnorms:

[P+(Hs−ε−2q)/Ms−ε−2q , P+(Hs+ε−2q)/Ms+ε−2q ]ψ

= [Hs−ε−2q/Ms−ε−2q , Hs+ε−2q/Ms+ε−2q ]ψ

∩ (P+(Hs−ε−2q)/Ms−ε−2q )

= (Hs−2q, ϕ/Ms−2q, ϕ) ∩ (P+(Hs−ε−2q)/Ms−ε−2q )

= (Hs−2q, ϕ ∩ P+(Hs−ε−2q) )/Ms−2q, ϕ

= P+(Hs−2q, ϕ)/Ms−2q, ϕ.

Thus, [P+(Hs−ε−2q)/Ms−ε−2q , P

+(Hs+ε−2q)/Ms+ε−2q]ψ

= P+(Hs−2q, ϕ)/Ms−2q, ϕ. (3.112)

Interpolation formulas (3.110) and (3.112) imply that mapping (3.109) gener-ates isomorphism (3.91).


Proof of Theorem 3.16. Recall that spaces N and N+ are finite-dimen-sional. First we show thatN is the kernel of operator (3.90). SinceN⊂C∞(b.c.),the image of any element u ∈ N under mapping (3.90) is the coset

Lu+ h : h ∈Ms−2q, ϕ = 0 + h : h ∈Ms−2q, ϕ,

i.e., it is zero element of the factor spaceHs−2q, ϕ/Ms−2q, ϕ. The inverse assertionis also true:

If an element u ∈ Hs,ϕ satisfies the condition Lu = 0, then due to theexpansion u = u0 + Pu stated by Lemma 3.2, where u0 ∈ N, we obtain theequality

0 = Lu = Lu0 + LPu = LPu.

Due to isomorphism (3.91), this yields Pu = 0, i.e., u = u0 ∈ N. Thus, N isthe finite-dimensional kernel of operator (3.90).


Isomorphism (3.91) also implies that the range of operator (3.90) coincideswith the factor space P+(Hs−2q, ϕ)/Ms−2q, ϕ, i.e., with (3.92). Therefore,the range is closed in the space Hs−2q, ϕ/Ms−2q, ϕ. The codimension of therange is equal to the dimension of the factor space

(Hs−2q, ϕ/Ms−2q, ϕ )/(P+(Hs−2q, ϕ)/Ms−2q, ϕ ) = Hs−2q, ϕ/P+(Hs−2q, ϕ).

By virtue of Lemma 3.2, this dimension coincides with dimN+, and therefore,it is finite. Thus, operator (3.90) is a Fredholm operator, and its index is equalto dimN − dimN+.

It remains to prove a priori estimate (3.93). Consider an arbitrary distri-bution u ∈ Hs,ϕ(b.c.) and fix a number ε > 0. By Lemma 3.2 the inclusionu − Pu ∈ N takes place. At the same time, N is a finite-dimensional sub-space of each of the spaces Hs,ϕ and Hs−ε. Hence, norms in these spaces areequivalent on N. In particular,

‖u− Pu‖Hs,ϕ ≤ c1 ‖u− Pu‖Hs−ε ,

where c1 > 0 is a constant that does not depend on u. This implies that

‖u‖Hs,ϕ ≤ ‖u− Pu‖Hs,ϕ + ‖Pu‖Hs,ϕ

≤ c1 ‖u− Pu‖Hs−ε + ‖Pu‖Hs,ϕ

≤ c1 ‖u‖Hs−ε + c1 ‖Pu‖Hs−ε + ‖Pu‖Hs,ϕ

≤ c1 ‖u‖Hs−ε + (c1c2 + 1) ‖Pu‖Hs,ϕ ,

where c2 is the norm of the embedding operator Hs,ϕ → Hs−ε (see Theo-rem 3.9(iv)). Thus,

‖u‖Hs,ϕ ≤ c1 ‖u‖Hs−ε + (c1c2 + 1) ‖Pu‖Hs,ϕ . (3.113)

Now, let Lu = [f ]. Since it was proved above that N is the kernel of opera-tor (3.90) and u − Pu ∈ N, we have LPu = [f ]. Thus, Pu is the preimage ofthe coset [f ] under isomorphism (3.91). Hence,

‖Pu‖Hs,ϕ ≤ c0 ‖f‖Hs−2q,ϕ ,

where c0 is the norm of the operator inverse to (3.91). This result and inequality(3.113) yield a priori estimate (3.93) immediately.



3.4.4 Local increase in smoothness of solutionsup to the boundary

Let H−∞ denote the union of all spaces Hs,ϕ, where s ∈ R and ϕ ∈M. We set

M−∞ := h ∈ H−∞ : (h,w)Ω = 0 for all w ∈ C(b.c.)+.

For s ∈ R and ϕ ∈M, operators (3.90) define a linear mapping

L : H−∞ → H−∞/M−∞

and an isomorphism

L : P (H−∞)↔ P+(H−∞)/M−∞. (3.114)

Consider the following problem:

Suppose that a distribution u ∈ H−∞ satisfies the equation

Lu = f + h : h ∈M−∞, (3.115)

where f has a given smoothness in the refined Sobolev scale on some set U ,which is open in Ω. Then what can be asserted on the smoothness of solutionson this set?

We now answer this question.Let U be an open set in the space Rn, and let Ω0 := Ω ∩ U 6= ∅. We set

Γ0 := Γ ∩ U (the case where Γ0 = ∅ is also possible). Let us introduce thefollowing spaces, which are characterized by a refined smoothness on Ω0.

Let s ∈ R and ϕ ∈M. We define

Hs,ϕ,(0)loc (Ω0) := u ∈ H−∞ : χu ∈ Hs,ϕ for all χ ∈ C∞(Ω), suppχ ⊂ Ω0.

Note that the operator of multiplication by a function χ ∈ C∞(Ω) is boundedon the space Hs,ϕ. This fact is well known in the Sobolev case where ϕ ≡ 1(see, e.g., [209, Sec 1.12, p. 57]); due to interpolation theorem 3.10, this is alsotrue for any ϕ ∈M.

In view of Theorem 3.17(ii), we set

Hs,ϕloc (Ω0, b.c.,Γ0) := u ∈ Hs,ϕ,(0)

loc (Ω0) : Bj u = 0 on Γ0

for each j ∈ 1, . . . , q such that s > mj + 1/2,

where we assume that s 6= mj + 1/2 for any j ∈ 1, . . . , q.

Theorem 3.19. Let u ∈ H−∞ be a solution of equation (3.115), where f ∈Hs−2q,ϕ,(0)loc (Ω0) for some s ∈ R satisfying (3.89), and some ϕ ∈ M. Then

u ∈ Hs,ϕloc (Ω0, b.c.,Γ0).


Theorem 3.19 is a statement on a local increase in the refined smoothness ofthe solution u to the boundary-value problem (3.82), (3.83) up to the boundaryof the domain Ω. Note that in the case where Ω ⊂ U, i.e., in the case whereΩ0 = Ω and Γ0 = Γ, the “local” spaces coincide with “global” ones:

Hs,ϕ,(0)loc (Ω0) = Hs,ϕ and Hs,ϕ

loc (Ω0, b.c.,Γ0) = Hs,ϕ(b.c.).

Therefore, Theorem 3.19 also states the global increase in smoothness, i.e.,it states that the smoothness increases globally in the whole closed domain Ω.At last, in addition, we emphasize the case where U ⊂ Ω, i.e., Γ0 = ∅, thatleads to a statement on increase in smoothness inside the domain Ω.

Remark 3.10. For the Sobolev case ϕ ≡ 1, Theorem 3.19 was proved byYu. M. Berezansky, S. G. Krein, and Ya. A. Roitberg (see papers [22, 201] andthe monograph [209, Theorem 7.3.1]).

Proof of Theorem 3.19. As it was mentioned in Remark 3.9, this theoremis known for the case where ϕ ≡ 1. By using this known result, we prove it forany ϕ ∈M with the help of Theorem 3.15 on isomorphism.

For any number ε > 0, we set

Uε := x ∈ U : dist(x, ∂U) > ε, Ωε := Ω ∩ Uε, Γε := Γ ∩ Uε.

Here, as usual, ∂U is the boundary of the set U. Then we consider a functionχε ∈ C∞(Ω) such that suppχε ⊂ Ω0 and χε ≡ 1 on Ωε.

Let us represent the distribution f in the form f = χεf + (1 − χε)f. Sincef ∈ Hs−2q,ϕ,(0)

loc (Ω0) and suppχε ⊂ Ω0, we have χεf ∈ Hs−2q,ϕ. Hence, due toisomorphism (3.91) from Theorem 3.15, we conclude that there exists a distri-bution uε ∈ P (Hs,ϕ(b.c.)) such that

Luε = [P+χεf ]. (3.116)

Since 1 − χε = 0 on the set Ωε, we have χ(1 − χε)f ≡ 0 for any function χ ∈C∞(Ω) with suppχ ⊂ Ωε. In particular, this yields (1−χε)f ∈ Hs+1−2q,(0)

loc (Ωε).By Lemma 3.2 we write

(1− χε)f = gε + P+(1− χε)f,

where gε ∈ N+ ⊂ C∞(b.c.). Hence,

P+(1− χε)f ∈ Hs+1−2q,(0)loc (Ωε).

In addition, due to (3.114), there exists vε ∈ P (H−∞) such that

Lvε = [P+(1− χε)f ]. (3.117)


Hence, by [209, Theorem 7.3.1], the inclusion vε ∈ Hs+1loc (Ωε, b.c.,Γε) holds.

Due to (3.116), (3.117), and the condition uε ∈ Hs,ϕ(b.c.), this implies theequality

L(uε + vε) = [P+χεf ] + [P+(1− χε)f ] = [P+f ],

where

uε + vε ∈ Hs,ϕ(b.c.) ∪Hs+1loc (Ωε, b.c.,Γε) ⊂ Hs,ϕ

loc (Ωε, b.c.,Γε).

To prove this, we used the embedding Hs+1 → Hs,ϕ and the definition of thespace Hs,ϕ

loc (Ωε, b.c.,Γε).At the same time, Lu = [f ]. Hence, due to (3.114), the relation f = P+f

holds. Therefore,L(u− uε − vε) = [f ]− [P+f ] = 0,

where uε + vε ∈ Hs,ϕloc (Ωε, b.c.,Γε). Due to Theorem 3.15, this implies

v := u− uε − vε ∈ N ⊂ C∞(b.c.).

Thus,u = uε + vε + v ∈ Hs,ϕ

loc (Ωε, b.c.,Γε).

Recall that the number ε > 0 in the proof is taken arbitraryly. It is clear that

Hs,ϕloc (Ω0, b.c.,Γ0) =

⋂ε>0

Hs,ϕloc (Ωε, b.c.,Γε).

Hence, u ∈ Hs,ϕloc (Ω0, b.c.,Γ0).


3.5 Some properties of Hörmander spaces

In the present section, we study some subspaces of the space Hs,ϕ(Ω) and givean equivalent description of this space. These results are used, in particular, inSection 4.5.

3.5.1 Space Hs,ϕ0 (Ω) and its properties

Let s ∈ R and ϕ ∈ M. By Hs,ϕ0 (Ω) we denote the closure of the set C∞0 (Ω)

in the space Hs,ϕ(Ω). We treat Hs,ϕ0 (Ω) as a Hilbert space with respect to

the inner product in Hs,ϕ(Ω). In this subsection, we study the properties ofthe space Hs,ϕ

0 (Ω) that lead to various equivalent descriptions of the spaceHs,ϕ(Ω) for negative nonhalf-integer indices s. These important results are usedin what follows. We delayed the representation of the proof for these resultsto the present subsection because our proof is based on a result obtained inSubsection 3.4.2.

Section 3.5 Some properties of Hörmander spaces 159

For k ∈ N, we set

C∞ν,k(Ω ) := u ∈ C∞(Ω ) : Dj−1ν u = 0 on Γ, j = 1, . . . , k. (3.118)

Here and in what follows, for the sake of convenience we use the notationDν := i ∂/∂ν. Let Hs,ϕ

ν,k (Ω) denote the closure of the set C∞ν,k(Ω ) in the spaceHs,ϕ(Ω).We treat Hs,ϕ

ν,k (Ω) as a Hilbert space with respect to the inner productin Hs,ϕ(Ω).

If s ≥ 0, then Hs,ϕν,k (Ω) is equal to the space Hs,ϕ(b.c.) introduced in Subsec-

tion 3.4.1, where (b.c.) stands for the Dirichlet homogeneous boundary condi-tions represented in (3.118). Therefore, by Theorem 3.17(ii),

Hs,ϕν,k (Ω) = u ∈ Hs,ϕ(Ω) : Dj−1

ν u = 0 on Γ

for all j = 1, . . . , k if s > k − 1/2. (3.119)

By virtue of Theorem 3.3(ii), we have

Hs,ϕ0 (Ω) = Hs,ϕ(Ω) for s < 1/2. (3.120)

If s > 1/2, then the inclusion Hs,ϕ0 (Ω) ⊂ Hs,ϕ(Ω) is proper. This is implied by

Theorem 3.5 on traces.

Theorem 3.20. Let s > 1/2 and s − 1/2 /∈ Z, and let ϕ ∈ M. Then thefollowing assertions hold:

(i) Hs,ϕ0 (Ω) coincides with space (3.119), where k := [s+ 1/2].

(ii) The norms in spaces Hs,ϕ0 (Ω) and Hs,ϕ

Ω(Rn) are equivalent on the dense

subset C∞0 (Ω), and therefore, Hs,ϕ0 (Ω) = Hs,ϕ

Ω(Rn) up to equivalence of

norms.

(iii) The spaces Hs,ϕ0 (Ω) and H−s,1/ϕ(Ω) are mutually dual (up to equivalence

of norms) with respect to the inner product in L2(Ω).

Proof. In the Sobolev case where ϕ ≡ 1, this theorem is well known.Its proof is represented, e.g, in the monograph by H. Triebel [258, Chap. 4].Namely, assertions (i), (ii), and (iii) are contained, respectively, in Theorems4.7.1(a), 4.3.2/1(c), and 4.8.2(a) of this monograph. From results for this case,we deduce assertions (i) – (iii) for arbitrary ϕ ∈M.

(i) Let k := [s+1/2]. Then k− 1/2 < s < k+1/2. Pick a number ε > 0 suchthat

k − 1/2 < s∓ ε < k + 1/2. (3.121)

Then we have the following dense continuous embedding:

Hs+ε0 (Ω) = Hs+ε

ν,k (Ω) → Hs,ϕν,k (Ω).


As it is indicated above, this equality for Sobolev spaces is true since [s+ ε+1/2] = k. Therefore, the set C∞0 (Ω) is dense in Hs,ϕ

ν,k (Ω). Hence, Hs,ϕ0 (Ω) =

Hs,ϕν,k (Ω), and assertion (i) is proved.

(ii) We deduce assertion (ii) from the Sobolev case (ϕ ≡ 1) by using theinterpolation. Let numbers k and ε be like those that considered in the previousparagraph. By O we denote the operator of extension by zero over the spaceRn for functions from the domain Ω. The mapping u 7→ Ou, where u ∈ C∞0 (Ω),can be extended by continuity to isomorphisms

O : Hs∓ε0 (Ω)↔ Hs∓ε

Ω(Rn).

Let us apply the interpolation with the functional parameter ψ from Theo-rem 1.14 taken for δ = ε. Due to Theorem 3.7, additionally we can obtain theisomorphism

O : [Hs−ε0 (Ω), Hs+ε

0 (Ω)]ψ↔ Hs,ϕ

Ω(Rn). (3.122)

Let us describe the domain of operator (3.122). Due to inequality (3.121) andassertion (i), we have

Hs∓ε0 (Ω) = Hs∓ε

ν,k (Ω) and Hs,ϕν,k (Ω) = Hs,ϕ

0 (Ω).

Hence, on the base of Theorem 3.17(i), we obtain

[Hs−ε0 (Ω), Hs+ε

0 (Ω)]ψ = [Hs−εν,k (Ω), Hs+ε

ν,k (Ω)]ψ

= Hs,ϕν,k (Ω)

= Hs,ϕ0 (Ω).

Here, the second equality for spaces is satisfied up to equivalence of norms.Hence, (3.122) means the isomorphism

O : Hs,ϕ0 (Ω)↔ Hs,ϕ

Ω(Rn).

Thus, assertion (ii) is proved.

Assertion (iii) is immediately implied by Theorem 3.8(iii) and assertion (ii).Theorem 3.20 is proved.

3.5.2 Equivalent description of Hs,ϕ(Ω)

By Theorem 3.20(iii) the space Hs,ϕ(Ω) with negative nonhalf-integer indices scan be defined as a space, which is dual to H−s,1/ϕ0 (Ω) with respect to the innerproduct in L2(Ω). In the Sobolev case, such a definition was used by J.-L. Lions

Section 3.5 Some properties of Hörmander spaces 161

and E. Magenes [119, 120, 121, 126] in their study of boundary-value problems(see Remark 3.6).

This yields a distinct (equivalent) description of the space Hs,ϕ(Ω). Let usrecall that by definition we have

Hs,ϕ(Ω) = w Ω : w ∈ Hs,ϕ(Rn).

It turns out that here we can restrict the study to the set of distributions wwith supports in Ω.

Theorem 3.21. Let s < 1/2 and s− 1/2 /∈ Z, and let ϕ ∈M. Then

Hs,ϕ(Ω) = Hs,ϕ

Ω(Rn)/Hs,ϕ

Γ(Rn) =

w Ω : w ∈ Hs,ϕ

Ω(Rn)

, (3.123)

‖u‖Hs,ϕ(Ω) inf‖w‖Hs,ϕ(Rn) : w ∈ Hs,ϕ

Ω(Rn), w = u in Ω

. (3.124)

Proof. By virtue of Theorem 3.20(iii), for each nonhalf-integer s < −1/2 wehave the equality

Hs,ϕ(Ω) = (H−s,1/ϕ0 (Ω))′ with equivalence of norms. (3.125)

Consider the duality of spaces with respect to the inner product in L2(Ω).If −1/2 < s < 1/2, then (3.125) is also true due to Theorem 3.8(ii), (iii) andequality (3.120). Namely,

Hs,ϕ(Ω) =(H−s,1/ϕΩ

(Rn))′

=(H−s,1/ϕ(Ω)

)′=(H−s,1/ϕ0 (Ω)

)′.

Let us describe (H−s,1/ϕ0 (Ω))′ as a space, which is dual to the subspace

H−s,1/ϕ0 (Ω) of the space H−s,1/ϕ(Ω). In view of Theorem 3.8(iii), we have

(H−s,1/ϕ0 (Ω))′ = (H−s,1/ϕ(Ω))′/Gs,ϕ

= Hs,ϕ

Ω(Rn)/Gs,ϕ.

Here,

Gs,ϕ :=w ∈ Hs,ϕ

Ω(Rn) : (w, v)Ω = 0 for all v ∈ H−s,1/ϕ0 (Ω)

=w ∈ Hs,ϕ

Ω(Rn) : (w, v)Ω = 0 for all v ∈ C∞0 (Ω)

= Hs,ϕ

Γ(Rn).


Thus, by virtue of (3.125),

Hs,ϕ(Ω) = Hs,ϕ

Ω(Rn)/Hs,ϕ

Γ(Rn)

up to equivalence of norms. This implies relations (3.123) and (3.124) immedi-ately.



Section 3.1. The notion of general elliptic boundary-value problem was firstformulated by Ya.B. Lopatinskii [123, 124]. In special cases, Z.Ya. Shapiro [230],N. Aronszajn and A. N. Milgram [16], and independently, M. Schechter [222]introduced the important condition of normality for a system of boundary ex-pressions. This condition ensures the existence of a formally adjoint boundary-value problem in the class of differential operators.

Systematic presentations of the theory of general elliptic boundary-valueproblems can be found, e.g., in the monographs by S.Agmon [3], Yu.M. Berezan-sky [21], L. Hörmander [81, 86], J.-L. Lions and E. Magenes [121], O. I. Panich[184], Ya. A. Roitberg [209], M. Schechter [227], H. Triebel [258], and in thesurvey by M. S. Agranovich [11].

Section 3.2. For open or closed Euclidean domains, the Hörmander innerproduct spaces were introduced and studied by L. R. Volevich and B. P. Paneah[269]. They used standard definitions from the theory of function spaces (see,e.g., the monograph by H. Triebel [258, Sec. 4.2.1 and 4.3.2]).

The refined scales over Euclidean domains were introduced in [143]. UnlikeL. R. Volevich and B. P. Paneah, we established the properties of Hörmanderspaces over the domains by using interpolation formulas that relate these spaceswith the Sobolev scale. Theorems 3.1–3.5 on the properties of the refined scaleover Euclidean domains were proved in [145, Sec. 3] in a more general situationof the refined scale over a manifold with boundary. Theorems 3.6–3.8 on theproperties of the refined scale over closed Euclidean domains were establishedin [149, Sec. 4]. Theorem 3.9 was also proved in the same work. The proof ofthe interpolation theorem 3.10 was not published earlier. The last two theoremsdescribe the properties of the rigging of the space L2(Ω) with Hörmander spaces.

The definition of Hilbert rigging and related notions can be found, e.g., in themonographs by Yu. M. Berezansky [21, Chap. 1, Sec. 1.1] and Yu. M. Berezan-sky, G. F. Us, and Z. G. Sheftel [23, Chap. 14, Sec. 1.1]. The rigging withSobolev spaces was introduced and studied by Yu. M. Berezansky (see [21,Chap. 1, Sec. 3] and [23, Chap. 14, Sec. 3 and 4]).

In [231, Sec. 2], G. Shlenzak proved an interpolation formula connecting theSobolev scale with some Hörmander spaces given over a Euclidean domain with


smooth boundary. They were applied by G. Shlenzak to the theory of generalelliptic boundary-value problems.

Section 3.3. All theorems of this section were proved by the authors in [148].The main result (Theorem 3.11) is new even in the Sobolev case for half integers < 0. For the remaining values of the parameter s, it is contained in theLions–Magenes theorem [121, Sec. 7.3] on solvability of regular inhomogeneouselliptic boundary-value problems in the two-sided scale of Sobolev spaces (seealso [119, 120]).

Section 3.4. The theorem on isomorphisms generated by an elliptic operatorin the two-sided Sobolev scale under homogeneous normal boundary conditionswas proved by Yu. M. Berezansky, S. G. Krein, and Ya. A. Roitberg [22, Theo-rem 2]. In the same paper, they also established the theorem on local increase insmoothness of solutions of elliptic equations up to the boundary of the domain.Independently, M. Schechter [225] established an appropriate a priori estimatefor the solutions of elliptic equations. The proofs of these results can be foundin the monographs by Yu. M. Berezansky [21, Chap. 3, Sec. 6.6 and 6.12] andYa. A. Roitberg [209, Sec. 5.5 and 7.3]. See also the survey by M. S. Agranovich[11, Sec. 7.9 c] and the book by Yu. M. Berezansky, G. F. Us, and Z. G. Sheftel[23, Chap. 16, Sec. 1.1 and 2.3] (in the last book, the authors consider the caseof second-order elliptic expressions with real coefficients).

A generalization of these results to the case of boundary conditions that arenot normal was obtained by Yu. V. Kostarchuk and Ya. A. Roitberg [97, Sec. 4].The case of elliptic systems was studied by I. Ya. Roitberg and Ya. A. Roitberg[200, Sec. 3.7]; see also the monograph by Ya. A. Roitberg [210, Sec. 1.3.7].

In connection with Subsection 3.4.2, we note that the interpolation with num-ber parameters between the Sobolev spaces satisfying homogeneous boundaryconditions was studied by P. Grisvard [66], R. T. Seeley [234], and J. Löfström[125] (the results of the first two authors can also be found in Triebel’s mono-graph [258, Sec. 4.3.3]).

All theorems in Section 3.4 (except Theorem 3.19) were proved in [149].Theorem 3.19 was announced, together with Theorems 3.15 and 3.16, in [143].

Section 3.5. The results obtained in this section give other equivalent defini-tions of the Hörmander spacesHs,ϕ(Ω) over Euclidean domains for any nonhalf-integer s < 0. In the Sobolev case, these definitions were used by J.-L. Lionsand E. Magenes [119, 120, 121, 126] in their investigation of boundary-valueproblems. The theorems presented in this section were not published earlier.

Chapter 4

Inhomogeneous elliptic boundary-valueproblems

4.1 Elliptic boundary-value problemsin the positive one-sided scale

In this section, we investigate inhomogeneous elliptic boundary-value prob-lems in Hörmander spaces that form the positive one-sided refined Sobolevscale. For these spaces, the number index defining the main smoothness ispositive. In Subsections 4.1.1 and 4.1.2, a regular elliptic boundary-value prob-lem is investigated. In the rest subsections, other important classes of ellipticboundary-value problems are considered.

4.1.1 Theorems on Fredholm property and isomorphisms

Consider the inhomogeneous regular elliptic boundary-value problem (3.1),(3.2), namely

Lu = f in Ω, Bj u = gj on Γ for j = 1, . . . , q. (4.1)

With it, we associate the linear mapping

u 7→ (Lu,Bu) := (Lu,B1u, . . . , Bqu), u ∈ C∞(Ω ). (4.2)

We study properties of the operator (L,B), which is an extension of this map-ping by continuity in the corresponding pairs of positive Hörmander spaces.Recall that the finite-dimensional spaces N,N+ ⊂ C∞(Ω ) are defined in Sec-tion 3.1

Theorem 4.1. For arbitrary parameters s > 2q and ϕ ∈ M, mapping (4.2)extends uniquely (by continuity) to the bounded linear operator

(L,B) : Hs,ϕ(Ω)→ Hs−2q,ϕ(Ω)⊕q⊕j=1

Hs−mj−1/2,ϕ(Γ) =: Hs,ϕ(Ω,Γ). (4.3)

This operator is Fredholm. Its kernel is equal to N and the range consists of allvectors (f, g1, . . . , gq) ∈ Hs,ϕ(Ω,Γ) such that

(f, v)Ω +

q∑j=1

(gj , C+j v)Γ = 0 for all v ∈ N+. (4.4)

166 Chapter 4 Inhomogeneous elliptic boundary-value problems


Proof. In the Sobolev case where ϕ ≡ 1 and s ≥ 2q, this theorem is theclassical result on solvability of regular elliptic boundary-value problems. It isproved, e.g., in the monographs by Yu. M. Berezansky [21, Chap. 3, Sec. 6] andJ.-L. Lions and E. Magenes [121, Part. 2, Sec. 5.2]. The general case of ϕ ∈Mis obtained from the Sobolev case with the use of interpolation with a functionparameter.

Namely, let s > 2q and let ε := s−2q.Mapping (4.2) is extended by continuityto the bounded and Fredholm operators

(L,B) : Hs∓ε(Ω)→ Hs∓ε−2q(Ω)⊕q⊕j=1

Hs∓ε−mj−1/2(Γ)

=: Hs∓ε(Ω,Γ). (4.5)

They have the common kernel N, the same index κ := dimN − dimN+, andthe range

(L,B)(Hs∓ε(Ω)) = (f, g1, . . . , gq) ∈ Hs∓ε(Ω,Γ) : (4.4) is true . (4.6)

We apply the interpolation with the function parameter ψ in Theorem 1.14,where ε = δ, to (4.5). We obtain the operator

(L,B) : [Hs−ε(Ω), Hs+ε(Ω)]ψ → [Hs−ε(Ω,Γ),Hs+ε(Ω,Γ)]ψ, (4.7)

which extends mapping (4.2) by continuity. By virtue of the interpolationtheorems 1.5, 2.2, and 3.2, this implies that mapping (4.2) extends by continuityto the bounded operator (4.3) equal to (4.7). By Theorem 1.7, the Fredholmproperty of operators (4.5) yields the Fredholm property of operator (4.3),which inherits their kernel N and index κ = dimN − dimN+. In addition,the range of operator (4.3) is equal to Hs,ϕ(Ω,Γ)∩ (L,B)(Hs−ε(Ω)). By virtueof (4.6), this implies that the range is the same as that in the statement of thetheorem being demonstrated.


By virtue of Theorem 4.1, for an arbitrary function u ∈ Hs,ϕ(Ω), where s >2q and ϕ ∈ M, the right-hand sides f ∈ Hs−2q,ϕ(Ω) and gj ∈ Hs−mj−1/2,ϕ(Γ)of the boundary-value problem (4.1) are determined, as images of u at themapping (4.3).

If N = N+ = 0 (the defect of the boundary-value problem is absent), thenoperator (4.3) is an isomorphism of Hs,ϕ(Ω) onto Hs,ϕ(Ω,Γ). This follows fromTheorem 4.1 and the Banach theorem on inverse operator. In the general case,

Section 4.1 Elliptic boundary-value problems in the positive one-sided scale 167

it is convenient to define the corresponding isomorphism with the use of thefollowing projectors.

Let s > 2q and ϕ ∈ M. We represent spaces in which operator (4.3) acts inthe form of the direct sums of (closed) subspaces

Hs,ϕ(Ω) = N u u ∈ Hs,ϕ(Ω) : (u,w)Ω = 0 for all w ∈ N, (4.8)

Hs,ϕ(Ω,Γ) = (v, 0, . . . , 0) : v ∈ N+u (L,B)(Hs,ϕ(Ω)). (4.9)

These decomposition in direct sums exist. Indeed, equality (4.8) is the re-striction of the decomposition of the space L2(Ω) in the orthogonal sum ofthe subspace N and its complement. Equality (4.9) follows from Theorem 4.1according to which the subspaces on its right-hand side have a trivial intersec-tion, while the (finite) dimension of the first of the subspaces coincides withthe codimension of the second.

Let P and Q+ denote the oblique projectors of the spaces Hs,ϕ(Ω) andHs,ϕ(Ω,Γ), respectively, onto the second summands in sums (4.8) and (4.9)parallel to the first ones. These projectors are independent of s and ϕ.

Theorem 4.2. For arbitrary parameters s > 2q and ϕ ∈M, the restriction ofmapping (4.3) onto the subspace P (Hs,ϕ(Ω)) is the isomorphism

(L,B) : P (Hs,ϕ(Ω))↔ Q+(Hs,ϕ(Ω,Γ)). (4.10)

Proof. By Theorem 4.1, N is the kernel and Q+(Hs,ϕ(Ω,Γ)) is the range ofoperator (4.3). Therefore, the bounded operator (4.10) is a bijection. Thus, byvirtue of the Banach theorem on inverse operator, it is an isomorphism.


The following a priori estimate for the solutions to the elliptic boundary-valueproblem (4.1) results from Theorem 4.2.

Theorem 4.3. Let s > 2q and ϕ ∈M. Suppose that a function u ∈ Hs,ϕ(Ω) isa solution to the boundary-value problem (4.1) with (f, g1, . . . , gq) ∈ Hs,ϕ(Ω,Γ).Then the following estimate is true:

‖u‖Hs,ϕ(Ω) ≤ c(‖(f, g1, . . . , gq)‖Hs,ϕ(Ω,Γ) + ‖u‖L2(Ω)

), (4.11)

where the number c = c(s, ϕ) > 0 is independent of u and (f, g1, . . . , gq).

Proof. We use decomposition (4.8) and rewrite the function u ∈ Hs, ϕ(Ω) inthe form u = u0 + u1, where u0 := (1− P )u ∈ N and u1 := Pu ∈ P (Hs,ϕ(Ω)).By virtue of Theorem 4.2, we have

‖u1‖Hs,ϕ(Ω) ≤ c1‖(L,B)u1‖Hs,ϕ(Ω,Γ)


= c1‖(L,B)u‖Hs,ϕ(Ω,Γ)

= c1‖(f, g1, . . . , gq)‖Hs,ϕ(Ω,Γ).

Here, c1 is the norm of the inverse to (4.10). In addition, since N is finite-dimensional and since 1− P is the orthoprojector of L2(Ω) onto N , we have

‖u0‖Hs,ϕ(Ω) ≤ c0‖u0‖L2(Ω) ≤ c0‖u‖L2(Ω).

Here, the number c0 > 0 is independent of u and (f, g1, . . . , gq). Summing theseinequalities, we obtain (4.11).


If N = 0, i.e., the boundary-value problem (4.1) has at most one solution,then the term ‖u‖L2(Ω) on the right-hand side of the a priori estimate (4.11)can be omitted.

At the end of this subsection, we discuss the connection between inhomoge-neous and semihomogeneous elliptic boundary-value problems. For the sake ofsimplicity, we suppose that N = N+ = 0. Let s > 2q and ϕ ∈ M. It fol-lows from Theorems 3.11 and 3.17(ii) that the space Hs,ϕ(Ω) is the direct sumof the subspaces Ks,ϕ

L (Ω) and Hs,ϕ(b.c.). Therefore, Theorems 3.11 and 3.15(on solvability of inhomogeneous problems) give the isomorphism

(L,B) : Hs,ϕ(Ω)↔ Hs,ϕ(Ω,Γ).

(Note that the antidual space (H2q−s,1/ϕ(b.c.)+)′ coincides with Hs−2q,ϕ(Ω)in view of Theorem 3.18(i) and (ii)). Thus, the inhomogeneous problem (4.1)can be immediately reduced to the semihomogeneous problems provided thats > 2q. The same conclusion is also true in more general case of s > m+ 1/2,we denoting m := maxm1, . . . ,mq.

This reduction fails for s < m + 1/2. Indeed, if 0 ≤ s < m + 1/2, thenthe operator (L,B) cannot be reasonably defined on Ks,ϕ

L (Ω) ∪Hs,ϕ(b.c.) be-cause Ks,ϕ

L (Ω) ∩ Hs,ϕ(b.c.) 6= ∅. This inequality follows from Theorems 3.11and 3.17(ii) if we note that the boundary-value problem (3.36), with gq ≡ 1 andgj ≡ 0 for j < q, has a nonzero solution u ∈ K∞L (Ω) belonging to Hs,ϕ(b.c.).Here we may suppose that mq = m.

So much the more, the above reduction is impossible for negative s. Note,if s < −1/2, then the solutions to the different semihomogeneous problemspertain to the spaces of distributions of different nature. Namely, the solutionsto the problem (3.36) belong to Ks,ϕ

L (Ω) ⊂ Hs,ϕ(Ω) and are distributions givenin the open domain Ω, whereas the solutions to the problem (3.82) and (3.83)belong to Hs,ϕ(b.c.) ⊂ Hs,ϕ

Ω(Rn) and are distributions supported on the closed

domain Ω.The same conclusions remain valid for nontrivial N and/or N+.


4.1.2 Smoothness of the solutions up to the boundary

Assume that the right-hand side of the elliptic boundary-value problem (4.1)has, in a set open in Ω, a certain smoothness on the positive refined Sobolevscale We study the smoothness of the solution u on this set. First, consider thecase of the global smoothness on the whole closed domain Ω.

Theorem 4.4. Suppose that a function u ∈ H2q(Ω) is a solution to the bound-ary-value problem (4.1) in which

f ∈ Hs−2q,ϕ(Ω) and gj ∈ Hs−mj−1/2,ϕ(Γ), j = 1, . . . , q,

for certain parameters s > 2q and ϕ ∈M. Then u ∈ Hs,ϕ(Ω).

Proof. According to Theorem 4.1, which is true in the Sobolev case of s = 2qand ϕ ≡ 1, the vector F := (f, g1, . . . , gq) ∈ Hs,ϕ(Ω,Γ) satisfies condition (4.4).Therefore, by virtue of the same theorem, F ∈ (L,B)(Hs,ϕ(Ω)). Thus, parallelwith condition (L,B)u = F, the equality (L,B)v = F is true for a certainv ∈ Hs,ϕ(Ω). This implies that (L,B)(u − v) = 0, which, by Theorem 4.1,yields the inclusion

w := u− v ∈ N ⊂ C∞(Ω ) ⊂ Hs,ϕ(Ω).

Thus, u = v + w ∈ Hs,ϕ(Ω).Theorem 4.4 is proved.

Now consider the case of local smoothness. Let U be an open set in Rnthat has a nonempty intersection with domain Ω. We set Ω0 := U ∩ Ω andΓ0 := U ∩ Γ (the case where Γ0 = ∅ is possible). For arbitrary parametersσ ∈ R and ϕ ∈ M, we introduce a local analog of the Hörmander space overthe Euclidean domain. Namely, we set

Hσ,ϕloc (Ω0,Γ0) := u ∈ D′(Ω) : χu ∈ Hσ,ϕ(Ω)

for all χ ∈ C∞(Ω ) with suppχ ⊂ Ω0 ∪ Γ0.

Recall that D′(Ω) is the topological linear space of all distributions defined inthe domain Ω. We also need the local space Hσ,ϕ

loc (Γ0) introduced in Sec. 2.2.3.As above, in the case of ϕ ≡ 1, we omit the index ϕ in the notation of thesespaces.

We note the inclusions

Hσ,ϕ(Ω) ⊂ Hσ,ϕloc (Ω0,Γ0) and Hσ,ϕ(Γ) ⊂ Hσ,ϕ

loc (Γ0). (4.12)

They follow from the fact that the multiplication by every function from theclass C∞(Ω ) (respectively, from C∞(Γ)) is a bounded operator on the space


Hσ,ϕ(Ω) (on Hσ,ϕ(Γ)). In the Sobolev case of ϕ ≡ 1, this fact is known [11,Sec. 2.1 c, p. 13]. The general case of an arbitrary ϕ ∈ M follows from theSobolev case by virtue of the interpolation theorems 2.2 and 3.2.

Theorem 4.5. Suppose that a function u ∈ H2q(Ω) is a solution to the bound-ary-value problem (4.1) in which

f ∈ Hs−2q,ϕloc (Ω0,Γ0), (4.13)

gj ∈ Hs−mj−1/2,ϕloc (Γ0), j = 1, . . . , q, (4.14)

for certain parameters s > 2q and ϕ ∈M. Then u ∈ Hs,ϕloc (Ω0,Γ0).

Proof. We follow the scheme of the proof given in [15]. Put

ϒ := χ ∈ C∞(Ω ) : suppχ ⊂ Ω0 ∪ Γ0.

Beforehand, we prove that, by virtue of the condition of this theorem, the fol-lowing implication holds for each r ≥ 0 :(

χu ∈ Hs,ϕ(Ω) +Hr+2q(Ω) for all χ ∈ ϒ)

⇒(χu ∈ Hs,ϕ(Ω) +Hr+1+2q(Ω) for all χ ∈ ϒ). (4.15)

Here and below in this proof, we use algebraic sums of sets.Let us choose r ≥ 0 arbitrarily and assume that the premise of implica-

tion (4.15) is true. Consider an arbitrary function χ ∈ ϒ; let another functionη ∈ ϒ be such that η = 1 in a neighborhood of suppχ. By the condition of thistheorem, we have χF ∈ Hs,ϕ(Ω,Γ), where F := (L,B)u = (f, g1, . . . , gq).

Interchanging the operator of multiplication by the function χ and the dif-ferential operators L and Bj , j = 1, . . . , q, we may write

χF = χ(L,B)(ηu) = (L,B)(χηu)− (L′, B′)(ηu)

and, hence,(L,B)(χu) = χF + (L′, B′)(ηu). (4.16)

Here, L′ is a certain linear differential expression on Ω, and B′ := (B′1, . . . , B′g),

where each B′j is a certain boundary linear differential expression on Γ. Thecoefficients of these expressions are infinitely smooth, while the orders satisfythe conditions ordL′ ≤ 2q − 1 and ordB′j ≤ mj − 1.

By the premise of implication (4.15), we have ηu = u1 + u2 for certainfunctions

u1 ∈ Hs,ϕ(Ω) and u2 ∈ Hr+2q(Ω).


Hence and by (4.16), we may write (L,B)(χu) = F1 + F2, with

F1 := χF + (L′, B′)u1 ∈ Hs,ϕ(Ω,Γ), (4.17)

F2 := (L′, B′)u2 ∈ Hr+2q+1(Ω,Γ). (4.18)

Here, similar to (4.3), we denote

Hσ(Ω,Γ) := Hσ−2q(Ω)⊕q⊕j=1

Hσ−mj−1/2(Γ) for every σ ∈ R.

Let us argue the inclusions appearing in (4.17) and (4.18). Since ordL′ ≤ 2q−1and ordB′j ≤ mj − 1, the mapping v 7→ (L′v,B′v), with v ∈ C∞(Ω ), extendsby continuity to the bounded operator

(L′, B′) : Hσ(Ω)→ Hσ+1(Ω,Γ) for every σ ≥ 2q − 1

(see, e.g., [11, Sec. 2.2, p. 16]). Hence, u2 ∈ Hr+2q(Ω)⇒ (4.18).Moreover, using the interpolation with the function parameter ψ, we obtain

the bounded operator

(L′, B′) : Hs,ϕ(Ω) = [Hs−ε(Ω), Hs+ε(Ω)]ψ

→ [Hs−ε+1(Ω,Γ),Hs+ε+1(Ω,Γ)]ψ = Hs+1,ϕ(Ω,Γ).

Here, ε := s − 2q, while ψ is the same as that in Theorem 1.14, where ε = δ.Therefore, (4.17) is a consequence of the inclusions χF ∈ Hs,ϕ(Ω,Γ) and u1 ∈Hs,ϕ(Ω).

Now, we use the projector Q+ and apply Theorem 4.2 (in the Sobolev caseas well). It follows from the equality (L,B)(χu) = F1+F2 and inclusions (4.17)and (4.18) that

(L,B)(χu) = Q+(L,B)(χu) = Q+F1 +Q+F2 = (L,B)v1 + (L,B)v2.

Here, the functions

v1 ∈ P(Hs,ϕ(Ω)

)and v2 ∈ P

(Hr+2q+1(Ω)

)(4.19)

are the (unique) solutions to the boundary value-problems

(L,B)v1 = Q+F1 ∈ Q+(Hs,ϕ(Ω,Γ)

)and

(L,B)v2 = Q+F2 ∈ Q+(Hr+2q+1(Ω,Γ)

).


Hence, it follows from the equality

(L,B)(χu) = (L,B)(v1 + v2)

thatχu = v1 + (v2 + w) for a certain w ∈ N ⊂ C∞(Ω ).

Therefore, by virtue of (4.19), we arrive at the inference of implication (4.15).This implication is proved for each r ≥ 0.

Now, we may complete the proof in the following way. The premise of im-plication (4.15) is true for r = 0 because of the condition u ∈ H2q(Ω). Usingthis implication in succession for values r = 0, r = 1, . . . , and r = [s− 2q], weconclude that

χu ∈ Hs,ϕ(Ω) +H [s]+1(Ω) = Hs,ϕ(Ω) for all χ ∈ ϒ.

Thus, u ∈ Hs,ϕloc (Ω0,Γ0).


In Theorem 4.5, we note the case Γ0 = ∅, which leads to the statement onthe increase in the local smoothness of the solution in neighborhoods of interiorpoints in the domain Ω.

As an application of Theorems 4.4 and 4.5, we establish a sufficient conditionfor the solution u of the elliptic boundary-value problem (4.1) to be classical,i.e., to belong to the class C2q(Ω)∩Cm(Ω ), where m := maxm1, . . . ,mq. If uis a solution in this class, then the left-hand sides of the equalities in (4.1) arecalculated with the use of classical derivatives, and these equalities are fulfilledat every point in the set Ω or Γ, respectively. Moreover, their right-hand sideshave the following smoothness:

f ∈ C(Ω) and gj ∈ Cm−mj (Γ) for each j ∈ 1, . . . , q. (4.20)

The converse is not true; namely, the condition (4.20) does not imply that thesolution u is classical [64, Chap. 4, Notes]. With the use of the refined Sobolevscale, we strengthen this condition so that it becomes sufficient for the solutionto be classical.

Theorem 4.6. Suppose that a function u ∈ H2q(Ω) ∩ H2q,ϕ(Ω) is a solutionto problem (4.1), where

f ∈ Hn/2,ϕloc (Ω,∅) ∩Hm−2q+n/2,ϕ(Ω), (4.21)

gj ∈ Hm−mj+(n−1)/2,ϕ(Γ) for all j = 1, . . . , q, (4.22)

and the function parameter ϕ ∈ M satisfies condition (1.37). Then the solu-tion u is classical, i.e., u ∈ C2q(Ω) ∩ Cm(Ω ).


Remark 4.1. Conditions (4.21) and (4.22) imply property (4.20). This followsfrom Theorems 2.8 and 3.4. We also note that, in Theorem 4.6, condition (1.37)not only is sufficient for the solution u to be classical but also is necessary onthe class of all the considered solutions to the elliptic boundary-value problem;see also Remark 1.9.

Proof of Theorem 4.6. In view of condition (4.21), for s := 2q + n/2, wehave the inclusions

f ∈ Hs−2q,ϕloc (Ω,∅),

gj ∈ D′(Γ) = Hs−mj−1/2,ϕloc (∅) for each j ∈ 1, . . . , q.

Based on Theorems 4.5 and 3.4, we obtain

u ∈ Hs,ϕloc (Ω,∅) = H

2q+n/2,ϕloc (Ω,∅) ⊂ C2q(Ω).

Now let us prove that u ∈ Cm(Ω ). If s := m+n/2 > 2q, then, in view of (4.21)and (4.22), the condition of Theorem 4.4 is satisfied. Therefore, by virtue ofTheorems 4.4 and 3.4,

u ∈ Hs,ϕ(Ω) = Hm+n/2,ϕ(Ω) ⊂ Cm(Ω ).

If m+ n/2 ≤ 2q, then, by the condition,

u ∈ H2q,ϕ(Ω) ⊆ Hm+n/2,ϕ(Ω) ⊂ Cm(Ω ).

Thus, u ∈ C2q(Ω) ∩ Cm(Ω ). Theorem 4.6 is proved.

Remark 4.2. If we restricted ourselves to the Sobolev scale in Theorem 4.6, wewould fix small ε > 0 and replace (4.21) and (4.22) with the stronger conditions

f ∈ Hn/2+εloc (Ω,∅) ∩Hm−2q+n/2+ε(Ω),

gj ∈ Hm−mj+(n−1)/2+ε(Γ) for each j ∈ 1, . . . , q.

These conditions set the main smoothness of the right-hand sides of the bound-ary-value problem (4.1) too high, which coarsens the result.

4.1.3 Nonregular elliptic boundary-value problems

In this section, we assume that the boundary-value problem (4.1) is ellipticin the domain Ω but not regular. This means that it satisfies conditions (i)and (ii) of Definition 3.1 but does not satisfy condition (iii) of this definition.For this problem, all results in Subsections 4.1.1 and 4.1.2 remain true (with


certain changes). Changes refer only to Theorems 4.1 and 4.2 and are causedby the fact that a nonregular elliptic boundary-value problem does not havea formally conjugate boundary-value problem in a class of differential ones.We give analogs of these theorems.

Theorem 4.7. For arbitrary parameters s > 2q and ϕ ∈ M, mapping (4.2)extends uniquely (by continuity) to the Fredholm bounded operator (4.3). Thekernel of this operator is equal to N and the range consists of all vectors(f, g1, . . . , gq) ∈ Hs,ϕ(Ω,Γ) such that

(f, v)Ω +

q∑j=1

(gj , vj)Γ = 0 for all (v, v1, . . . , vq) ∈W. (4.23)

Here, W is a certain finite-dimensional space that lies in C∞(Ω )× (C∞(Γ))q

and is independent of s and ϕ. The index of operator (4.3) is equal to dimN −dimW and does not depend on s and ϕ.

In the Sobolev case of ϕ ≡ 1, this theorem is known (see, e.g., [86, The-orems 20.1.2 and 20.1.8] or [11, Sec. 2.4 a]). The general case of ϕ ∈ M isdeduced from the latter with the use of interpolation by analogy with the proofof Theorem 4.1.

By virtue of Theorem 4.7, for arbitrary s > 2q and ϕ ∈M, we can write

Hs,ϕ(Ω,Γ) =W u (L,B)(Hs,ϕ(Ω)).

Let Q denote the oblique projector of the space Hs,ϕ(Ω,Γ) onto the subspace(L,B)(Hs,ϕ(Ω)) parallel to W. This projector is independent of s and ϕ.

Theorem 4.8. For arbitrary parameters s > 2q and ϕ ∈M, the restriction ofmapping (4.3) to the subspace P (Hs,ϕ(Ω)) is the isomorphism

(L,B) : P (Hs,ϕ(Ω))↔ Q(Hs,ϕ(Ω,Γ)).

Theorem 4.8 directly follows from Theorem 4.7 and the Banach theorem oninverse operator.

Example 4.1. The oblique derivative problem for the Laplace equation

∆u = f in Ω,∂u

∂τ= g on Γ.

Here, τ is an infinitely smooth field of unit vectors tangent to the boundary Γ.If dim Ω = 2, then this problem is elliptic but not regular. If dim Ω ≥ 3, thenit is not elliptic at all [11, Sec. 1.4].


Example 4.2. Let τ1, . . . , τn−1 be a linearly independent system of infinitelysmooth fields of nonzero vectors tangent to the boundary Γ. (Recall that n =dim Ω.) The following boundary-value problem is elliptic but not regular:

∆n−1u = f in Ω,

∂u

∂τj= gj on Γ for j = 1, . . . , n− 1.

Example 4.3. Let dim Ω = 2. Consider the boundary-value problem

∆2u = f in Ω,

∂u

∂ν+∂u

∂τ= g1,

∂u

∂ν− ∂u

∂τ= g2 on Γ.

Here, τ is an infinitely smooth field of unit vectors tangent to the boundary Γ.This problem is elliptic but not regular.

Other examples of elliptic but nonregular boundary-value problems are given,e.g., in the paper by Ya. A. Roitberg [205].

In conclusion of this section, we recall the following important fact [11,Sec. 2.4 a]: if, for the boundary-value problem (4.1), the bounded operator (4.3)is Fredholm for a certain s > 2q under the condition that ϕ ≡ 1, then thisproblem is elliptic in the domain Ω, i.e., satisfies conditions (i) and (ii) of Def-inition 3.1.

4.1.4 Parameter-elliptic boundary-value problems

S. Agmon, L. Nirenberg [2, 6] and M. S. Agranovich, M. I. Vishik [13] selectedan important subclass of elliptic boundary-value problems, called parameter-elliptic (see also review [11, Sec. 4]). These problems depend on a complex-valued parameter and characterized by the following important property. Forsufficiently large absolute values of the parameter, the corresponding operatorsets isomorphisms between appropriate Sobolev spaces, and, furthermore, thenorm of the operator admits a two-sided estimate with constants independentof the parameter. In this section, we prove that this property is preserved forthe refined Sobolev scale.

We give the definition of a parameter-elliptic boundary-value problem. Con-sider the inhomogeneous boundary-value problem

L(λ)u = f in Ω, (4.24)

Bj(λ)u = gj on Γ for j = 1, . . . , q (4.25)

that depends on the complex-valued parameter λ as follows:

L(λ) :=2q∑r=0

λ2q−rLr and Bj(λ) :=mj∑r=0

λmj−rBj,r. (4.26)


Here, Lr is a linear differential expression on Ω, and Bj,r is a boundary lineardifferential expression on Γ; the coefficients of these expressions are infinitelysmooth complex-valued functions, and the orders do not exceed the number r.As above, the fixed integers q and mj satisfy the conditions q ≥ 1 and 0 ≤mj ≤ 2q − 1. Note that L(0) = L2q and Bj(0) = Bj,mj .

We associate the differential expressions (4.26) with certain homogeneouspolynomials in (ξ, λ) ∈ Cn+1. We set

L(0)(x; ξ, λ) :=2q∑r=0

λ2q−rL(0)r (x, ξ) for x ∈ Ω, ξ ∈ Cn, λ ∈ C.

Here, L(0)r (x, ξ) is the principal symbol of the expression Lr in the case where

ordLr = r, otherwise L(0)r (x, ξ) ≡ 0. By analogy, for each j ∈ 1, . . . , q, we set

B(0)j (x; ξ, λ) :=

mj∑r=0

λmj−rB(0)j,r (x, ξ) for x ∈ Γ, ξ ∈ Cn, λ ∈ C.

Here, B(0)j,r (x, ξ) is the principal symbol of the boundary differential expres-

sion Bj,r in the case where ordBj,r = r, otherwise B(0)j,r (x, ξ) ≡ 0. Note that

L(0)(x; ξ, λ) and B(0)j (x; ξ, λ) are homogeneous polynomials in (ξ, λ) ∈ Cn+1 of

degrees 2q and mj , respectively.Let K be a fixed closed angle on the complex plane with vertex at the origin

(the case where K degenerates into a ray is not excluded).

Definition 4.1. The boundary-value problem (4.24),(4.25) is called parameter-elliptic in the angle K if the following conditions are satisfied:

(i) For all x ∈ Ω, ξ ∈ Rn, and λ ∈ K such that |ξ| + |λ| 6= 0, the inequalityL(0)(x; ξ, λ) 6= 0 is true.

(ii) For arbitrarily fixed point x ∈ Γ, vector ξ ∈ Rn tangent to the boundaryΓ at the point x, and the parameter λ ∈ K such that |ξ| + |λ| 6= 0, thepolynomials B(0)

j (x; ξ+τν(x), λ), j = 1, . . . , q, in τ are linearly independentmodulo the polynomial

q∏j=1

(τ − τ+j (x; ξ;λ)).

Here, τ+1 (x; ξ;λ), . . . , τ+q (x; ξ;λ) are all τ -roots of the polynomial L(0)(x; ξ+τν(x), λ) that have the positive imaginary part and are written with regardfor their multiplicity.

Remark 4.3. Condition (ii) of Definition 4.1 is correctly defined in the sensethat the polynomial L(0)(x; ξ + τν(x), λ) has exactly q τ -roots with positive


imaginary part and the same number of roots with negative imaginary part(with regard for their multiplicity). Indeed, it follows from condition (i) that,for every point x ∈ Ω, the differential expression

L(x;D,Dt) :=2q∑r=0

D2q−rt Lr(x,D)

is elliptic. Since it contains the operators of differentiation with respect ton+1 ≥ 3 real variables x1, . . . , xn, t, its ellipticity leads to the proper ellipticity[11, Sec. 1.2, p. 7]. Therefore, the τ -roots of the polynomial L(0)(x; ξ+τν(x), λ)have the indicated property.

We give some examples of parameter-elliptic boundary-value problems [11,Sec. 3.1 b].

Example 4.4. Let the differential expression L(λ) satisfy condition (i) ofDefinition 4.1. Then the Dirichlet boundary-value problem for the equationL(λ) = f is parameter-elliptic in the angle K. Here, the boundary conditionsare independent of the parameter λ.

Example 4.5. The boundary-value problem

∆u+ λ2u = f in Ω,∂u

∂ν− λu = g on Γ

is parameter-elliptic in each angle

Kε := λ ∈ C : ε ≤ |argλ| ≤ π − ε,

where 0 < ε < π/2 and the complex plane is slitted along the negative ray.

Further, in this section, we assume that the boundary-value problem (4.24),(4.25) is parameter-elliptic in the angle K.

For λ = 0, condition (i) of Definition 4.1 implies that the differential ex-pression L(0) is properly elliptic on Ω (see Remark 4.3). Condition (ii) meansthat the collection B1(0), . . . , Bq(0) of boundary differential expressions sat-isfies the complementing condition with respect to L(0) on Γ. Therefore, forλ = 0, the boundary-value problem (4.24), (4.25) is elliptic (not necessarilyregular) in the domain Ω. Since the parameter λ affects only the lower termsof the differential expressions L(λ) and Bj(λ), this problem is elliptic for allλ ∈ C. According to Theorem 4.7, the mapping

u 7→ (L(λ)u,B(λ)u) := (L(λ)u,B1(λ)u, . . . , Bq(λ)u), u ∈ C∞(Ω ),

extends by continuity to the Fredholm bounded operator

(L(λ), B(λ)) : Hs,ϕ(Ω)→ Hs,ϕ(Ω,Γ) (4.27)


for arbitrary parameters s > 2q, ϕ ∈ M, and λ ∈ C. Its index is independentof both s, ϕ, and λ because λ affects only the lower terms [86, Theorem 20.1.8].

Since the boundary-value problem (4.24), (4.25) is parameter-elliptic in theangle K, operator (4.27) has the following important properties.


(i) There exists a number λ0 > 0 such that, for each λ ∈ K with |λ| ≥ λ0and for arbitrary s > 2q and ϕ ∈ M, operator (4.27) is an isomorphismof Hs,ϕ(Ω) onto Hs,ϕ(Ω,Γ).

(ii) For arbitrarily fixed parameters s > 2q and ϕ ∈ M, there exists a numberc = c(s, ϕ) ≥ 1 such that, for each λ ∈ K with |λ| ≥ maxλ0, 1 and forevery function u ∈ Hs,ϕ(Ω), we have the two-sided estimate

c−1(‖u‖Hs,ϕ(Ω)+|λ|sϕ(|λ|) ‖u‖L2(Ω)

)≤ ‖L(λ)u‖Hs−2q,ϕ(Ω) + |λ|s−2qϕ(|λ|) ‖L(λ)u‖L2(Ω)

+

q∑j=1

(‖Bj(λ)u‖Hs−mj−1/2,ϕ(Γ)

+ |λ|s−mj−1/2ϕ(|λ|) ‖Bj(λ)u‖L2(Γ)

)≤ c

(‖u‖Hs,ϕ(Ω) + |λ|sϕ(|λ|) ‖u‖L2(Ω)

). (4.28)

Here, the number c is independent of u and λ.

Remark 4.4. Assertion (ii) of Theorem 4.9 needs commenting. For a fixed λ,estimate (4.28) is written for norms equivalent to the norms ‖u‖Hs,ϕ(Ω) and‖(L(λ), B(λ))u‖Hs,ϕ(Ω,Γ). To avoid awkward expressions, we write this estimatefor non-Hilbert norms. It is also true for the corresponding Hilbert norms (gen-erating inner products in Hs,ϕ(Ω) and Hs,ϕ(Ω,Γ)) because they are estimatedvia the used norms with constants independent of s, ϕ, and λ. The additionalcondition |λ| ≥ 1 is caused by the fact that the function ϕ(t) is defined onlyfor t ≥ 1. Note that estimate (4.28) is of interest only for |λ| 1.

In the Sobolev case of ϕ ≡ 1 and s ≥ 2q, Theorem 4.9 is proved by M. S. Agra-novich and M. I. Vishik in [13, Sec. 4 and 5] (see also [11, Sec. 3.2]). The two-sided a priori estimate (4.28) has a number of applications, specifically, in thetheory of parabolic problems [13]. Note [13, Proposition 4.1] that the right-hand side of estimate (4.28) is true without assumption that problem (4.24),(4.25) is parameter-elliptic.

We separately prove assertions (i) and (ii) of Theorem 4.9.


Proof of assertion (i) of Theorem 4.9. We take the number λ0 > 0 fromthe statement of this theorem in the case of ϕ ≡ 1. Let λ ∈ K, |λ| ≥ λ0, andlet s > 2q, ϕ ∈M. We set ε := s− 2q. We have the isomorphisms

(L(λ), B(λ)) : Hs∓ε(Ω)↔ Hs∓ε(Ω,Γ).

Using the interpolation with the function parameter ψ from Theorem 1.14,where ε = δ, and applying the interpolation theorems 1.5, 2.2, and 3.2, we ob-tain the isomorphism

(L(λ), B(λ)) : Hs,ϕ(Ω)↔ Hs,ϕ(Ω,Γ).


Prior to the proof of assertion (ii), it is useful to introduce several Hörman-der spaces whose norms depend on the additional parameter %. We need oneinterpolation property of these spaces.

Let σ > 0, ϕ ∈ M, and % ≥ 1. Assume that G ∈ Rk,Ω,Γ, where k ∈ N.Let Hσ,ϕ(G, %) denote the space Hσ,ϕ(G) endowed with the norm that dependson the parameter % as follows:

‖u‖Hσ,ϕ(G,%) :=(‖u‖2Hσ,ϕ(G) + %2σϕ2(%)‖u‖2L2(G)

)1/2. (4.29)

This norm is equivalent to the norm in the space Hσ,ϕ(G) for every value ofthe parameter %. Therefore, the space Hσ,ϕ(G, %) is complete. It is a Hilbertspace because norm (4.29) is generated by the inner product

(u1, u2)Hσ,ϕ(G,%) := (u1, u2)Hσ,ϕ(G) + %2σϕ2(%)(u1, u2)L2(Ω).

As usual, in the Sobolev case of ϕ ≡ 1, we omit the index ϕ in notation.By virtue of Theorems 1.14, 2.2, and 3.2, the spaces

[Hσ−ε(G, %), Hσ+δ(G, %)]ψ and Hσ,ϕ(G, %)

are equal up to equivalence of norms. It turns out that the constants in esti-mates for norms of these spaces can be chosen so that they are independent ofthe parameter %.

Lemma 4.1. Let a function ϕ ∈M and positive numbers σ, ε, and δ be givenand, furthermore, let σ−ε > 0. Then there exists a number c ≥ 1 such that, forarbitrary % ≥ 1 and u ∈ Hσ,ϕ(G), the following two-sided estimate for norms istrue:

c−1 ‖u‖Hσ,ϕ(G,%) ≤ ‖u‖[Hσ−ε(G,%),Hσ+δ(G,%)]ψ≤ c ‖u‖Hσ,ϕ(G,%). (4.30)

Here, G ∈ Rk,Ω,Γ, ψ is the interpolation parameter from Theorem 1.14, andthe number c is independent of both % and u.


Proof. First, we prove Lemma 4.1 for G = Rk, where k ∈ N. Using thisresult, we prove the lemma in the required cases G = Ω and G = Γ.

The case G = Rk. Let % ≥ 1 and let u ∈ Hσ,ϕ(Rk). Using the definition ofnorms in the spaces Hσ,ϕ(Rk, %) and Hσ,ϕ(Rk), we write

‖u‖Hσ,ϕ(Rk,%) =

( ∫Rk

(〈ξ〉2σϕ2(〈ξ〉) + (2π)−k%2σϕ2(%)

)|u(ξ)|2 dξ

)1/2

. (4.31)

Parallel with (4.31), consider one more Hilbert norm of the function u:( ∫Rk

(〈ξ〉+ %)2σ ϕ2(〈ξ〉+ %) |u(ξ)|2 dξ)1/2

. (4.32)

Norms (4.31) and (4.32) are equivalent and, furthermore, the constants wherebyone norm is estimated in terms of the second depend only on σ and ϕ and,hence, do not depend on the parameter %. This follows from Lemma 4.2, whichwill be established after the proof of this lemma. Let Hσ,ϕ(Rk, %, 1) denote theHilbert space Hσ,ϕ(Rk) endowed with norm (4.32) and the corresponding innerproduct

(u1, u2)Hσ,ϕ(Rk,%,1) :=∫Rk

(〈ξ〉+ %)2σ ϕ2(〈ξ〉+ %) u1(ξ) u2(ξ) dξ.

We have the equivalence of the norms

c−10 ‖u‖Hσ,ϕ(Rk,%) ≤ ‖u‖Hσ,ϕ(Rk,%,1) ≤ c0 ‖u‖Hσ,ϕ(Rk,%). (4.33)

Here, the number c0 = c0(σ, ϕ) ≥ 1 does not depend on u and %.We interpolate the pair [Hσ−ε(Rk, %, 1), Hσ+δ(Rk, %, 1)] with the parameter ψ

from Theorem 1.14. The PsDO J% with symbol (〈ξ〉 + %)ε+δ is the generatingoperator for this pair. With the use of the Fourier transform

F : Hσ−ε(Rk, %, 1)↔ L2

(Rk, (〈ξ〉+ %)2(σ−ε)dξ

),

the operator ψ(J%) is reduced to the form of multiplication by the functionψ((〈ξ〉+ %)(ε+δ)) = (〈ξ〉+ %)εϕ(〈ξ〉+ %) of argument ξ ∈ Rk. Therefore,

‖u‖2[Hσ−ε(Rk,%,1),Hσ+δ(Rk,%,1)]ψ = ‖ψ(J%)u‖2Hσ−ε(Rk,%,1)

=

∫Rk

(〈ξ〉+ %)2(σ−ε) ψ2((〈ξ〉+ %)ε+δ) |u(ξ)|2 dξ

=

∫Rk

(〈ξ〉+ %)2σ ϕ2(〈ξ〉+ %) |u(ξ)|2 dξ

= ‖u‖2Hσ,ϕ(Rk,%,1).


Thus,‖u‖[Hσ−ε(Rk,%,1),Hσ+δ(Rk,%,1)]ψ = ‖u‖Hσ,ϕ(Rk,%,1). (4.34)

Note that

c−11 ‖u‖[Hσ−ε(Rk,%),Hσ+δ(Rk,%)]ψ ≤ ‖u‖[Hσ−ε(Rk,%,1),Hσ+δ(Rk,%,1)]ψ

≤ c1 ‖u‖[Hσ−ε(Rk,%),Hσ+δ(Rk,%)]ψ , (4.35)

where the number c1 ≥ 1 does not depend on u and %. Indeed, the identityoperator I defines the isomorphisms

I : Hσ−ε(Rk, %)↔ Hσ−ε(Rk, %, 1),

I : Hσ+δ(Rk, %)↔ Hσ+δ(Rk, %, 1).

Here, the norms of direct and inverse operators are uniformly bounded in theparameter %. By virtue of Theorem 1.8, this yields the isomorphism

I : [Hσ−ε(Rk, %), Hσ+δ(Rk, %)]ψ ↔ [Hσ−ε(Rk, %, 1), Hσ+δ(Rk, %, 1)]ψ

such that the norms of the direct and inverse operators are uniformly boundedin the parameter %. (Note that the pairs of spaces written above are normal.)This means the two-sided estimate for norms (4.35).

Now, relations (4.33)–(4.35) yield the required estimate (4.30) for G = Rk,where the number c := c0c1 does not depend on the parameter %.

The case G = Ω. We derive it from the previous case, in which k := n. Let% ≥ 1. Let RΩ denote the linear operator that restricts a distribution from thespace Rn to the domain Ω. We have the bounded operators

RΩ : Hσ,ϕ(Rn, %)→ Hσ,ϕ(Ω, %), (4.36)

RΩ : Hα(Rn, %)→ Hα(Ω, %), α > 0. (4.37)

It is obvious that their norms does not exceed number 1. We apply theinterpolation with parameter ψ to spaces in which operators (4.37), whereα ∈ σ − ε, σ + δ, act. By virtue of Theorem 1.8, we conclude that thenorm of the operator

RΩ : [Hσ−ε(Rn, %), Hσ+δ(Rn, %)]ψ → [Hσ−ε(Ω, %), Hσ+δ(Ω, %)]ψ

is uniformly bounded in the parameter %. (Since, here, the left pair of spacesis normal, the right pair is also normal.) Using inequality (4.30) for the caseG = Rn, we conclude that the norm of the operator

RΩ : Hσ,ϕ(Rn, %)→ [Hσ−ε(Ω, %), Hσ+δ(Ω, %)]ψ (4.38)

is uniformly bounded in the parameter %.


We need a linear bounded operator which is a right inverse to (4.38). In mono-graph [258, Sec. 4.2.2], for each l ∈ N, it is constructed a linear map Tl thatextends an arbitrary distribution u ∈ H−l(Ω) to the space Rn and is a boundedoperator

Tl : Hα(Ω)→ Hα(Rn) for each α ∈ R, |α| < l. (4.39)

(This operator has been used in Section 3.2.1.) We choose the integer l > σ+δ.Applying the interpolation with the parameter ψ to (4.39), where α ∈ σ − ε,σ + δ, and using Theorems 1.14 and 3.2, we obtain the bounded operator

Tl : Hσ,ϕ(Ω)→ Hσ,ϕ(Rn). (4.40)

It follows from the boundedness of operators (4.39) and (4.40) that the normsof the operators

Tl : Hα(Ω, %)→ Hα(Rn, %), 0 < α < l, (4.41)

Tl : Hσ,ϕ(Ω, %)→ Hσ,ϕ(Rn, %) (4.42)

are uniformly bounded in the parameter %.We apply the interpolation with theparameter ψ to the spaces in which operators (4.41), where α ∈ σ− ε, σ+ δ,act. Using Theorem 1.8 and inequality (4.30) for G = Rn, we conclude thatthe norm of the operator

Tl : [Hσ−ε(Ω, %), Hσ+δ(Ω, %)]ψ → Hσ,ϕ(Rn, %) (4.43)

is uniformly bounded in %.The operator RΩTl = I is the identity mapping It follows from the uniform

boundedness in the parameter % of the norms of operators (4.43), (4.36) and(4.42), (4.38) that the norms of the embedding operators

I = RΩTl : [Hσ−ε(Ω, %), Hσ+δ(Ω, %)]ψ → Hσ,ϕ(Ω, %),

I = RΩTl : Hσ,ϕ(Ω, %)→ [Hσ−ε(Ω, %), Hσ+δ(Ω, %)]ψ

are uniformly bounded in %. This immediately gives the two-sided estimate(4.30) for G = Ω.

The case G = Γ. We derive it from the first case G = Rk with k := n − 1.Let % ≥ 1. We reason by analogy with the proof of Lemma 2.6. We use thelocal definition 2.1 of the spaces Hs,ϕ(Γ), s ∈ R, ϕ ∈ M, for fixed finite atlasand partition of unity on Γ. Consider the linear mapping of “flattening” of themanifold Γ:

T : u 7→ ((χ1u) α1, . . . , (χru) αr), u ∈ D′(Γ).


It is directly verified that this mapping defines the isometric operators

T : Hσ,ϕ(Γ, %)→ (Hσ,ϕ(Rn−1, %))r, (4.44)

T : Hα(Γ, %)→ (Hα(Rn−1, %))r, α > 0. (4.45)

We apply the interpolation with the parameter ψ to the spaces in which oper-ators (4.45), where α ∈ σ − ε, σ + δ, act. Using Theorem 1.8, we concludethat the norm of the operator

T : [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ →[(Hσ−ε(Rn−1, %))r, (Hσ+δ(Rn−1, %))r]ψ

is uniformly bounded in the parameter %. (It is obvious that the pairs of spaceswritten here are normal.) By virtue of Theorem 1.5 and relation (4.30) provedfor G = Rn−1, we conclude that the norm of the operator

T : [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ → (Hσ,ϕ(Rn−1, %))r (4.46)

is also uniformly bounded in %.Parallel with T, consider the linear mapping of “sewing”

K : (w1, . . . , wr) 7→r∑j=1

Θj((ηjwj) α−1j ),

where w1, . . . , wr are distributions in Rn−1. Here, the function ηj ∈ C∞(Rn−1)is finite and equal to 1 on the set α−1j (suppχj), and Θj is the operator ofextension by zero to Γ. By virtue of (2.8), we have the bounded operators

K : (Hσ,ϕ(Rn−1))r → Hσ,ϕ(Γ),

K : (Hα(Rn−1))r → Hα(Γ), α ∈ R.

It is directly verified that the norm of each of the operators

K : (Hσ,ϕ(Rn−1, %))r → Hσ,ϕ(Γ, %), (4.47)

K : (Hα(Rn−1, %))r → Hα(Γ, %), α > 0, (4.48)

is uniformly bounded in the parameter %.We apply the interpolation with the parameter ψ to the spaces in which oper-

ators (4.48), where α ∈ σ− ε, σ+ δ, act. Based on Theorem 1.8, we concludethat the norm of the operator

K : [(Hσ−ε(Rn−1, %))r, (Hσ+δ(Rn−1, %))r]ψ → [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ


is uniformly bounded in %. By virtue of Theorem 1.5 and relation (4.30) provedfor G = Rn−1, we conclude that the norm of the operator

K : (Hσ,ϕ(Rn−1, %))r → [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ (4.49)

is also uniformly bounded in %.By virtue of (2.7), the product KT = I is the identity operator. Therefore,

the uniform boundedness in the parameter % of the norms of operators (4.44),(4.49) and (4.46), (4.47) leads to the uniform boundedness in % of the norms ofthe embedding operators

I = KT : Hσ,ϕ(Γ, %)→ [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ,

I = KT : [Hσ−ε(Γ, %), Hσ+δ(Γ, %)]ψ → Hσ,ϕ(Γ, %).

This immediately yields the two-sided estimate (4.30) for G = Γ.Lemma 4.1 is proved.

In the proof of Lemma 4.1, we used the following result.

Lemma 4.2. Let σ > 0, ϕ ∈ M, and ϕσ(t) := tσϕ(t) for t ≥ 1. Then thereexists a number c = c(σ, ϕ) ≥ 1 such that

c−1ϕσ(t1 + t2) ≤ ϕσ(t1) + ϕσ(t2) ≤ c ϕσ(t1 + t2) (4.50)

for all t1, t2 ≥ 1.

Proof. Since ϕσ ∈ RO (see Subsection 2.8.1), we have

ϕσ(2t) ϕσ(t) for t ≥ 1. (4.51)

In addition, since the function ϕσ has the order of variation σ > 0, it is (weakly)equivalent to a certain increasing positive function ψ:

ϕσ(t) ψ(t) for t ≥ 1. (4.52)

Now (4.50) follows from relations (4.51) and (4.52) because the function ψincreases. Indeed, for each number j = 1, 2, we can write

ϕσ(tj) ψ(tj) ≤ ψ(t1 + t2) ϕσ(t1 + t2) for t1, t2 ≥ 1.

Therefore, there exists a number c1 > 0 such that

ϕσ(t1) + ϕσ(t2) ≤ c1ϕσ(t1 + t2) for t1, t2 ≥ 1. (4.53)


Conversely, assuming without loss of generality that t1 ≤ t2, we have

ϕσ(t1 + t2) ψ(t1 + t2) ≤ ψ(2t2) ϕσ(2t2) ϕσ(t2)

≤ ϕσ(t1) + ϕσ(t2) for t1, t2 ≥ 1.

Therefore, there exists a number c2 > 0 such that

ϕσ(t1 + t2) ≤ c2 (ϕσ(t1) + ϕσ(t2)) for t1, t2 ≥ 1. (4.54)

Relations (4.53) and (4.54) mean the two-sided inequality (4.50).Lemma 4.2 is proved.

Using Lemma 4.1, we can give the following:

Proof of assertion (ii) of Theorem 4.9. Let s > 2q and ϕ ∈M, and letthe parameter λ ∈ K be such that |λ| ≥ maxλ0, 1, where the number λ0 > 0is taken from assertion (i) of this theorem. We set ε = δ = (s − 2q)/2 > 0.As indicated above, Theorem 4.9 is true in the Sobolev case of ϕ ≡ 1. Therefore,we have the isomorphisms

(L(λ), B(λ)) : Hs∓ε(Ω, |λ|)↔ Hs∓ε−2q(Ω, |λ|)⊕q⊕j=1

Hs∓ε−mj−1/2(Γ, |λ|)

=: Hs∓ε(Ω,Γ, |λ|) (4.55)

such that the norms of the direct and inverse operators are uniformly boundedin the parameter λ. [Note that we passed to Hilbert norms in estimate (4.28).]Let ψ be the interpolation parameter from Theorem 1.14. Applying the inter-polation with this parameter to (4.55), we obtain one more isomorphism

(L(λ), B(λ)) : [Hs−ε(Ω, |λ|), Hs+ε(Ω, |λ|)]ψ

↔ [Hs−ε(Ω,Γ, |λ|),Hs+ε(Ω,Γ, |λ|)]ψ. (4.56)

Here, the norms of the direct and inverse operators are uniformly bounded in λby virtue of Theorem 1.8. [Note that, the pairs of spaces written in (4.56) arenormal.]

Further, according to Theorem 1.5 on interpolation of orthogonal sums ofspaces, we can write

[Hs−ε(Ω,Γ, |λ|),Hs+ε(Ω,Γ, |λ|)]ψ

=[Hs−ε−2q(Ω, |λ|), Hs+ε−2q(Ω, |λ|)]ψ

⊕q⊕j=1

[Hs−ε−mj−1/2(Γ, |λ|), Hs+ε−mj−1/2(Γ, |λ|)]ψ


with equality of norms. By virtue of Lemma 4.1 and (4.56), this yields theisomorphism

(L(λ), B(λ)) : Hs,ϕ(Ω, |λ|)

↔ Hs−2q,ϕ(Ω, |λ|)⊕q⊕j=1

Hs−mj−1/2,ϕ(Γ, |λ|)

such that the norms of the direct and inverse operators are uniformly boundedin the parameters λ. This immediately gives the two-sided estimate (4.28).

Assertion (ii) is proved.Thus, Theorem 4.9 is proved.

Corollary 4.1. Let the boundary-value problem (4.24), (4.25) be parameter-elliptic on a certain closed beam K := λ ∈ C : argλ = const. Then boundedoperator (4.27) has the zero index for arbitrary s > 2q, ϕ ∈M, and λ ∈ C.

Proof. As indicated above, operator (4.27) is Fredholm, and its index isindependent of the mentioned parameters s, ϕ, and λ. By virtue of Theorem 4.9,the index of operator (4.27) is equal to 0 for |λ| 1. Therefore, it is equal to 0for every λ ∈ C. Corollary 4.1 is proved.

4.1.5 Formally mixed elliptic boundary-value problem

In this section, we consider the elliptic boundary-value problem for the lineardifferential equation Lu = f in the multiply connected domain Ω. Unlike theprevious sections, it is assumed that orders of boundary expressions are differenton different connected components of the boundary Γ. For example, for theLaplace equation in a ring, one can define the Dirichlet boundary condition onone component of the boundary and the Neumann boundary condition on theother. The considered problem relates to the class of mixed elliptic boundary-value problems. The theory of such problems is more complicated than that ofunmixed problems (see, e.g., papers [223, 186, 265, 236, 76, 43], monograph [77]and references therein). In the considered problem, parts of the boundary onwhich the orders of the boundary expression are different do not border on eachother. We formally call this problem mixed. With the use of local constructions,this problem can be reduced to an elliptic model problem in the half-space.

In this subsection we assume that the boundary Γ of the domain Ω consistsof r ≥ 2 nonempty connected components Γ1, . . . ,Γr. Consider the formallymixed boundary-value problem in the domain Ω

Lu = f in Ω (4.57)

Bk,ju = gk,j on Γk for j = 1, . . . , q and k = 1, . . . , r. (4.58)


Here, as above, L is a linear differential expression on Ω of even order 2q andBk,j : j = 1, . . . , q is a system of linear differential expressions defined onthe component Γk. It is assumed that all orders mk,j := ordBk,j ≤ 2q − 1.The coefficients of the differential expressions L and Bk,j are infinitely smoothcomplex-valued functions. We set

Λ := (L,B1,1, . . . , B1,q, . . . , Br,1, . . . , Br,q),

NΛ := u ∈ C∞(Ω ) : Λu = 0.

Definition 4.2. The formally mixed boundary-value problem (4.57), (4.58) iscalled elliptic in the multiply connected domain Ω if the following conditionsare satisfied:

(i) The differential expression L is properly elliptic on Ω.

(ii) For each k ∈ 1, . . . , r, the system of boundary expressions

Bk,j : j = 1, . . . , q

satisfies the complementing condition with respect to L on Γk.

Theorem 4.10. Suppose that the boundary-value problem (4.57), (4.58) is el-liptic in the domain Ω. Let s > 2q and ϕ ∈ M. Then the mapping u 7→ Λu,where u ∈ C∞(Ω ), extends uniquely (by continuity) to the Fredholm boundedoperator

Λ : Hs,ϕ(Ω)→ Hs−2q, ϕ(Ω)⊕r⊕k=1

q⊕j=1

Hs−mk,j−1/2,ϕ(Γk) (4.59)

=: Hs,ϕ.

The kernel of this operator coincides with NΛ. Its range consists of all vectors

(f, g1,1, . . . , g1,q, . . . , gr,1, . . . , gr,q) ∈ Hs,ϕ

such that

(f, w0)Ω +r∑k=1

q∑j=1

(gk,j , wk,j)Γk= 0

for every vector-valued function

(w0, w1,1, . . . , w1,q, . . . , wr,1, . . . , wr,q) ∈WΛ.

Here, WΛ is a certain finite-dimensional subspace of

C∞(Ω )×r∏j=1

(C∞(Γj))q,


this subspace not depending on s and ϕ. The index of operator (4.59) is equalto dimNΛ − dimWΛ and does not depend on s and ϕ as well.

In the Sobolev case of ϕ ≡ 1, this theorem is a special case of Theorem 1in [163], where the boundedness and Fredholm property of the operator Λ areestablished for the scale of Lizorkin–Triebel spaces, which contains the Sobolevscale. The general case of ϕ ∈ M is derived with the use of interpolation byanalogy with the proof of Theorem 4.1.

4.2 Elliptic boundary-value problems in thetwo-sided scale

In this section, we study the regular elliptic boundary-value problem (4.1) inthe modified two-sided scale of Hörmander inner product spaces. Now, unlikethe previous Section 4.1, the numerical index defining the main smoothnesspasses through the entire real axis.

4.2.1 Preliminary remarks

Theorems on solvability of elliptic boundary-value problems proved in Sec-tion 4.1 are, generally speaking, not true for an arbitrary real parameter s thatdefines the main smoothness of a solution to the problem. This is caused by thefact that the mapping u 7→ uΓ, where u ∈ C∞(Ω ), cannot be extended to thecontinuous trace operator RΓ : Hs,ϕ(Ω)→ D′(Γ) for s < 1/2 (see Remark 3.5).Therefore, operator (4.3) corresponding to the boundary-value problem (4.1)is not defined for s < m + 1/2, where m is the maximum of the orders ofthe boundary expressions B1, . . . , Bq. For the other boundary-value problemsconsidered in Section 4.1, the situation is analogous.

To obtain the bounded operator (L,B) for any s < 2q, it is necessary to takeanother space, somewhat different from Hs,ϕ(Ω), as a domain of this operator.There are two essentially different methods for the construction of the domainproposed by Ya. A. Roitberg [202, 203, 209] and J.-L. Lions and E. Magenes[121, 126, 119, 120] in the Sobolev case. These methods lead to different typesof the theorems on solvability of elliptic boundary-value problems (the generaltheorem and the individual theorems). In the general theorem, the domain ofthe operator (L,B) does not depend on coefficients of the elliptic expression Land is unique for all boundary-value problems of the same order. In individualtheorems, the domain depends on coefficients of the expression L (even on thecoefficients in lower order derivatives).

Note that the theorems (about solvability of elliptic boundary-value prob-lems) proved in Section 4.1 are general (for the corresponding classes of prob-lems).

Section 4.2 Elliptic boundary-value problems in the two-sided scale 189

In Section 4.2, we extend Ya. A. Roitberg’s approach over the two-sidedrefined Sobolev scale. Namely, we modify this scale in the sense of Roitbergand then prove the general theorem on the solvability of the regular ellipticboundary-value problem (4.1) in this modified scale. (The theory developedbelow can be easily extend over the other elliptic boundary-value problemsconsidered in Section 4.1. We do not dwell on it.)

J.-L. Lions and E. Magenes’ approach leading to the individual theorems onsolvability will be considered in Sections 4.4 and 4.5.

Note again (see the end of Subsection 4.1.1) that, in the case of s < m +1/2, the properties of the inhomogeneous elliptic boundary-value problem (4.1)can not be directly derived from the properties of the two semihomogeneousboundary-value problems studied in Sections 3.3 and 3.4.

4.2.2 The refined scale modified in the sense of Roitberg

First, following Ya. A. Roitberg [202, 203, 209], we introduce the notion of thegeneralized solution to the boundary-value problem (4.1).

In a neighborhood of the boundary Γ, we write the differential expressions Land Bj in the form

L =

2q∑k=0

LkDkν and Bj =

mj∑k=0

Bj,kDkν . (4.60)

Here, Dν := i ∂/∂ν as above, and Lk and Bj,k are certain tangential differentialexpressions (with respect to the boundary Γ). Integrating by parts, we writethe following Green formula:

(Lu, v)Ω = (u, L+v)Ω − i2q∑k=1

(Dk−1ν u, L(k)v)Γ (4.61)

for arbitrary functions u, v ∈ C∞(Ω ). Here,

L(k) :=2q∑r=k

Dr−kν L+

r ,

where L+r denotes the differential expression which is formally conjugate to Lr.

Passing to the limit, we establish that relation (4.61) is true for each functionu ∈ H2q(Ω). Denote

u0 := u and uk := (Dk−1ν u)Γ for k = 1, . . . 2q. (4.62)

By virtue of (4.60) and (4.61), the boundary-value problem (4.1) for the un-known function u ∈ H2q(Ω) is equivalent to the system of conditions

(u0, L+v)Ω − i

2q∑k=1

(uk, L(k)v)Γ = (f, v)Ω for all v ∈ C∞(Ω ), (4.63)


mj∑k=0

Bj,k uk+1 = gj on Γ for j = 1, . . . , q. (4.64)

Note that these conditions make sense in the case of arbitrary distributions

u0 ∈ D′(Rn), suppu0 ⊆ Ω, u1, . . . , u2q ∈ D′(Γ), (4.65)

f ∈ D′(Rn), supp f ⊆ Ω, g1, . . . , gq ∈ D′(Γ).

For this reason, it is useful to introduce the following definition.

Definition 4.3. A vector u = (u0, u1, . . . , u2q) satisfying (4.65) is said to bea Roitberg generalized solution to the boundary-value problem (4.1) if conditions(4.63) and (4.64) are fulfilled.

We introduce Hilbert spaces whose elements can be considered as Roitberggeneralized solutions.

Let r ∈ N, s ∈ R, and ϕ ∈M. We set

Er := k − 1/2 : k = 1, . . . , r.

Definition 4.4. In the case of s ∈ R \ Er, the linear space Hs,ϕ,(r)(Ω) is,by definition, the completion of C∞(Ω ) with respect to the Hilbert norm

‖u‖Hs,ϕ,(r)(Ω) :=(‖u‖2

Hs,ϕ,(0)(Ω)+

r∑k=1

‖(Dk−1ν u)Γ‖2

Hs−k+1/2,ϕ(Γ)

)1/2

. (4.66)

In the case s ∈ Er, we define the space Hs,ϕ,(r)(Ω) by the interpolation

Hs,ϕ,(r)(Ω) := [Hs−ε,ϕ,(r)(Ω), Hs+ε,ϕ,(r)(Ω)]t1/2 for 0 < ε < 1. (4.67)

Remark 4.5. In relation (4.67), we use the interpolation of Hilbert spaces withthe power parameter ψ(t) = t1/2. In what follows, we will show that the pair ofspaces on the right-hand side of this relation is admissible [Theorem 4.12(i), (iv)]and that the result of this interpolation is independent of ε up to equivalenceof norms (Section 4.2.5, Theorem 4.21).

In the Sobolev case of ϕ ≡ 1, the space Hs,ϕ,(r)(Ω) was introduced byYa. A. Roitberg in [202, 203] (see also his monograph [209, Sec. 2.1]). As usual,we set Hs,(r)(Ω) := Hs,1,(r)(Ω).

Definition 4.5. The family of Hilbert spaces

Hs,ϕ,(r)(Ω) : s ∈ R, ϕ ∈M (4.68)

is said to be the refined Sobolev scale modified in the sense of Roitberg, or,briefly, the modified refined scale. The number r is called the order of modifi-cation.


From the point of viewof the application to the boundary-value problem (4.1),we are interested in the case of even order of modification r = 2q. By virtue ofDefinition 4.4, the mapping u 7→ (Dk−1

ν u) Γ, where u ∈ C∞(Ω ), extends bycontinuity to a bounded operator from Hs,ϕ,(r)(Ω) to Hs−k+1/2,ϕ(Γ) for arbi-trary s ∈ R and ϕ ∈ M provided that k ∈ 0, 1, . . . , 2q. Therefore, for everyelement u ∈ Hs,ϕ,(2q)(Ω), the vector

(u0, u1, . . . , u2q) ∈ Hs,ϕ,(0)(Ω)⊕2q⊕k=1

Hs−k+1/2,ϕ(Γ) (4.69)

is well-defined by relations (4.62) whereby the closure. Thus, the element u canbe regarded as the Roitberg generalized solution (4.69) of the boundary-valueproblem (4.1).

We study properties of the modified refined scale (4.68). For arbitrary s ∈ Rand ϕ ∈M, we set

Πs,ϕ,(r)(Ω,Γ) := Hs,ϕ,(0)(Ω)⊕r⊕k=1

Hs−k+1/2,ϕ(Γ).

In addition, denote

Ks,ϕ,(r)(Ω,Γ) := (u0, u1, . . . , ur) ∈ Πs,ϕ,(r)(Ω,Γ) : uk = (Dk−1ν u0)Γ

for all k ∈ 1, . . . r such that s > k − 1/2.

By virtue of Theorem 3.5, Ks,ϕ,(r)(Ω,Γ) is closed in Πs,ϕ,(r)(Ω,Γ). We con-sider Ks,ϕ,(r)(Ω,Γ) as a Hilbert space with respect to the inner product inΠs,ϕ,(r)(Ω,Γ). In the Sobolev case ϕ ≡ 1, we omit the index ϕ in notation ofthe spaces introduced in this chapter.

Theorem 4.11. Let r ∈ N, s ∈ R \ Er, and ϕ ∈ M. The following assertionsare true:

(i) The linear mapping

Tr : u 7→ (u, uΓ, . . . , (Dr−1ν u)Γ), u ∈ C∞(Ω ), (4.70)

extends uniquely (by continuity) to the isometric isomorphism

Tr : Hs,ϕ,(r)(Ω)↔ Ks,ϕ,(r)(Ω,Γ). (4.71)

(ii) For arbitrary positive numbers ε and δ such that all the numbers s, s− ε,and s+ δ belong to one of the intervals

α0 := (−∞, 1/2),

αk := (k − 1/2, k + 1/2), k = 1, . . . , r − 1,

αr := (r − 1/2,+∞),

(4.72)


we have[Hs−ε,(r)(Ω), Hs+δ,(r)(Ω)]ψ = Hs,ϕ,(r)(Ω) (4.73)

up to equivalence of norms. Here, ψ is the interpolation parameter fromTheorem 1.14.

Proof. In the case of ϕ ≡ 1, assertion (i) is established by Ya. A. Roitberg[209, Lemma 2.2.1]. Using this result and interpolation with a function param-eter, we first derive assertion (ii) and then prove assertion (i) for an arbitraryϕ ∈M.

Let Xψ denote the left-hand side of equality (4.73). (The pair of spacesin (4.73) is obviously admissible.) Consider the isometric operators

Tr : Hσ,(r)(Ω)→ Πσ,(r)(Ω,Γ), σ ∈ s− ε, s+ δ.

Using the interpolation with the parameter ψ, we obtain the bounded operator

Tr : Xψ → [Πs−ε,(r)(Ω,Γ),Πs+δ,(r)(Ω,Γ)]ψ.

By virtue of the interpolation theorems 1.5, 2.2, and 3.10, we obtain the fol-lowing equalities of spaces with equivalence of norms:

[Πs−ε,(r)(Ω,Γ),Πs+δ,(r)(Ω,Γ)]ψ

= [Hs−ε,(0)(Ω), Hs+δ,(0)(Ω)]ψ ⊕r⊕k=1

[Hs−ε−k+1/2(Γ), Hs+δ−k+1/2(Γ)]ψ

= Hs,ϕ,(0)(Ω)⊕r⊕k=1

Hs−k+1/2,ϕ(Γ) = Πs,ϕ,(r)(Ω,Γ).

Therefore, the operator

Tr : Xψ → Πs,ϕ,(r)(Ω,Γ) (4.74)

is bounded. This yields the estimate

‖u‖Hs,ϕ,(r)(Ω) = ‖Tr u‖Πs,ϕ,(r)(Ω,Γ) ≤ c1 ‖u‖Xψ (4.75)

for any u ∈ C∞(Ω ). Here, c1 is the norm of operator (4.74).We prove the inequality inverse to (4.75). By condition, s, s − ε, s + δ ∈ αp

for a certain number p ∈ 0, 1, . . . , r. Consider the linear mapping

Tr,p : u 7→(u, (Dk−1

ν u)Γ : p+ 1 ≤ k ≤ r), u ∈ C∞(Ω ).

(As above, the index k is integer.)


This mapping extends by continuity to the isomorphism

Tr,p : Hσ,(r)(Ω) ↔ Hσ,(0)(Ω) ⊕⊕

p+1≤k≤rHσ−k+1/2(Γ) (4.76)

for each σ ∈ s− ε, s+ δ.

Indeed, the boundedness of operator (4.76) follows from the definition of thespace Hσ,(r)(Ω).We show that this operator is bijective. Let u ∈ Hσ,(r)(Ω) andlet (

u0, uk : p+ 1 ≤ k ≤ r)∈ Hσ,(0)(Ω) ⊕

⊕p+1≤k≤r

Hσ−k+1/2(Γ).

We set uk := (Dk−1ν u0)Γ for 1 ≤ k ≤ p. By virtue of Theorem 3.5, the distri-

bution uk is well-defined because σ > k−1/2 for the indicated numbers k. Notethat σ < k − 1/2 for p + 1 ≤ k ≤ r. Therefore, (u0, u1, . . . , ur) ∈ Kσ,(r)(Ω,Γ).As was mentioned above, assertion (i) of the theorem is true in the case ofϕ ≡ 1. Therefore, we have the isomorphisms

Tr : Hσ,(r)(Ω)↔ Kσ,(r)(Ω,Γ), σ ∈ s− ε, s+ δ.

Since

Tr u = (u0, u1, . . . , ur) ⇔ Tr,p u = (u0, uk : p+ 1 ≤ k ≤ r ),

we establish that the bounded operator (4.76) is bijective. Therefore, it is anisomorphism (by the Banach theorem on inverse operator).

We apply the interpolation with the parameter ψ to (4.76). By virtue ofTheorems 1.5, 2.2, and 3.10, we obtain the isomorphism

Tr,p : Xψ ↔ Hs,ϕ,(0)(Ω) ⊕⊕

p+1≤k≤rHs−k+1/2,ϕ(Γ). (4.77)

This yields the following inequality, which is inverse to (4.75):

‖u‖Xψ ≤ c2(‖u‖2

Hs,ϕ,(0)(Ω)+

∑p+1≤k≤r

‖(Dk−1ν u)Γ‖2

Hs−k+1/2,ϕ(Γ)

)1/2

≤ c2 ‖u‖Hs,ϕ,(r)(Ω)

for all u ∈ C∞(Ω ). Here, c2 is the norm of the inverse to (4.77). Thus, thenorms in the spaces Xψ and Hs,ϕ,(r)(Ω) are equivalent on the set C∞(Ω ).It is dense in Hs,ϕ,(r)(Ω) by definition and in Xψ by virtue of Theorem 1.1.Therefore, Xψ = Hs,ϕ,(r)(Ω) up to equivalent norms. Assertion (ii) is proved.


Let us prove assertion (i). According to the definition of the spaceHs,ϕ,(r)(Ω),mapping (4.70) extends by continuity to the isometric operator

Tr : Hs,ϕ,(r)(Ω)→ Πs,ϕ,(r)(Ω,Γ). (4.78)

On the basis of Theorem 3.5, we have the inclusion

Tr(Hs,ϕ,(r)(Ω)) ⊆ Ks,ϕ,(r)(Ω,Γ).

We prove the inverse inclusion. Let

(u0, u1, . . . , ur) ∈ Ks,ϕ,(r)(Ω,Γ).

By virtue of (4.77) and the equality Xψ = Hs,ϕ,(r)(Ω), we obtain the isomor-phism

Tr,p : Hs,ϕ,(r)(Ω) ↔ Hs,ϕ,(0)(Ω) ⊕⊕

p+1≤k≤rHs−k+1/2,ϕ(Γ).

Therefore, there exists u ∈ Hs,ϕ,(r)(Ω) such that

Tr,p u = (u0, uk : p+ 1 ≤ k ≤ r ).

In view of Theorem 3.5, this yields the equality Tr u = (u0, u1, . . . , ur). There-fore, the inclusion

Ks,ϕ,(r)(Ω,Γ) ⊆ Tr(Hs,ϕ,(r)(Ω))

is proved. Thus,Tr(H

s,ϕ,(r)(Ω)) = Ks,ϕ,(r)(Ω,Γ),

which, together with the isometric operator (4.78), implies the isometric iso-morphism (4.71). Assertion (i) is proved.


Assertion (i) of Theorem 4.11 gives the useful isometric representation of thespace Hs,ϕ,(r)(Ω) for s /∈ Er, namely

Tr(Hs,ϕ,(r)(Ω)) = Ks,ϕ,(r)(Ω,Γ).

Remark 4.6. If s ∈ Er, then mapping (4.70) extends by continuity to thebounded operator (4.78). However, we can only state that

Tr(Hs,ϕ,(r)(Ω)) ⊆ Ks,ϕ,(r)(Ω,Γ).

This follows from relation (4.67), Theorem 4.11(i), and the interpolation lemmagiven below.


Lemma 4.3. For arbitrary σ ∈ R, ε > 0, and ϕ ∈M, we have

[Hσ−ε,ϕ,(0)(Ω), Hσ+ε,ϕ,(0)(Ω)]t1/2 = Hσ,ϕ,(0)(Ω) (4.79)

[Hσ−ε,ϕ(Γ), Hσ+ε,ϕ(Γ)]t1/2 = Hσ,ϕ(Γ) (4.80)

up to equivalence of norms.

Proof. We derive equality (4.79) from the interpolation theorems 3.10 and1.3. According to the first of them,

Hσ∓ε,ϕ,(0)(Ω) = [Hσ−2ε,(0)(Ω), Hσ+2ε,(0)(Ω)]ψ∓ .

Here, the interpolation parameters ψ∓ are defined by the relations

ψ−(t) := t1/4ϕ(t1/(4ε)), ψ+(t) := t3/4ϕ(t1/(4ε)) for t ≥ 1,

and ψ∓(t) := 1 for 0 < t < 1. By virtue of Theorem 1.3 on reiterated interpo-lation, we obtain

[Hσ−ε,ϕ,(0)(Ω), Hσ+ε,ϕ,(0)(Ω)]t1/2

=[[Hσ−2ε,(0)(Ω), Hσ+2ε,(0)(Ω)]ψ− , [H

σ−2ε,(0)(Ω), Hσ+2ε,(0)(Ω)]ψ+

]t1/2

= [Hσ−2ε,(0)(Ω), Hσ+2ε,(0)(Ω)]ψ. (4.81)

Here, the interpolation parameter ψ is defined by the formulas

ψ(t) := ψ−(t) (ψ+(t)/ψ−(t))1/2 = t1/2ϕ(t1/(4ε)) for t ≥ 1

and ψ(t) = 1 for 0 < t < 1. Therefore, on the basis of Theorem 3.10, we have

[Hσ−2ε,(0)(Ω), Hσ+2ε,(0)(Ω)]ψ = Hσ,ϕ,(0)(Ω). (4.82)

Now equalities (4.81) and (4.82) yield (4.79). Since the obtained equalities ofspaces hold up to equivalence of norms, so does equality (4.79). Equality (4.80)is a special case of Theorem 2.4.


The proved lemma will also be useful in what follows (in the cases where wewill refer to the interpolation formula (4.67)).

We continue the study of properties of the modified refined scale.

Theorem 4.12. Let r ∈ N, s ∈ R, and ϕ,ϕ1 ∈ M. The following assertionsare true:

(i) The Hilbert space Hs,ϕ,(r)(Ω) is separable.


(ii) The set C∞(Ω ) is dense in the space Hs,ϕ,(r)(Ω).

(iii) If s > r − 1/2, then the norms in the spaces Hs,ϕ,(r)(Ω) and Hs,ϕ(Ω) areequivalent on the dense set C∞(Ω ) and, hence, these spaces are equal upto equivalence of norms.

(iv) For an arbitrary number ε > 0, we have the continuous and dense embed-ding Hs+ε,ϕ1,(r)(Ω) → Hs,ϕ,(r)(Ω). This embedding is compact.

(v) If the function ϕ/ϕ1 is bounded in the neighborhood of ∞, then we havethe continuous and dense embedding Hs,ϕ1,(r)(Ω) → Hs,ϕ,(r)(Ω). Thisembedding is compact if ϕ(t)/ϕ1(t)→ 0 as t→∞.

Remark 4.7. The continuous embedding Hs+ε,ϕ1,(r)(Ω) → Hs,ϕ,(r)(Ω), ap-peared in Theorem 4.12(iv), (v) for ε ≥ 0, is understood in the following sense[23, Chap. 14, Sec. 7]:

(i) there exists a number c > 0 such that

‖u‖Hs+ε,ϕ1,(r)(Ω) ≤ c ‖u‖Hs,ϕ,(r)(Ω) for all u ∈ C∞(Ω );

(ii) the identity mapping given on the functions u ∈ C∞(Ω ) extends by con-tinuity to an injective operator from Hs+ε,ϕ1,(r)(Ω) to Hs,ϕ,(r)(Ω).

Proof of Theorem 4.12. (i) For the parameter s /∈ Er, the separability ofthe space Hs,ϕ,(r)(Ω) follows from Theorem 4.11(i) and the separability of thespace Ks,ϕ,(r)(Ω,Γ). If s ∈ Er, then the space Hs,ϕ,(r)(Ω) is separable by virtueof (4.67) as the result of interpolation of separable Hilbert spaces.

(ii) Assertion (ii) for s /∈ Er is contained in the definition of the spaceHs,ϕ,(r)(Ω). If s ∈ Er, then, by virtue of (4.67) and Theorem 1.1, the dense con-tinuous embedding Hs+ε,ϕ,(r)(Ω) → Hs,ϕ,(r)(Ω) is true for a sufficiently smallε > 0. Since s + ε /∈ Er, the set C∞(Ω ) is dense in the space Hs+ε,ϕ,(r)(Ω).Therefore, this set is also dense in the space Hs,ϕ,(r)(Ω).

(iii) If s > r − 1/2 and k ∈ 1, . . . , r, then, by virtue of Theorem 3.5, wehave

‖(Dk−1ν u)Γ‖Hs−k+1/2,ϕ(Γ) ≤ c ‖u‖Hs,ϕ(Ω)

for any u ∈ C∞(Ω ), where the number c > 0 is independent of u. Therefore,norms in the spaces Hs,ϕ,(r)(Ω) and Hs,ϕ,(0)(Ω) = Hs,ϕ(Ω) are equivalent onthe dense linear manifold C∞(Ω ). Therefore, these spaces are equal.

(iv) By virtue of Theorems 3.9(iv) and 2.3(iii), the compact embedding

Πs+ε,ϕ1,(r)(Ω,Γ) → Πs,ϕ,(r)(Ω,Γ)


is true. This implies the compact embedding of subspaces

Ks+ε,ϕ1,(r)(Ω,Γ) → Ks,ϕ,(r)(Ω,Γ).

Based on Theorem 4.11(i), we obtain, in the case of s, s+ ε /∈ Er, the compactembedding of the space

Hs+ε,ϕ1,(r)(Ω) = T−1r (Ks+ε,ϕ1,(r)(Ω,Γ))

in the spaceHs,ϕ,(r)(Ω) = T−1r (Ks,ϕ,(r)(Ω,Γ)).

If s, s+ ε∩Er 6= ∅, then, by virtue of (4.67) and Theorem 1.1, the followingembeddings are true for a sufficiently small number ε0 > 0:

Hs+ε,ϕ1,(r)(Ω) → Hs+ε−ε0,ϕ1,(r)(Ω) → Hs+ε0,ϕ,(r)(Ω) → Hs,ϕ,(r)(Ω).

Here, the number ε0 must satisfy the conditions

0 < ε0 < 1, ε0 < ε/2, s+ ε− ε0 /∈ Er, and s+ ε0 /∈ Er.

By the proved result, the mean embedding is compact. Therefore, the embed-ding of the flanked spaces is also compact. This embedding is dense by virtueof assertion (ii).

(v) Assume that the function ϕ/ϕ1 is bounded in a neighborhood of ∞.Then, by virtue of assertion (iv) of Theorems 2.3, 3.3, and 3.8, we obtain thecontinuous embedding

Ks,ϕ1,(r)(Ω,Γ) → Ks,ϕ,(r)(Ω,Γ).

Based on Theorem 4.11(i), we obtain the required continuous embedding in thecase s /∈ Er, namely,

Hs,ϕ1,(r)(Ω) = T−1r (Ks,ϕ1,(r)(Ω,Γ))

→ T−1r (Ks,ϕ,(r)(Ω,Γ)) = Hs,ϕ,(r)(Ω). (4.83)

If s ∈ Er, then, by virtue of the interpolation formula (4.67), we have

Hs∓1/2,ϕ1,(r)(Ω) → Hs∓1/2,ϕ,(r)(Ω) ⇒ Hs,ϕ1,(r)(Ω) → Hs,ϕ,(r)(Ω). (4.84)

Here, the left continuous embeddings has already been proved because

s∓ 1/2 /∈ Er.

Therefore, the right continuous embedding is true. It is dense for any s ∈ Raccording to assertion (ii).


Further, if ϕ(t)/ϕ1(t) → 0 as t → ∞, then, by virtue of assertion (iv) ofTheorems 2.3, 3.3, and 3.8, the embedding

Πs,ϕ1,(r)(Ω,Γ) → Πs,ϕ,(r)(Ω,Γ)

is compact. This yields the compactness of the embedding of subspaces

Ks,ϕ1,(r)(Ω,Γ) → Ks,ϕ,(r)(Ω,Γ).

Therefore, embedding (4.83) is compact in the case of s /∈ Er. Now if s ∈ Er,then the left embeddings in implication (4.84) are compact. This yields thecompactness of the right embedding by virtue of (4.67) and the theorem whichstates that the compactness of operators is preserved in the case of interpolationwith power parameter [258, Sec. 1.16.4]. Assertion (v) is proved.

Theorem (4.12) is proved.

In conclusion of this section, we study properties of differential expressionsas operators on the space Hs,ϕ,(r)(Ω). Let K = K(x,D) be a linear differentialexpression defined on Ω, and let R = R(x,D) be a boundary linear differen-tial expression defined on Γ. The coefficients of these expressions are infinitelysmooth complex-valued functions, and orders are arbitrary.

Theorem 4.13. Let r ∈ N, s ∈ R, and ϕ ∈ M. The following assertions aretrue:

(i) If κ := ordK ≤ r, then

‖Ku‖Hs−κ,ϕ,(0)(Ω) ≤ c1 ‖u‖Hs,ϕ,(r)(Ω) for all u ∈ C∞(Ω ),

where the number c1 > 0 is independent of u. Therefore, the mapping

u 7→ Ku, where u ∈ C∞(Ω ),

extends uniquely (by continuity) to the bounded linear operator

K : Hs,ϕ,(r)(Ω)→ Hs−κ,ϕ,(0)(Ω).

(ii) If % := ordR ≤ r − 1, then

‖Ru‖Hs−%−1/2,ϕ(Γ) ≤ c2 ‖u‖Hs,ϕ,(r)(Ω) for all u ∈ C∞(Ω ),

where the number c2 > 0 is independent of u. Therefore, the mapping

u 7→ Ru, where u ∈ C∞(Ω ),

extends uniquely (by continuity) to the bounded linear operator

R : Hs,ϕ,(r)(Ω)→ Hs−%−1/2,ϕ(Γ).


Proof. In the Sobolev case, where ϕ ≡ 1, this theorem is proved byYa. A. Roitberg [209, Lemma 2.3.1]. The case of an arbitrary ϕ ∈M is derivedwith the use of interpolation. We perform it, e.g., for assertion (i). The proofof assertion (ii) is analogous.

First, assume that s /∈ Er. Let the positive number ε = δ be the same as inTheorem 4.11(ii). The mapping u 7→ Ku, where u ∈ C∞(Ω ), is extended bycontinuity to the bounded linear operators

K : Hs∓ε,(r)(Ω)→ Hs∓ε−κ,(0)(Ω).

Now we apply the interpolation with the function parameter ψ from Theo-rem 1.14. By virtue of Theorems 4.11(ii) and 3.10, we obtain assertion (i) inthe case of s /∈ Er.

Now assume that s ∈ Er. We choose an arbitrary number ε ∈ (0, 1). Takings ∓ ε /∈ Er into account and using the proved result, we obtain the boundedlinear operators

K : Hs∓ε,ϕ,(r)(Ω)→ Hs∓ε−κ,ϕ,(0)(Ω).

Applying the interpolation with the power parameter t1/2 and using (4.67) andLemma 4.3, we obtain assertion (i) in the case of s /∈ Er.


It follows from Theorem 4.13(i) for κ = 0 that the multiplication by anarbitrary function from the class C∞(Ω ) is a bounded linear operator on eachof the spaces Hs,ϕ,(r)(Ω).

4.2.3 Roitberg-type theorems on solvability.The complete collection of isomorphisms

We study the regular elliptic boundary-value problem (4.1) in the modifiedrefined scale, for which the order of modification r = 2q. It is useful to com-pare the results of this subsection and their proof with the results given inSubsection 4.2.1.

According to Theorem 4.13, mapping (4.2) is extended by continuity to thebounded linear operator

(L,B) : Hs,ϕ,(2q)(Ω)→ Hs−2q,ϕ,(0)(Ω)⊕q⊕j=1

Hs−mj−1/2,ϕ(Γ)

=: Hs,ϕ,(0)(Ω,Γ). (4.85)

Therefore, for an arbitrary element u ∈ Hs,ϕ,(2q)(Ω), the right-hand sidesf ∈ Hs−2q,ϕ,(0)(Ω) and gj ∈ Hs−mj−1/2,ϕ(Γ) of the boundary-value prob-lem (4.1) are defined by means of closure. By virtue of Theorem 4.11(i),


the equality (L,B)u = (f, g1, . . . , gq) is equivalent to the statement that the vec-tor (u0, u1, . . . , u2q) := T2qu is a Roitberg generalized solution to this boundary-value problem. We identify the element u with the vector (u0, u1, . . . , u2q) andalso call this vector a Roitberg generalized solution to the boundary-value prob-lem (4.1).

We study properties of operator (4.85).

Theorem 4.14. For arbitrary parameters s ∈ R and ϕ ∈ M, the boundedoperator (4.85) is Fredholm. Its kernel is equal to N and the range consists ofall vectors (f, g1, . . . , gq) ∈ Hs,ϕ,(0)(Ω,Γ) that satisfy condition (4.4). The indexof operators (4.85) is equal to dimN−dimN+ and does not depend on s and ϕ.

Proof. In the Sobolev case, where ϕ ≡ 1, this theorem was established byYa. A. Roitberg in [202], [203]; see also his monograph [209, Theorems 4.1.1and 5.3.1]. Based on this result and applying interpolation with a functionparameter, we will prove the theorem in the general case of arbitrary ϕ ∈M.

First, assume that s /∈ E2q. Let a positive number ε = δ be the same as inTheorem 4.11(ii). Mapping 4.2 is extended by continuity to the bounded andFredholm operators

(L,B) : Hs∓ε,(2q)(Ω)→ Hs∓ε,(0)(Ω,Γ). (4.86)

They have the common kernel N, the same index κ := dimN − dimN+, andthe range

(L,B)(Hs∓ε(Ω)) = (f, g1, . . . , gq) ∈ Hs∓ε,(0)(Ω,Γ) : (4.4) is true. (4.87)

We apply the interpolation with the parameter ψ to the spaces in whichoperators (4.86) act. By virtue of Theorem 1.7, we obtain the bounded andFredholm operator

(L,B) : [Hs−ε,(2q)(Ω), Hs+ε,(2q)(Ω)]ψ

→ [Hs−ε,(0)(Ω,Γ),Hs+ε,(0)(Ω,Γ)]ψ, (4.88)

which is extension of mapping (4.2) by continuity. Here, by virtue of the inter-polation theorems 1.5, 2.2, and 3.10, we have

[Hs−ε,(0)(Ω,Γ),Hs+ε,(0)(Ω,Γ)]ψ = Hs,ϕ,(0)(Ω,Γ) (4.89)

with equivalence of norms. It follows from this and Theorem 4.11(ii) that map-ping (4.2) extends by continuity to the bounded operator (4.85), which is equalto (4.88). According to Theorem 1.7, the Fredholm property of operators (4.86)implies the Fredholm property of operator (4.85), which inherits their kernel Nand the index κ. In addition, the range of operator (4.85) is equal to

Hs,ϕ,(0)(Ω,Γ) ∩ (L,B)(Hs−ε,(2q)(Ω)).


By virtue of (4.87), this implies that the range is the same as in the statementof the theorem.

Now assume that s ∈ E2q.We arbitrarily choose a number ε ∈ (0, 1). Accord-ing to the proved result and in view of s∓ ε /∈ E2q, mapping (4.2) is extendedby continuity to the bounded and Fredholm operators

(L,B) : Hs∓ε,ϕ,(2q)(Ω)→ Hs∓ε,ϕ,(0)(Ω,Γ).

They have the common kernel N and index κ. Applying the interpolationwith the power parameter t1/2 and using Theorem 1.7, equality (4.67), andLemma 4.3, we obtain the bounded and Fredholm operator (4.85). It extendsmapping (4.2) by continuity and has the same kernel N and index κ. In addi-tion, the range of this operator is equal to

Hs,ϕ,(0)(Ω,Γ) ∩ (L,B)(Hs−ε,ϕ,(2q)(Ω)).

It follows directly from this and the results proved above that the range is thesame as in the statement of the theorem.


Theorem 4.14 is a general theorem on the solvability of the elliptic boundary-value problem (4.1) because the space Hs,ϕ,(2q)(Ω) used as a domain of theoperator (L,B) is independent of the elliptic expression L.

If N = N+ = 0 (the defect of the boundary-value problem is absent),then operator (4.85) is an isomorphism of Hs,ϕ,(2q)(Ω) onto Hs,ϕ,(0)(Ω,Γ). Thisfollows from Theorem 4.14 and the Banach theorem on inverse operator. In thegeneral case, it is convenient to define an isomorphism with the help of thefollowing projectors.

Lemma 4.4. For arbitrary parameters s ∈ R and ϕ ∈M, the following decom-positions of the spaces Hs,ϕ,(2q)(Ω) and Hs,ϕ,(0)(Ω,Γ) in direct sums of (closed)subspaces are true:

Hs,ϕ,(2q)(Ω) = N uu ∈ Hs,ϕ,(2q)(Ω) : (u0, w)Ω = 0 for all w ∈ N

, (4.90)

Hs,ϕ,(0)(Ω,Γ) = (v, 0, . . . , 0) : v ∈ N+u (L,B)(Hs,ϕ,(2q)(Ω)). (4.91)

Here, u0 is the initial component of the vector (u0, u1, . . . , u2q) := T2qu. Let P2q

denote the oblique projector of the space Hs,ϕ,(2q)(Ω) onto the second summandin (4.90), and let Q+

0 denote the oblique projector of the space Hs,ϕ,(0)(Ω,Γ)onto the second summand in (4.91), both parallel to the first summand. Theseprojectors are independent of s and ϕ.

Proof. First, we prove equality (4.90). It follows from the definition of thespace Hs,ϕ,(2q)(Ω) that the mapping u 7→ u0 is the bounded operator

T0 : Hs,ϕ,(2q)(Ω)→ Hs,ϕ,(0)(Ω).


Therefore, the second summand in (4.90) is closed in Hs,ϕ,(2q)(Ω) and has thetrivial intersection with N. Therefore, the following decomposition is true:

Hs,ϕ,(0)(Ω) = N u u0 ∈ Hs,ϕ,(0)(Ω) : (u0, w)Ω = 0 for all w ∈ N. (4.92)

Indeed, the space dual to the subspace N of H−s,1/ϕ,(0)(Ω) is isomorphic to thefactor space of the space Hs,ϕ,(0)(Ω) by the second summand in (4.92). Hence,the codimension of this summand is equal to dimN, which yields (4.92).

Let Π denote the oblique projector of the space Hs,ϕ,(0)(Ω) onto the firstsummand in (4.92) parallel to the second summand. For a arbitrary elementu ∈ Hs,ϕ,(2q)(Ω), we write u = u′ + u′′, where u′ := Πu0 ∈ N and

u′′ := u−Πu0 ∈ Hs,ϕ,(2q)(Ω)

satisfies the condition

(u′′0, w)Ω = (u0 −Πu0, w)Ω = 0 for any w ∈ N.

Equality (4.90) is proved.Equality (4.91) follows from the fact that, by virtue of Theorem 4.14, the

linear manifolds on the right-hand side of this equality are closed, have thetrivial intersection, and the finite dimension of the first of them coincides withthe codimension of the second. Finally, inclusions N,N+ ⊂ C∞(Ω ) impliesthat the projectors P2q and Q+

0 are independent of the parameters s and ϕ.Lemma 4.4 is proved.

Theorem 4.15. For arbitrary parameters s ∈ R and ϕ ∈ M, the restrictionof mapping (4.85) to the subspace P2q(H

s,ϕ(Ω)) is the isomorphism

(L,B) : P2q(Hs,ϕ(Ω))↔ Q+

0 (Hs,ϕ,(0)(Ω,Γ)). (4.93)

Proof. According to Theorem 4.14, N is the kernel and Q+0 (Hs,ϕ,(0)(Ω,Γ)) is

the range of operator (4.85). Hence, the bounded operator (4.93) is a bijection.Therefore, it is an isomorphism by the Banach theorem on inverse operator.


For each fixed ϕ ∈ M, the collection of isomorphisms (4.93) is completebecause s runs through the whole real axis. This property differs this collectionfrom the one-sided collection of isomorphisms in Theorem 4.2.

The following a priori estimate for the solution to the elliptic boundary-valueproblem (4.1) ensues from Theorem 4.15.

Theorem 4.16. Let s ∈ R and ϕ ∈M, and let the number ε > 0. Suppose thatthe element u ∈ Hs,ϕ,(2q)(Ω) is a Roitberg generalized solution to the boundary-value problem (4.1), where (f, g1, . . . , gq) ∈ Hs,ϕ,(0)(Ω,Γ).


Then we have the estimate

‖u‖Hs,ϕ,(2q)(Ω) ≤ c(‖(f, g1, . . . , gq)‖Hs,ϕ,(0)(Ω,Γ) + ‖u‖Hs−ε,ϕ,(2q)(Ω)

), (4.94)

where the number c = c(s, ϕ, ε) > 0 is independent of u and (f, g1, . . . , gq).

Proof. We use decomposition (4.90) and write the element u ∈ Hs,ϕ,(2q)(Ω)in the form u = u′ + u′′, where

u′ := (1− P2q)u ∈ N and u′′ := P2qu ∈ P2q(Hs,ϕ(Ω)).

By virtue of Theorem 4.15,

‖u′′‖Hs,ϕ,(2q)(Ω) ≤ c2‖(L,B)u′′‖Hs,ϕ,(0)(Ω,Γ)

= c2‖(L,B)u‖Hs,ϕ,(0)(Ω,Γ) = c2‖(f, g1, . . . , gq)‖Hs,ϕ,(0)(Ω,Γ).

Here, c2 is the norm of the inverse to (4.93). In addition, since the space N isfinite-dimensional and 1−P2q is a projector of the space Hs−ε,ϕ,(2q)(Ω) onto N(Lemma 4.4), we have

‖u′‖Hs,ϕ,(2q)(Ω) ≤ c0‖u′‖Hs−ε,ϕ,(2q)(Ω) ≤ c1‖u‖Hs−ε,ϕ,(2q)(Ω).

Here, the positive numbers c0 and c1 are independent of u′, u, and (f, g1, . . . , gq).Summing these inequalities, we obtain estimate (4.94).


If N = 0, i.e., the boundary-value problem (4.1) has at most one solution,the norm ‖u‖Hs−ε,ϕ,(2q)(Ω) is absent on the right-hand side of (4.94).

In the Sobolev case of ϕ ≡ 1, Theorems 4.14–4.16 were proved by Ya. A. Roit-berg in [202, 203]; see also monographs [209, Secs. 3.3 and 5.3] and [21, Chap. 3,Sec. 6, Subsec. 8] and review [11, Sec. 7.9 b]. For an arbitrary ϕ ∈ M, thesetheorems refines Roitberg’s results for the scale (4.68).

By virtue of Theorems 4.12(iii) and 3.9(i), the following equalities of spacesare true up to equivalence of norms in them:

Hs,ϕ,(2q)(Ω) = Hs,ϕ(Ω) and Hs−2q,ϕ,(0)(Ω) = Hs−2q,ϕ(Ω) (4.95)

for each s > 2q − 1/2.

Therefore, Theorems 4.14–4.16 include Theorems 4.1–4.3 on solvability of theelliptic boundary-value problem (4.1) in the scale of positive Hörmander spaces.Moreover, the following result is true.

Theorem 4.17. Theorems 4.1–4.3 are true for arbitrary parameters s > 2q −1/2 and ϕ ∈M.

This theorem follows from Theorems 4.14–4.16, relations (4.95), and theequality of the projectors P2q = P on Hs,ϕ,(2q)(Ω).


4.2.4 Smoothness of generalized solutions up to the boundary

We study the smoothness of Roitberg generalized solutions to the boundary-value problem (4.1), considered in the modified refined Sobolev scale. It is usefulto compare the results of this section with results obtained in Subsection 4.1.2.

For integer r≥0, letH−∞,(r)(Ω) denote the union of all the spacesHs,ϕ,(r)(Ω)with s ∈ R and ϕ ∈ M. The topology of inductive limit is introduced in thespace H−∞,(r)(Ω).

The bounded operators (4.85) considered for all parameters s ∈ R and ϕ ∈Mgenerate the continuous linear operator

(L,B) : H−∞,(2q)(Ω)→ H−∞,(0)(Ω)× (D′(Γ))q

=: H−∞,(0)(Ω,Γ). (4.96)

By virtue of Theorem 4.14, the kernel of this operator is equal to N and therange consists of all vectors (f, g1, . . . , gq) ∈ H−∞,(0)(Ω,Γ) that satisfy condi-tion (4.4).

Theorem 4.18. Suppose that u ∈ H−∞,(2q)(Ω) is a generalized solution to theboundary-value problem (4.1) in which

f ∈ Hs−2q,ϕ,(0)(Ω) and gj ∈ Hs−mj−1/2,ϕ(Γ), j = 1, . . . , q,

for some parameters s ∈ R and ϕ ∈M. Then u ∈ Hs,ϕ,(2q)(Ω).

Proof. By virtue of the above-mentioned properties of operator (4.96),the vector F := (f, g1, . . . , gq) = (L,B)u satisfies (4.4). By condition, F ∈Hs,ϕ(Ω,Γ); hence, by Theorem 4.14, the inclusion F ∈ (L,B)(Hs,ϕ(Ω)) is true.Therefore, parallel with condition (L,B)u = F , the equality (L,B)v = F istrue for a certain element v ∈ Hs,ϕ(Ω). This yields, (L,B)(u − v) = 0, whichimplies the inclusion

w := u− v ∈ N ⊂ Hs,ϕ(Ω).

Thus, u = v + w ∈ Hs,ϕ(Ω).Theorem 4.18 is proved.

Theorem 4.18 is a statement about the global smoothness of generalizedsolutions (i.e., on the whole closed domain Ω). In the Sobolev case of ϕ ≡ 1, thistheorem was proved by Ya. A. Roitberg in [202, 203] (see also his monograph[209, Theorem 7.1.1]).

Now consider the case of local smoothness. Let U be an open set in Rnthat has a nonempty intersection with domain Ω. As in Section 4.1.2, we setΩ0 := U ∩Ω and Γ0 := U ∩ Γ (the case Γ0 = ∅ is possible). We introduce the


following local analog of the space Hσ,ϕ,(r)(Ω), where σ ∈ R, ϕ ∈ M, and theinteger r ≥ 0. We set

Hσ,ϕ,(r)loc (Ω0,Γ0) := u ∈ H−∞,(r)(Ω) : χu ∈ Hσ,ϕ,(r)(Ω)

for all χ ∈ C∞(Ω ) such that suppχ ⊂ Ω0 ∪Γ0. By virtue of Theorem 4.13(i),the multiplication by the function χ ∈ C∞(Ω ) is a bounded operator on thespace Hσ,ϕ,(r)(Ω). Therefore, for each element u ∈ H−∞,(r)(Ω), the productχu ∈ H−∞,(r)(Ω) is well defined. In addition,

Hσ,ϕ,(r)(Ω) ⊂ Hσ,ϕ,(r)loc (Ω0,Γ0).

We also need the local space Hσ,ϕloc (Γ0) introduced in Sec. 2.2.3.

Theorem 4.19. Suppose that u ∈ H−∞,(2q)(Ω) is a generalized solution to theboundary-value problem (4.1) in which

f ∈ Hs−2q,ϕ,(0)loc (Ω0,Γ0) and

gj ∈ Hs−mj−1/2,ϕloc (Γ0), j = 1, . . . , q,

(4.97)

for some parameters s ∈ R and ϕ ∈M. Then u ∈ Hs,ϕ,(2q)loc (Ω0,Γ0).

Proof. First we prove that, by virtue of the condition of this theorem, thefollowing implication holds for each r ≥ 1:

u ∈ Hs−r,ϕ,(2q)loc (Ω0,Γ0) ⇒ u ∈ Hs−r+1,ϕ,(2q)

loc (Ω0,Γ0). (4.98)

We choose r ≥ 1 arbitrarily and assume that the premise of implication (4.98)is true. Let functions χ, η ∈ C∞(Ω ) be such that suppχ, supp η ⊂ Ω0 ∪ Γ0

and η = 1 in a neighborhood of suppχ. Interchanging the operator of multipli-cation by χ with the differential operators L and Bj , j = 1, . . . , q, we obtainequality (4.16), where the differential expressions L′ and B′j are the same asthose used in the proof of Theorem 4.5. Recall that ordL′ ≤ 2q − 1 and everyordB′j ≤ mj − 1. Thus,

(L,B)(χu) = χF + (L′, B′)(ηu),

where χF ∈ Hs,ϕ,(0)(Ω,Γ) by condition (4.97), and

(L′, B′)(ηu) ∈ Hs−r+1,ϕ,(0)(Ω,Γ)

by Theorem 4.13 and the premise of implication (4.98). Since (L,B)(χu) ∈Hs−r+1,ϕ,(0)(Ω,Γ), we conclude that χu ∈ Hs−r+1,ϕ,(2q)(Ω) by virtue of Theo-rem 4.18. Thus, u ∈ Hs−r+1,ϕ,(2q)

loc (Ω0,Γ0); implication (4.98) is proved.


Now we may complete the proof of the theorem in the following way. Sinceu ∈ H−∞,(2q)(Ω), there exists an integer k ≥ 1 such that u ∈ Hs−k,ϕ,(2q)(Ω).Therefore, the premise of implication (4.98) is true for r = k. Using thisimplication in succession for values r = k, r = k − 1, . . . , and r = 1, weconclude that

u ∈ Hs−k,ϕ,(2q)(Ω) ⇒ u ∈ Hs−k+1,ϕ,(2q)loc (Ω0,Γ0)

⇒ . . .⇒ u ∈ Hs−1,ϕ,(2q)loc (Ω0,Γ0)

⇒ u ∈ Hs,ϕ,(2q)loc (Ω0,Γ0).

Thus, u ∈ Hs,ϕ,(2q)loc (Ω0,Γ0).


Theorems 4.18 and 4.19 refines the Roitberg theorems [202, 203] on increasein smoothness of generalized solutions to the elliptic boundary-value problemconsidered on the modified Sobolev scale (see also Ya. A. Roitberg’s monograph[209, Sections 7.1 and 7.2]). It follows from equality (4.95) that Theorems 4.18and 4.19 contain Theorems 4.4 and 4.5. In Theorem 4.19, we note the Γ0 = ∅case, which leads to the statement about increase in local smoothness of thegeneralized solutions in neighborhoods of inner points of the domain Ω.

As an application of Theorems 4.18 and 4.19, we establish a sufficient con-dition for a Roitberg generalized solution u ∈ H−∞,(2q)(Ω) of the ellipticboundary-value problem (4.1) to be classical, i.e., to satisfy the condition

u ∈ Hσ+2q(Ω) ∩ C2q(Ω) ∩ Cm(Ω ), (4.99)

where σ > −1/2 and m := maxm1, . . . ,mq. Let us explain why this conditionappears. By virtue of Theorems 4.12(iii) and 3.5, it follows from the inclusion

u ∈ Hσ+2q,(2q)(Ω) = Hσ+2q(Ω)

that the element u is a solution to problem (4.1) in the sense of theory ofdistributions defined in the domain Ω. Not it is reasonable to consider theinclusion u ∈ C2q(Ω) ∩ Cm(Ω ). It implies that the functions Lu and Bju arecalculated in (4.1) with the use of classical derivatives, i.e., the solution u isclassical.

Theorem 4.20. Suppose that u ∈ H−∞,(2q)(Ω) is a Roitberg generalized solu-tion to the boundary-value problem (4.1) in which

f ∈ Hn/2,ϕ,(0)loc (Ω,∅) ∩Hm−2q+n/2,ϕ,(0)(Ω) ∩Hσ,(0)(Ω), (4.100)

gj ∈ H m−mj+(n−1)/2,ϕ(Γ) ∩Hσ+2q−mj−1/2(Γ), j = 1, . . . , q, (4.101)


for some number σ > −1/2 and a certain function parameter ϕ ∈ M thatsatisfies condition (1.37). Then the solution u is classical, i.e., u satisfies in-clusion (4.99).

Proof. Applying Theorems 4.18 and 4.19 and using conditions (4.100) and(4.101), we obtain the inclusion

u ∈ H2q+n/2,ϕ,(2q)loc (Ω,∅) ∩Hm+n/2,ϕ,(2q)(Ω) ∩Hσ+2q,(2q)(Ω).

Hence, by Theorems 4.12(iii) and 3.4, we have

u ∈ Hm+n/2,ϕ,(2q)(Ω) ∩Hσ+2q,(2q)(Ω) = Hm+n/2,ϕ(Ω) ∩Hσ+2q(Ω)

⊆ Cm(Ω ) ∩Hσ+2q(Ω).

(The last equality becomes clear if we separately consider the cases m+n/2 ≥σ + 2q and m+ n/2 < σ + 2q.) In addition,

χu ∈ H2q+n/2,ϕ,(2q)(Ω) = H2q+n/2,ϕ(Ω) ⊂ C2q(Ω )

for any function χ ∈ C∞0 (Ω), which yields the inclusion u ∈ C2q(Ω). Thus,condition (4.99) is satisfied, i.e., u is a classical solution.


4.2.5 Interpolation in the modified refined scale

We study two problems concerning interpolation in the modified refined scale.First, we prove that the right-hand side of equality (4.67) is independent of thechoice of the parameter ε. Second, we establish that the interpolation formula(4.73) is true under essentially weaker conditions on the parameters involved.

Theorem 4.21. Let r ∈ N, s ∈ Er, and ϕ ∈M. The space

Hs,ϕ,(r)(Ω, ε) :=[Hs−ε,ϕ,(r)(Ω), Hs+ε,ϕ,(r)(Ω)

]t1/2

,

is independent of the parameter ε ∈ (0, 1) up to equivalence of norms.

Proof. First, assume that r = 2q is an even number. Consider the regularelliptic boundary-value problem in the domain Ω:

Lu ≡ (1− ∆)q u = f in Ω, (4.102)

Bju ≡ Dj−1ν u = gj on Γ, j = 1, . . . , q. (4.103)


As usual, ∆ is the Laplace operator. This problem is formally self-adjoint andN = N+ = 0 for it. According to Theorem 4.14, we have the isomorphism

(L,B) : Hs,ϕ,(2q)(Ω, ε)↔ Hs−2q,ϕ,(0)(Ω)⊕q⊕j=1

Hs−j+1/2,ϕ(Γ)

for 0 < ε < 1. This immediately yields the conclusion of theorem for evenr = 2q.

Further, assume that the number r is odd. By virtue of Theorem 4.11 (i),for any number σ ∈ (−∞, r + 1/2) \ Er, we have the isometric isomorphisms

Tr : Hσ,ϕ,(r)(Ω)↔ Kσ,ϕ,(r)(Ω,Γ),

Tr+1 : Hσ,ϕ,(r+1)(Ω)↔ Kσ,ϕ,(r+1)(Ω,Γ) = Kσ,ϕ,(r)(Ω,Γ)⊕Hσ−r−1/2,ϕ(Γ).

Therefore, the composition of the mappings

u 7→ Tr+1 u =: (u0, u1, . . . , ur, ur+1) 7→ (T−1r (u0, u1, . . . , ur), ur+1),

where u ∈ Hσ,ϕ,(r+1)(Ω), defines the isometric isomorphism

T : Hσ,ϕ,(r+1)(Ω)↔ Hσ,ϕ,(r)(Ω)⊕Hσ−r−1/2,ϕ(Γ). (4.104)

Here, we take σ = s∓ ε, where 0 < ε < 1, and apply the interpolation with thepower parameter t1/2. In view of Lemma 4.3, we obtain the isomorphism

T : Hs,ϕ,(r+1)(Ω, ε)↔ Hs,ϕ,(r)(Ω, ε)⊕Hs−r−1/2,ϕ(Γ) =: X(ε).

Therefore,

‖u‖Hs,ϕ,(r)(Ω,ε) = ‖(u, 0)‖X(ε) ‖T−1(u, 0)‖Hs,ϕ,(r+1)(Ω,ε).

It follows from this and the theorem proved for even r + 1 that the norms inthe spaces Hs,ϕ,(r)(Ω, ε), where 0 < ε < 1, are equivalent. Therefore, thesespaces are equal because the set C∞(Ω ) is dense in each of them according toTheorem 4.12(ii).


Theorem 4.22. Let r ∈ N, s ∈ R, ϕ ∈ M and positive numbers ε and δ bearbitrarily chosen. If the number r is odd, then we additionally suppose that atleast one of the inequalities s − ε > r − 1/2 and s + δ < r + 1/2 is satisfied.Then the interpolation formula (4.73) is true, namely,

[Hs−ε,(r)(Ω), Hs+δ,(r)(Ω)]ψ = Hs,ϕ,(r)(Ω) (4.105)

with equivalence of norms. Here, ψ is the interpolation parameter in Theo-rem 1.14.


Proof. First, assume that r = 2q is an even number. By virtue of Theo-rem 4.14, for the elliptic boundary-value problem (4.102), (4.103), we have theisomorphisms

(L,B) : Hs,ϕ,(2q)(Ω)↔ Hs−2q,ϕ,(0)(Ω)⊕q⊕j=1

Hs−j+1/2,ϕ(Γ), (4.106)

(L,B) : Hσ,(2q)(Ω)↔ Hσ−2q,(0)(Ω)⊕q⊕j=1

Hσ−j+1/2(Γ), σ ∈ R. (4.107)

We apply the interpolation with the parameter ψ to the spaces in which iso-morphisms (4.107), with σ ∈ s − ε, s + δ, act. Taking into account that ψis an interpolation parameter and using Theorems 3.10 and 2.2, we obtain onemore isomorphism

(L,B) : [Hs−ε,(2q)(Ω), Hs+δ,(2q)(Ω)]ψ

↔ Hs−2q,ϕ,(0)(Ω)⊕q⊕j=1

Hs−j+1/2,ϕ(Γ). (4.108)

Now, using isomorphisms (4.106) and (4.108), we obtain the equality of spaces(4.105) up to equivalence of norms.

Further, assume that the number r is odd. We separately consider the caseof s− ε > r − 1/2 and the case of s+ δ < r + 1/2.

If s−ε > r−1/2, then, by virtue of Theorems 4.12(iii) and 3.2, the followingequalities of spaces are true up to equivalence of norms:

[Hs−ε,(r)(Ω), Hs+δ,(r)(Ω)]ψ = [Hs−ε(Ω), Hs+δ(Ω)]ψ

= Hs,ϕ(Ω) = Hs,ϕ,(r)(Ω).

Relation (4.105) is proved in the considered case.Now consider the case where s+ δ < r+1/2. Note that isomorphism (4.104)

is true for any σ < r + 1/2. Indeed, if σ /∈ Er, then this isomorphism wasestablished in the proof of Theorem 4.21. Hence, we deduce it for every σ ∈ Erif we apply interpolation and use equalities (4.67) and (4.80). In particular, wehave the isomorphisms

T : Hs,ϕ,(r+1)(Ω)↔ Hs,ϕ,(r)(Ω)⊕Hs−r−1/2,ϕ(Γ), (4.109)

T : Hσ,(r+1)(Ω)↔ Hσ,(r)(Ω)⊕Hσ−r−1/2(Γ), σ ∈ s− ε, s+ δ. (4.110)


We apply the interpolation with the parameter ψ to (4.110). By virtue ofthe result proved above (since the number r + 1 is even) and Theorem 2.2, weobtain one more isomorphism

T : Hs,ϕ,(r+1)(Ω)↔ [Hs−ε,(r)(Ω), Hs+δ,(r)(Ω)]ψ ⊕Hs−r−1/2,ϕ(Γ). (4.111)

Now, using homeomorphisms (4.109) and (4.111), we can write the following:

u ∈ [Hs−ε,(r)(Ω), Hs+δ,(r)(Ω)]ψ ⇔ T−1(u, 0) ∈ Hs,ϕ,(r+1)(Ω)

⇔ u ∈ Hs,ϕ,(r)(Ω).

Therefore, the equality of spaces (4.105) is true in the case of s+ δ < r + 1/2.The norms in these spaces are equivalent:

‖u‖[Hs−ε,(r)(Ω),Hs+δ,(r)(Ω)]ψ ‖T−1(u, 0)‖Hs,ϕ,(r+1)(Ω) ‖u‖Hs,ϕ,(r)(Ω),

where u ∈ Hs,ϕ,(r)(Ω).Theorem 4.22 is proved.

4.3 Some properties of the modified refined scale

In this section, we formulate and prove two important properties of the mod-ified refined Sobolev scale, whose order of modification is an arbitrary evennumber 2q, with q ∈ N. They will be used in Section 4.5.

4.3.1 Statement of results

The first property gives us an equivalent alternative definition of the spaceHs,ϕ,(2q)(Ω). It turns out that the norm in this space is equivalent to a certainnorm involving the properly elliptic expression L and not using any boundaryvalues of functions. Unlike (4.66), it is not necessary to eliminate the caseof s ∈ E2q.

Theorem 4.23. Let s ∈ R and ϕ ∈M. The following assertions are true:

(i) On the set of all functions u ∈ C∞(Ω ), the norm in the space Hs,ϕ,(2q)(Ω)is equivalent to the graph norm(

‖u‖2Hs,ϕ,(0)(Ω)

+ ‖Lu‖2Hs−2q,ϕ,(0)(Ω)

)1/2. (4.112)

Therefore, the space Hs,ϕ,(2q)(Ω) coincides with the completion of C∞(Ω )with respect to the norm (4.112).

Section 4.3 Some properties of the modified refined scale 211

(ii) The mappingIL : u 7→ (u, Lu), u ∈ C∞(Ω ), (4.113)

extends uniquely (by continuity) to the isomorphism

IL : Hs,ϕ,(2q)(Ω)↔ Ks,ϕ,L(Ω). (4.114)

Here,

Ks,ϕ,L(Ω) :=(u0, f) : u0 ∈ Hs,ϕ,(0)(Ω), f ∈ Hs−2q,ϕ,(0)(Ω),

(u0, L+w)Ω = (f, w)Ω for all w ∈ C∞ν,2q(Ω )

(4.115)

is a (closed) subspace of

Hs,ϕ,(0)(Ω)⊕Hs−2q,ϕ,(0)(Ω).

Recall (see Section 3.5.1) that

C∞ν,2q(Ω ) :=u ∈ C∞(Ω ) : Dj−1

ν u = 0 on Γ, j = 1, . . . , 2q.

If s > −1/2, then, for the distributions u0 and f in Theorem 4.23(ii), condition(4.115) is equivalent to the condition

(u0, L+w)Ω = (f, w)Ω for all w ∈ C∞0 (Ω). (4.116)

The latter means that Lu0 = f in the domain Ω. (The equivalence will beproved in Lemma 4.5.) If s < −1/2, then this is not true and we need thesecond property.

Theorem 4.24. Let s < −1/2, s+ 1/2 /∈ Z, and ϕ ∈ M. Then, for arbitrarydistributions

u0 ∈ Hs,ϕ,(0)(Ω), f ∈ Hs−2q,ϕ,(0)(Ω) (4.117)

that satisfy condition (4.116), there exists a unique pair (u∗0, f) ∈ Ks,ϕ,L(Ω)such that

(u0, w)Ω = (u∗0, w)Ω for all w ∈ C∞0 (Ω). (4.118)

Moreover,

‖u∗0‖Hs,ϕ,(0)(Ω) ≤ c(‖u0‖2Hs,ϕ,(0)(Ω)

+ ‖f‖2Hs−2q,ϕ,(0)(Ω)

)1/2, (4.119)

where the number c = c(s, ϕ) > 0 is independent of u0 and f.


In the Sobolev case of ϕ ≡ 1, Theorems 4.23 and 4.24 were proved byYa.A.Roitberg in [207] (see also his monograph [209, Theorems 6.1.1 and 6.2.1]).Note that Ya. A. Roitberg made use of somewhat other statements, which areequivalent to ours. We describe their differences.

In the statement of Theorem 4.23, Ya. A. Roitberg [209, Theorem 6.1.1],instead of (4.115), uses the condition (for ϕ ≡ 1)

(u0, L+w)Ω = (f, w)Ω for all w ∈ H2q,1/ϕ

0 (Ω) ∩H2q−s,1/ϕ,(0)(Ω). (4.120)

This is a tantamount change, as we will show in Lemma 4.5.Further, using Theorem 4.23 and setting u∗ := I−1L (u∗0, f), we can refor-

mulate Theorem 4.24 in the following equivalent form: For arbitrary distri-butions (4.117) that satisfy condition (4.118), there exists a unique elementu∗ ∈ Hs,ϕ,(2q)(Ω) such that

u∗0 = u0 in Ω and Lu∗ = f in Hs−2q,ϕ,(0)(Ω).

Moreover,

‖u∗‖Hs,ϕ,(2q)(Ω) ≤ c(‖u0‖2Hs,ϕ,(0)(Ω)

+ ‖f‖2Hs−2q,ϕ,(0)(Ω)

)1/2.

Here, u∗0 is the initial component of the vector T2qu∗. This equivalent statementof Theorem 4.24 is used by Ya. A. Roitberg in [209, Theorem 6.2.1] for ϕ ≡ 1.

4.3.2 Proof of results

We derive Theorems 4.23 and 4.24 from the Sobolev case of ϕ ≡ 1 with the useof interpolation. First, we prove Lemma 4.5 mentioned above.

Lemma 4.5. Let s ∈ R and ϕ ∈ M. Then, for arbitrarily given distributions(4.117), conditions (4.115) and (4.120) are equivalent. If s > −1/2, then condi-tions (4.115) and (4.116) are equivalent. They are also equivalent for s = −1/2in the case of ϕ ≡ 1.

Proof. First, we show that (4.115)⇔ (4.120). By virtue of Theorem 3.20(i),we have (4.120) ⇒ (4.115). We prove the converse. Assume that condition(4.115) is satisfied. We separately consider the case of s ≥ 0 and the caseof s < 0.

The case of s ≥ 0. By virtue of Theorem 3.20(i), we have

H2q,1/ϕ0 (Ω) ∩H2q−s,1/ϕ,(0)(Ω) = H

2q,1/ϕ0 (Ω) = H

2q,1/ϕν,2q (Ω). (4.121)

We approximate an arbitrary distribution w belonging to space (4.121) by a cer-tain sequence of functions wj ∈ C∞ν,2q(Ω ) with respect to the norm in the spaceH2q,1/ϕ(Ω). According to condition (4.115), we have

(u0, L+wj)Ω = (f, wj)Ω for all j ∈ N. (4.122)


Passing to the limit as j →∞, we obtain condition (4.120). This follows frominclusions (4.117) and the following limits:

limj→∞

L+wj = L+w in H0,1/ϕ(Ω) → H−s,1/ϕ,(0)(Ω),

limj→∞

wj = w in H2q,1/ϕ(Ω) → H2q−s,1/ϕ,(0)(Ω).

The case of s < 0. By virtue of Theorem 3.20(i) and relation (3.119), we have

H2q,1/ϕ0 (Ω) ∩H2q−s,1/ϕ,(0)(Ω) = H

2q−s,1/ϕν,2q (Ω). (4.123)

We approximate an arbitrary distribution w belonging to space (4.123) by a cer-tain sequence of functions wj ∈ C∞ν,2q(Ω ) with respect to the norm in the spaceH2q−s,1/ϕ(Ω). According to condition (4.115), equalities (4.122) hold. Passingin them to the limit as j → ∞, we again obtain (4.120). This follows from(4.117) and the limits

limj→∞

L+wj = L+w in H−s,1/ϕ(Ω) = H−s,1/ϕ,(0)(Ω),

limj→∞

wj = w in H2q−s,1/ϕ(Ω) = H2q−s,1/ϕ,(0)(Ω).

We have proved that (4.115)⇔ (4.120) for any s ∈ R.Now, assuming that s > −1/2, we prove the equivalence of conditions (4.115)

and (4.116). Implication (4.115) ⇒ (4.116) is obvious. Let us prove the con-verse. Assume that condition (4.116) is satisfied. By virtue of the result provedabove, it suffices to show that (4.116) ⇒ (4.120). Using Theorem 3.20(i) andthe inequality s > −1/2, we obtain the equality

H2q,1/ϕ0 (Ω) ∩H2q−s,1/ϕ,(0)(Ω) = H

λ,1/ϕ0 (Ω), (4.124)

where λ := max2q, 2q − s. Therefore, each distribution w belonging tospace (4.124) can be approximated by a certain sequence of functions wj ∈C∞0 (Ω) with respect to the norm in the space Hλ,1/ϕ(Ω). According to condi-tion (4.116), equality (4.122) is true. Hence, passing to the limit as j → ∞,we obtain condition (4.120). This follows from inclusions (4.117) and the limits

limj→∞

L+wj = L+w in Hλ−2q,1/ϕ(Ω) → H−s,1/ϕ,(0)(Ω),

limj→∞

wj = w in Hλ,1/ϕ(Ω) → H2q−s,1/ϕ,(0)(Ω).

Thus, we have proved that (4.115)⇔ (4.116) for s > −1/2.


Finally, if s = −1/2 and ϕ ≡ 1, then equality (4.124) remains true by virtueof [258, Theorem 4.7.1 (a)]. Repeating the reasonings given in the previousparagraph, we obtain equivalence (4.115)⇔ (4.116) in this case as well.


To prove Theorem 4.23, we need the following lemma on projectors.

Lemma 4.6. For each number σ ∈ R, there exists a projector Πσ of the spaceHσ,(0)(Ω)×Hσ−2q,(0)(Ω) onto the subspace Kσ,L(Ω) such that Πσ is an extensionof the map Πλ for σ < λ.

Proof. Let σ ∈ R. First, we establish a useful equality. Passing to the limitand using Theorem 4.13, we conclude that the Green formula (3.6) remainstrue for arbitrary distributions u ∈ Hσ,(2q)(Ω) and v ∈ H2q−σ,(2q)(Ω), namely

(Lu, v0)Ω +

q∑j=1

(Bju, C+j v)Γ = (u0, L

+v)Ω +

q∑j=1

(Cju, B+j v)Γ. (4.125)

Here, the components of the sesquilinear forms (·, ·)Ω and (·, ·)Γ belong to thefollowing spaces, which are mutually dual with respect to these forms:

Lu ∈ Hσ−2q,(0)(Ω), (v0, v1, . . . , v2q) := T2qv, v0 ∈ H2q−σ,(0)(Ω),

(u0, u1, . . . , u2q) := T2qu, u0 ∈ Hσ,(0)(Ω), L+v ∈ H−σ,(0)(Ω),

Bju ∈ Hσ−mj−1/2(Γ), C+j v ∈ H

−σ+mj+1/2(Γ),

Cju ∈ Hσ−2q+m+j +1/2(Γ), B+

j v ∈ H2q−σ−m+

j −1/2(Γ).

Recall that ordL = ordL+ = 2q and, by virtue of (3.7),

ordBj = mj ≤ 2q − 1, ordC+j = 2q − 1−mj ,

ordB+j =: m+

j ≤ 2q − 1, ordCj = 2q − 1−m+j .

Further, since B+1 , . . . , B

+q , C

+1 , . . . , C

+q is the Dirichlet system of order 2q

(see [121, Part 2, Theorem 2.1]), the following statement [209, Lemma 6.1.2] istrue for it. The bounded operator(B+

1 , . . . , B+q , C

+1 , . . . , C

+q

): H2q−σ,(2q)(Ω)

→q⊕j=1

H2q−σ−m+j −1/2(Γ)⊕

q⊕j=1

H−σ+mj+1/2(Γ)


has the linear bounded right inverse operator

Φσ :q⊕j=1

H2q−σ−m+j −1/2(Γ)⊕

q⊕j=1

H−σ+mj+1/2(Γ)→ H2q−σ,(2q)(Ω) (4.126)

such that Φσ is a restriction of the operator Φλ for σ < λ.For an arbitrary vector

h := (0, . . . , 0, h1, . . . , hq) ∈ 0q ⊕q⊕j=1

H−σ+mj+1/2(Γ), (4.127)

we set v := Φσh ∈ H2q−σ,(2q)(Ω) in the Green formula (4.125). Since

B+j v = 0, C+

j v = hj for each j ∈ 1, . . . , q,

we obtain the useful equality

(u0, L+

Φσh)Ω − (Lu, (Φσh)0)Ω =

q∑j=1

(Bju, hj)Γ. (4.128)

Here, the element u ∈ Hσ,(2q)(Ω) and the vector h of the form (4.127) arearbitrary. As before,

u0 ∈ Hσ,(0)(Ω) and (Φσh)0 = v0 ∈ H2q−σ,(0)(Ω)

are the initial components of the vectors T2qu and T2qΦσh = T2qv, respectively.Now we construct the projector Πσ. To this end, we use the following five

mappings. Let an arbitrary vector

(u0, f) ∈ Hσ,(0)(Ω)⊕Hσ−2q,(0)(Ω) (4.129)

be chosen. Let ‖ · ‖σ denote the norm in space (4.129).Mappings 1 and 2. Consider the following decomposition of the space

Hσ,(0)(Ω)

in the direct sum of (closed) subspaces (see the proof of Lemma 4.4):

Hσ,(0)(Ω) = N uu′0 ∈ Hσ,(0)(Ω) : (u′0, w)Ω = 0 for all w ∈ N

. (4.130)

It exists because N is a finite-dimensional subspace of Hσ,(0)(Ω). Let Ψσ denotethe projector of the space Hσ(0)(Ω) onto the subspace N parallel to the second


term of sum (4.130). Since N ⊂ C∞(Ω ), Ψσ is the extension of the projectorΨλ for σ < λ. We define linear mappings 1 and 2 by the relations

(u0, f) 7→ (Ψσu0, 0) ∈ Kσ,L(Ω), (4.131)

(u0, f) 7→ (u0 −Ψσu0, f) =: (u′0, f). (4.132)

The inclusion in (4.131) follows from the inclusions Ψσu0 ∈ N ⊂ C∞(Ω ) andthe definition of the space Kσ,L(Ω). Namely,(

(Ψσu0, L+w)Ω = (LΨσu0, w)Ω = (0, w)Ω for all w ∈ C∞ν,2q(Ω)

)⇒ (Ψσu0, 0) ∈ Kσ,L(Ω).

Here, the first equality is obtained by the Green formula (4.61), integrals overΓ being absent because of w ∈ C∞ν,2q(Ω ).

Mapping 3. Consider relation (4.128). For the pair (u′0, f), we construct thefunctional

lσ(h) := (u′0, L+

Φσh)Ω − (f, (Φσh)0)Ω, h being vector (4.127). (4.133)

This functional is bounded by virtue of the following chain of inequalities:

|lσ(h)| ≤ |(u′0, L+Φσh)Ω|+ |(f, (Φσh)0)Ω|

≤ ‖u′0‖Hσ,(0)(Ω) ‖L+

Φσh‖H−σ,(0)(Ω)

+ ‖f‖Hσ−2q,(0)(Ω) ‖(Φσh)0‖H−σ+2q,(0)(Ω)

≤(c1 ‖u′0‖Hσ,(0)(Ω) + c2 ‖f‖Hσ−2q,(0)(Ω)

)‖Φσh‖H−σ+2q,(2q)(Ω)

≤ (c1 + c2) c3 ‖(u′0, f)‖σ( q∑j=1

‖hj‖2H−σ+mj+1/2(Γ)

)1/2

.

Here, c1, c2, and c3 are, respectively, the norms of the operators

L+ : H−σ+2q,(2q)(Ω)→ H−σ,(0)(Ω),

T2q : H−σ+2q,(2q)(Ω)→ H−σ+2q,(0)(Ω)⊕2q⊕j=1

H−σ+2q−j+1/2(Γ),

and operator (4.126). Thus, lσ is an antilinear bounded functional on thespace

⊕qj=1 H

−σ+mj+1/2(Γ) and, furthermore, its norm satisfies the inequality


‖lσ‖ ≤ c4 ‖(u′0, f)‖σ, where c4 := (c1 + c2) c3. By virtue of Theorem 2.3(v),(∃! g = (g1, . . . , gq) ∈

q⊕j=1

Hσ−mj−1/2(Γ)

): lσ(h) =

q∑j=1

(gj , hj)Γ. (4.134)

Moreover,( q∑j=1

‖gj‖2Hσ−mj−1/2(Γ)

)1/2

‖lσ‖ ≤ c4 ‖(u′0, f)‖σ. (4.135)

We define linear mapping 3 by the relation

Rσ : (u′0, f) 7→ lσ 7→ g. (4.136)

We show that Rσ is an extension of the mapping Rλ for σ < λ. If

(u′0, f) ∈ Hλ,(0)(Ω)⊕Hλ−2q,(0)(Ω),

then, parallel with lσ, the functional lλ is defined. Moreover, lσ is a restrictionof the operator lλ because Φσ is a restriction of the operator Φλ. In particular,lσ(h) = lλ(h) for any vector (h1, . . . , hq) ∈ (C∞(Γ))q. By virtue of (4.134)and (4.136), this yields Rσ(u′0, f) = Rλ(u

′0, f), i.e., Rσ is an extension of the

mapping Rλ.We define mapping 4 with the use of isomorphism (4.93) (Theorem 4.15) as

follows:(f, g) 7→ (L,B)−1Q+(f, g) =: ω ∈ P (Hσ,(2q)(Ω)). (4.137)

Recall that f ∈ Hσ−2q,(0)(Ω) and that the vector g satisfies condition (4.134).The linear mapping (4.137) does not depend on σ and is bounded; namely,

‖ω‖Hσ,(2q)(Ω) ≤ c5 ‖Q+(f, g)‖Hσ,(0)(Ω,Γ) ≤ c5 c6 ‖(f, g)‖Hσ,(0)(Ω,Γ). (4.138)

Here, c5 is the norm of the inverse of (4.93), and c6 is the norm of the projectorQ+ acting on the space Hσ,(0)(Ω,Γ).

We construct mapping 5 based on the ϕ ≡ 1 case of Theorem 4.23 proved byYa. A. Roitberg [209, Theorem 6.1.1]:

ω 7→ IL ω ∈ Kσ,L(Ω), ω ∈ P (Hσ,(2q)(Ω)). (4.139)

This mapping is independent of σ and satisfies the two-sided estimate

‖IL ω‖σ ‖ω‖Hσ,(2q)(Ω). (4.140)

Now, using mappings 1–5, we define the operator Πσ on vectors (4.129) asfollows:

Πσ : (u0, f) 7→ (Ψσu0, 0) + IL ω. (4.141)


Here, we recall that

ω = (L,B)−1Q+(f, g), g = Rσ(u′0, f), and u′0 = u0 −Ψσu0. (4.142)

The operator Πσ is linear because mappings 1–5 are linear. It is boundedon space (4.129) by virtue of estimates (4.135), (4.138), and (4.140) and theboundedness of the projector Ψσ on the space Hσ,(0)(Ω). In addition, inclusions(4.131) and (4.139) yield the property Πσ(u0, f) ∈ Kσ,L(Ω). Thus, we have thebounded linear operator

Πσ : Hσ,(0)(Ω)⊕Hσ−2q,(0)(Ω)→ Kσ,L(Ω).

Since, with the decrease in the parameter σ, mappings 1–5 extend, Πσ is anextension of the operator Πλ for σ < λ.

It remains to show that Πσ is a projector onto the subspace Kσ,L(Ω), i.e.,Πσ(u0, f) = (u0, f) for any vector (u0, f) ∈ Kσ,L(Ω). We arbitrarily choosesuch a vector. By virtue of (4.131) and (4.132), we have the inclusion (u′0, f) ∈Kσ,L(Ω). As above, u′0 = u0 −Ψσu0. Therefore, according to the ϕ ≡ 1 case ofTheorem 4.23, (

∃! u′ ∈ Hσ,(2q)(Ω))

: IL u′ = (u′0, f). (4.143)

This equality means the following:

u′0 is the initial component of T2qu′ = (u′0, u′1, . . . , u

′2q), (4.144)

Lu′ = f in Hσ−2q,(0)(Ω). (4.145)

We show that u′ = ω, where the element ω ∈ P (Hσ,(2q)(Ω)) is defined accordingto (4.142). Recall that, by virtue of (4.133) and (4.134), the following inequalityis true for an arbitrary vector h of the form (4.127):

lσ(h) := (u′0, L+

Φσh)Ω − (f, (Φσh)0)Ω =

q∑j=1

(gj , hj)Γ.

On the other hand, setting u := u′ in formula (4.128) and substituting relations(4.144) and (4.145) into this formula we obtain one more equality

(u′0, L+

Φσh)Ω − (f, (Φσh)0)Ω =

q∑j=1

(Bju′, hj)Γ.

It follows from these equalities that Bju′ = gj for every j ∈ 1, . . . , q. Togetherwith (4.145), this means the equality (L,B)u′ = (f, g). Therefore, by virtue ofTheorem 4.15, the equality Q+(f, g) = (f, g) is true. In addition, by virtueof Theorem 4.15, it follows from relations (4.144) and (4.132) and the definition


of the projector Ψσ that Pu′ = u′. Using isomorphism (4.93), we can write theequality u′ = (L,B)−1Q+(f, g). Together with (4.142), it yields the requiredequality u′ = ω.

Now, by virtue of (4.143), we have IL ω = IL u′ = (u′0, f). It follows from

this result and relations (4.141) and (4.142) that

Πσ(u0, f) = (Ψσu0, 0) + IL ω = (Ψσu0, 0) + (u′0, f) = (u0, f)

for an arbitrary vector (u0, f) ∈ Kσ,L(Ω). Thus, the required projector Πσ isconstructed.


Based on this lemma, we prove Theorem 4.23.

Proof of Theorem 4.23. Let s ∈ R, ϕ ∈ M, and ε > 0. In the Sobolevcase of ϕ ≡ 1, this theorem is proved by Ya. A. Roitberg [209, Theorem 6.1.1].Thus, mapping (4.113) extends by continuity to the isomorphisms

IL : Hs∓ε,(2q)(Ω)↔ Ks∓ε,L(Ω).

Let us use the interpolation with the function parameter ψ from Theorem 1.14,where ε = δ. We obtain one more isomorphism

IL : [Hs−ε,(2q)(Ω), Hs+ε,(2q)(Ω)]ψ ↔ [Ks−ε,L(Ω),Ks+ε,L(Ω)]ψ. (4.146)

We describe spaces in which it acts. By Theorem 4.22,

[Hs−ε,(2q)(Ω), Hs+ε,(2q)(Ω)]ψ= Hs,ϕ,(2q)(Ω). (4.147)

Further, by virtue of Lemma 4.6 and Theorem 1.6, we can interpolate a pair ofsubspaces Ks∓ε,L(Ω) as follows:

[Ks−ε,L(Ω),Ks+ε,L(Ω)]ψ

= [Hs−ε,(0)(Ω)⊕Hs−ε−2q,(0)(Ω), Hs+ε,(0)(Ω)⊕Hs+ε−2q,(0)(Ω)]ψ

∩Ks−ε,L(Ω)

= Hs,ϕ,(0)(Ω)⊕Hs−2q,ϕ,(0)(Ω) ∩Ks−ε,L(Ω) = Ks,ϕ,L(Ω).

Here,we also used Theorems 1.5 and 3.10 and the definition of the setsKs−ε,L(Ω)and Ks,ϕ,L(Ω). By virtue of Theorem 1.6, Ks,ϕ,L(Ω) is a subspace of the spaceHs,ϕ,(0)(Ω)⊕Hs−2q,ϕ,(0)(Ω), and, moreover, equalities (4.147) and

[Ks−ε,L(Ω),Ks+ε,L(Ω)]ψ= Ks,ϕ,L(Ω) (4.148)


are true up to equivalence of norms. Substituting these equalities into (4.146),we obtain isomorphism (4.114). Assertion (ii) of Theorem 4.23 is proved and,hence, assertion (i) is also proved.


To prove Theorem 4.24, we need one more lemma on projectors. Let σ ∈ Rand ϕ ∈M. We set

K0σ,ϕ,L(Ω) :=

(u0, f) ∈ Hσ,ϕ,(0)(Ω)⊕Hσ−2q,ϕ,(0)(Ω) : (4.116) is true

.

By virtue of Theorem 3.9(iii), K0σ,ϕ,L(Ω) is a subspace of the spaceHσ,ϕ,(0)(Ω)⊕

Hσ−2q,ϕ,(0)(Ω). Moreover, Kσ,ϕ,L(Ω) ⊆ K0σ,ϕ,L(Ω).

Lemma 4.7. Let numbers r ∈ N and σ ∈ R satisfy the condition

−r − 1/2 ≤ σ < −r + 1/2. (4.149)

Then there exist a projector Π(r)σ of the space Hσ,(0)(Ω) ⊕ Hσ−2q,(0)(Ω) onto

the subspace K0σ,L(Ω) such that Π

(r)σ is an extension of the mapping Π

(r)λ if

σ < λ < −r + 1/2.

Proof. First, we establish a useful equality. Let (u0, f) ∈ K0σ,L(Ω). In the

Sobolev case of ϕ ≡ 1, Theorem 4.24 is proved by Ya. A. Roitberg [209, The-orem 6.2.1] for any parameter s < −1/2. Therefore, there exists a uniquepair (u∗0, f) ∈ Kσ,L(Ω) such that u0 = u∗0 in the domain Ω. Since u0, u∗0 ∈Hσ,(0)(Ω) = Hσ

Ω(Rn) and supp(u0 − u∗0) ⊆ Γ, inequality (4.149) yields the fol-

lowing representation for the distribution u0−u∗0 (see, e.g., [209, Lemma 6.2.2]).There exists a unique vector

ω∗ = (ω∗1, . . . , ω∗r ) ∈

r⊕j=1

Hσ+j−1/2(Γ) (4.150)

such that

(u0 − u∗0, w)Ω =r∑j=1

(ω∗j , Dj−1ν w)Γ (4.151)

for any w ∈ H−σ,(0)(Ω) = H−σ(Ω).In the Green formula (4.125), we set

u := u∗ := I−1L (u∗0, f) ∈ Hσ,(2q)(Ω)

and use the property

Lu = Lu∗ = f ∈ Hσ−2q,(0)(Ω) = Hσ−2qΩ

(Rn).


For any v ∈ H2q−σ,(2q)(Ω) = H2q−σ(Ω) (see Theorem 4.12(iii)), we obtain theequality

(f, v)Ω +

q∑j=1

(Bju∗, C+

j v)Γ = (u∗0, L+v)Ω +

q∑j=1

(Cju∗, B+

j v)Γ. (4.152)

Here, u∗0 and v = v0 are the initial components of the vectors T2qu∗ and T2qv,respectively.

Using relations (4.152) and (4.151) with w := L+v ∈ H−σ(Ω), we obtain theequality

(f, v)Ω+

q∑j=1

(Bju∗, C+

j v)Γ

= (u0, L+v)Ω −

r∑j=1

(ω∗j , Dj−1ν L+v)Γ +

q∑j=1

(Cju∗, B+

j v)Γ. (4.153)

The boundary operators B+j , C

+j , with j = 1, . . . , q, and (Dj−1

ν L+·) Γ, withj = 1, . . . , r, form a Dirichlet system of order 2q + r. According to [209,Lemma 6.1.2], this system has the following property. The bounded linearoperator

(B+1 , . . . , B

+q , C

+1 , . . . , C

+q , (L

+·) Γ, . . . , (Dr−1ν L+·) Γ) : H2q−σ,(2q+r)(Ω)

→q⊕j=1

H2q−σ−m+j −1/2(Γ)⊕

q⊕j=1

H−σ+mj+1/2(Γ)⊕r⊕j=1

H−σ−j+1/2(Γ)

has the bounded right inverse operator

Φ(r)σ :

q⊕j=1

H2q−σ−m+j −1/2(Γ)⊕

q⊕j=1

H−σ+mj+1/2(Γ)⊕r⊕j=1

H−σ−j+1/2(Γ)

→ H2q−σ,(2q+r)(Ω) = H2q−σ(Ω) (4.154)

such that Φ(r)σ is a restriction of the operator Φ

(r)λ for σ < λ < −r+ 1/2. Here,

ordC+j = 2q − 1 − mj and ordB+

j =: m+j . Moreover, the equality in (4.154)

follows from Theorem 4.12(iii) in view of the condition σ < −r + 1/2.For an arbitrary vector

h := (0, . . . , 0, h1, . . . , hr) ∈ 02q ⊕r⊕j=1

H−σ−j+1/2(Γ), (4.155)


we set v := Φ(r)σ h ∈ H2q−σ(Ω) in relation (4.153). Since

B+j v = 0, C+

j v = 0 on Γ for every j ∈ 1, . . . , q,

Dj−1ν L+v = hj on Γ for every j ∈ 1, . . . , r,

we obtain the equality

(u0, L+

Φ(r)σ h)Ω − (f,Φ(r)

σ h)Ω =r∑j=1

(ω∗j , hj)Γ. (4.156)

Recall that, here, the pair (u0, f) ∈ K0σ,L(Ω) and the vector h of the form

(4.155) are arbitrary, while the vector (ω∗1, . . . , ω∗r ) satisfies conditions (4.150)

and (4.151) and is uniquely defined by them on the basis of the pair (u0, f).Now we proceed to the construction of the projector Π

(r)σ .We arbitrarily chose

a vector (u0, f) that satisfies inclusion (4.129). As before, let ‖ · ‖σ denote thenorm in space (4.129). Consider the functional

l(r)σ (h) := (u0, L+


σ h)Ω, h being vector (4.155). (4.157)

The functional is bounded by virtue of the chain of inequalities

|l(r)σ (h)| ≤ |(u0, L+Φ

(r)σ h)Ω|+ |(f,Φ(r)

σ h)Ω|

≤ ‖u0‖Hσ,(0)(Ω) ‖L+

Φ(r)σ h‖H−σ,(0)(Ω)

+ ‖f‖Hσ−2q,(0)(Ω) ‖Φ(r)σ h‖H−σ+2q,(0)(Ω)

≤(c1 ‖u0‖Hσ,(0)(Ω) + ‖f‖Hσ−2q,(0)(Ω)

)‖Φ(r)

σ h‖H−σ+2q(Ω)

≤ (c1 + 1) c2 ‖(u0, f)‖σ( r∑j=1

‖hj‖2H−σ−j+1/2(Γ)

)1/2

.

Here, c1 is the norm of the operator

L+ : H−σ+2q(Ω)→ H−σ(Ω) = H−σ,(0)(Ω),

and c2 is the norm of operator (4.154). Thus, l(r)σ is a bounded antilinearfunctional on the space

⊕rj=1 H

−σ−j+1/2(Γ) and, furthermore, its norm satisfies

the inequality ‖l(r)σ ‖ ≤ c3 ‖(u0, f)‖σ, where c3 := (c1 + 1) c2. By virtue ofTheorem 2.3(v),(

∃! (ω1, . . . , ωr) ∈r⊕j=1

Hσ+j−1/2(Γ)

): l(r)σ (h) =

r∑j=1

(ωj , hj)Γ. (4.158)


Moreover, ( r∑j=1

‖ωj‖2Hσ+j−1/2(Γ)

)1/2

‖l(r)σ ‖ ≤ c3 ‖(u0, f)‖σ. (4.159)

Based on the vector (ω1, . . . , ωr), we form the distribution ω′ ∈ Hσ,(0)(Ω) bythe formula

(ω′, w)Ω :=r∑j=1

(ωj , Dj−1ν w)Γ, (4.160)

where w ∈ H−σ,(0)(Ω) = H−σ(Ω) is an arbitrary distribution. This definitionis reasonable by virtue of Theorems 2.3(v), 3.5, and 3.8(iii). Indeed, since−σ > r − 1/2, by virtue of condition (4.149), we have the bounded operators

Dj−1ν : H−σ(Ω)→ H−σ−j+1/2(Γ) for j = 1, . . . , r.

Therefore, (ω′, ·)Ω is a bounded antilinear functional on the space H−σ(Ω),which yields the inclusion

ω′ ∈ Hσ,(0)(Ω) = HσΩ(Rn).

Moreover, the inclusion suppω′ ⊆ Γ and the two-sided estimate

‖ω′‖Hσ,(0)(Ω) ( r∑j=1

‖ωj‖2Hσ+j−1/2(Γ)

)1/2

(4.161)

are true (see, e.g., [209, Lemma 6.2.2]).On vectors (4.129), we define the linear mapping ϒ

(r)σ by the formula

ϒ(r)σ : (u0, f) 7→ l(r)σ 7→ (ω1, . . . , ωr) 7→ ω′. (4.162)

By virtue of estimates (4.159) and (4.161), we have the bounded operator

ϒ(r)σ : Hσ,(0)(Ω)⊕Hσ−2q,(0)(Ω)→ Hσ,(0)(Ω). (4.163)

Let us show that ϒ(r)σ is an extension of the mapping ϒ

(r)λ for σ < λ < −r+1/2.

If(u0, f) ∈ Hλ,(0)(Ω)⊕Hλ−2q,(0)(Ω),

then, parallel with l(r)σ , the functional l(r)λ is defined. Moreover, l(r)σ is a re-

striction of the functional l(r)λ because Φ(r)σ is a restriction of the operator Φ

(r)λ .

In particular, l(r)σ (h) = l(r)λ (h) for any vector (h1, . . . , hq) ∈ (C∞(Γ))q. By virtue

of (4.158) and (4.160), this yields ϒ(r)σ (u0, f) = ϒ

(r)λ (u0, f), i.e., ϒ

(r)σ is an ex-

tension of the mapping ϒ(r)λ .


We define the linear mapping Π(r)σ with the use of the operator ϒ

(r)σ and the

projector Πσ in Lemma 4.6 as follows:

Π(r)σ (u0, f) := (ω′, 0) + Πσ(u0 − ω′, f), ω′ := ϒ

(r)σ (u0, f). (4.164)

Here, (u0, f) is an arbitrary vector (4.129). As was mentioned above, the in-clusions ω′ ∈ Hσ,(0)(Ω) and suppω′ ⊆ Γ are true. Therefore, (ω′, 0) ∈ K0

σ,L(Ω).This fact, together with the boundedness of operator (4.163), Lemma 4.6, andthe inclusion Kσ,L(Ω) ⊆ K0

σ,L(Ω), yields the boundedness of the operator

Π(r)σ : Hσ,(0)(Ω)⊕Hσ−2q,(0)(Ω)→ K0

σ,L(Ω).

Moreover, since both the operators ϒ(r)σ and Πσ extend provided the parameter

σ decreases, we conclude that Π(r)σ is an extension of the operator Π

(r)λ for

σ < λ < −r + 1/2.

It remains to show that Π(r)σ is a projector onto the subspace K0

σ,L(Ω), i.e.,

Π(r)σ (u0, f) = (u0, f)

for any vector (u0, f)∈K0σ,L(Ω).We arbitrarily choose such a vector. According

to Theorem 4.24 in the case of ϕ ≡ 1, we state that, for (u0, f), there existsa unique pair (u∗0, f) ∈ Kσ,L(Ω) satisfying condition (4.118). Let us show thatu0−u∗0 = ω′, where ω′ := ϒ

(r)σ (u0, f). For an arbitrary distribution w ∈ H−σ(Ω),

we define the vector h by formula (4.155) with

hj := (Dj−1ν w)Γ ∈ H−σ−j+1/2(Γ) for each j ∈ 1, . . . , r. (4.165)

As was mentioned above, the traces hj exist in view of condition (4.149). Wealso recall that condition (4.118) means the inclusion supp(u0−u∗0) ⊆ Γ, whichyields equality (4.151) for a certain (unique) vector (4.150). Now, using rela-tions (4.151), (4.156), and (4.165), we obtain the equality

(u0 − u∗0, w)Ω = (u0, L+


σ h)Ω.

On the other hand, by virtue of (4.157), (4.158), (4.160), and (4.165), we get

(ω′, w)Ω = l(r)σ (h) = (u0, L+


σ h)Ω.

Therefore,

(u0 − u∗0, w)Ω = (ω′, w)Ω for any w ∈ H−σ(Ω).

By virtue of Theorem 3.8(iii), this is equivalent to the equality

u0 − u∗0 = ω′ := ϒ(r)σ (u0, f) in Hσ,(0)(Ω).


Using this result and (4.164), we obtain

Π(r)σ (u0, f) = (ω′, 0) + Πσ(u0 − ω′, f)

= (ω′, 0) + Πσ(u∗0, f)

= (ω′, 0) + (u∗0, f)

= (u0, f)

for an arbitrary vector (u0, f) ∈ K0σ,L(Ω). Here, we also used the inclusion

(u∗0, f) ∈ Kσ,L(Ω) and the property of Πσ to be a projector onto the subspaceKσ,L(Ω). Thus, the required projector Π

(r)σ is constructed.


Using this lemma, we prove Theorem 4.24.

Proof of Theorem 4.24. Let s < −1/2, s + 1/2 /∈ Z, and ϕ ∈ M. Wechoose numbers r ∈ N and ε ∈ (0, 1/2) such that

−r − 1/2 < s∓ ε < −r + 1/2. (4.166)

As was mentioned above, Theorem 4.24 was proved by Ya. A. Roitberg [209,Theorem 6.2.1] in the Sobolev case of ϕ ≡ 1,. Therefore, we can introduce thelinear mapping G : (u0, f) 7→ (u∗0, f), where (u0, f) ∈ K0

s−ε,L(Ω) and, more-over, (u∗0, f) ∈ Ks−ε,L(Ω) satisfies (4.118). This mapping defines the boundedoperators

G : K0s∓ε,L(Ω)→ Ks∓ε,L(Ω). (4.167)

Theorem 4.24 will be proved for any ϕ ∈M if we establish that G is a boundedoperator from K0

s,ϕ,L(Ω) to Ks,ϕ,L(Ω). We do this using the interpolation withthe parameter ψ in Theorem 1.14 for ε = δ. Applying this theorem to (4.167),we obtain one more bounded operator

G : [K0s−ε,L(Ω),K0

s+ε,L(Ω)]ψ → [Ks−ε,L(Ω),Ks+ε,L(Ω)]ψ. (4.168)

Let us describe the spaces between which this operator acts. Based onLemma 4.7, inequality (4.166), and Theorem 1.6, we can interpolate a pairof the subspaces K0

s∓ε,L(Ω) as follows:

[K0s−ε,L(Ω),K0

s+ε,L(Ω)]ψ

= [Hs−ε,(0)(Ω)⊕Hs−ε−2q,(0)(Ω), Hs+ε,(0)(Ω)⊕Hs+ε−2q,(0)(Ω)]ψ

∩K0s−ε,L(Ω)


= Hs,ϕ,(0)(Ω)⊕Hs−2q,ϕ,(0)(Ω) ∩K0s−ε,L(Ω)

= K0s,ϕ,L(Ω).

Here, we also used Theorems 1.5 and 3.10 and the definition of the spacesK0s−ε,L(Ω) and K0

s,ϕ,L(Ω). The obtained equality of spaces

[K0s−ε,L(Ω),K0

s+ε,L(Ω)]ψ = K0s,ϕ,L(Ω) (4.169)

is true up to equivalence of norms. Substituting equalities (4.169) and (4.148)into relation (4.167), we obtain the bounded operator

G : K0s,ϕ,L(Ω)→ Ks,ϕ,L(Ω),

which is what to be proved.Theorem 4.24 is proved.

Remark 4.8. Theorems 4.23, 4.24 and Lemmas 4.6, 4.7 are proved under theassumption that problem (4.1) is regularly elliptic. Actually, they are true foran arbitrary differential expression L which is properly elliptic in the domain Ω.This follows from the fact that, for L, there exists a regular elliptic boundary-value problem, e.g., the Dirichlet problem,.

4.4 Generalization of the Lions–Magenes theorems

In this section, we establish individual theorems on solvability of the regularelliptic boundary-value problem (4.1) in scales of Sobolev spaces. Unlike thegeneral theorems 4.1 and 4.14, in individual theorems, the domain of the oper-ator (L,B) depends on the coefficients of the elliptic expression L. J.-L. Lionsand E. Magenes [119, 120, 121, 126], proved a number of individual theoremsfor the operator (L,B) acting in Sobolev spaces containing nonregular distri-butions. We prove a certain general form of the Lions–Magenes theorems anddetermine a general condition, for the space of the right-hand sides of the ellip-tic equation, under which the operator (L,B) is bounded and Fredholm in thecorresponding pair of Hilbert spaces. We also indicate wide classes of spacesthat satisfy this condition. They contain the spaces used by J.-L. Lions andE. Magenes and many other spaces including some Hördmander spaces. Un-like the general theorem 4.14, we consider individual theorems in which thesolution and the right-hand side of the elliptic equation are distributions in thedomain Ω.

The results of this section simulate statements and methods for the proof ofindividual theorems on solvability of elliptic boundary-value problems in scalesof Hörmander spaces. These theorems will be established in Section 4.5.

Section 4.4 Generalization of the Lions–Magenes theorems 227

4.4.1 Lions–Magenes theorems

First, we indicate an important difference in the definitions of the negativeorder Sobolev spaces over a Euclidean domain used here and in J.-L. Lions andE. Magenes’ papers cited above. Recall that, following [258, 256], we definethe Sobolev space of an arbitrary order s ∈ R over the domain Ω by theformulas:

Hs(Ω) := u := w Ω : w ∈ Hs(Rn), (4.170)

‖u‖Hs(Ω) := inf‖w‖Hs(Rn) : w ∈ Hs(Rn), w = u on Ω

(4.171)

(see Definition 3.2 in the case of ϕ ≡ 1). J.-L. Lions and E. Magenes use thisdefinition only for s ≥ 0. For s < 0, they consider the dual space (H−s0 (Ω))′ asthe Sobolev space of order s over the domain Ω. Recall that H−s0 (Ω) is a closureof the set C∞0 (Ω) in the space H−s(Ω), the duality of spaces being consideredwith respect to the inner product in L2(Ω). It is known [258, Theorem 4.8.2]that these definitions give the same space (up to equivalence of norms) if theorder s < 0 is not half-integer. For half-integer s < 0, different spaces areobtained.

In this section, we follow J.-L. Lions and E. Magenes and define the Sobolevspaces over the domain Ω by the formula

Hs(Ω) :=

Hs(Ω) for s ≥ 0,

(H−s0 (Ω))′ for s < 0.(4.172)

On the left-hand side of the definition, we use the Roman upright type ofletter H rather than italic used in definition (4.170). Here, the dual space(H−s0 (Ω))′ consists of antilinear functionals.

The functionals belonging to the space Hs(Ω) with s < 0 are uniquely definedby their values on the test functions in C∞0 (Ω). Therefore, it is correctly toidentify these functionals with distributions given in the domain Ω. Moreover,for any s < 0,

Hs(Ω) =w Ω : w ∈ Hs

Ω(Rn)

, (4.173)

‖u‖Hs(Ω) = inf‖w‖Hs(Rn) : w ∈ Hs

Ω(Rn), w = u in Ω

(4.174)

(see [121, Chap. 1, Remark 12.5]). It follows from these relations and formulas(4.170) and (4.171) that Hs(Ω) → Hs(Ω) continuously and that C∞0 (Ω) isdense in Hs(Ω) for each s < 0.

As was mentioned above,

Hs(Ω) = Hs(Ω) ⇔ s ∈ R \ −1/2,−3/2,−5/2, . . .. (4.175)


If the parameter s < 0 is half-integer, then the space Hs(Ω) is narrower thanthe space Hs(Ω).

Note that, for arbitrary s ∈ R and ε > 0, the compact and dense embeddingHs+ε(Ω) → Hs(Ω) is true.

Now we proceed to the Lions–Magenes theorems. To this end, we first recallthe classical general theorem (Proposition 3.1) on solvability of the boundary-value problem (4.1) in the scale of positive Sobolev spaces.

Theorem A. The mapping

u 7→ (Lu,Bu), u ∈ C∞(Ω ), (4.176)

extends uniquely (by continuity) to the Fredholm bounded operator

(L,B) : Hσ+2q(Ω)→ Hσ(Ω)⊕q⊕j=1

Hσ+2q−mj−1/2(Γ) =: Hσ(Ω,Γ) (4.177)

for any real σ ≥ 0. The kernel of this operator coincides with N and the rangeconsists of all vectors (f, g1, . . . , gq) ∈ Hσ(Ω,Γ) that satisfy condition (4.4).The index of operator (4.177) is equal to dimN − dimN+ and does not dependon σ.

Here and below, unlike the previous theorems on solvability of the boundary-value problem (4.1), it is convenient to represent the index s, which defines thesmoothness of the domain of (L,B), in the form of the sum s = σ + 2q. Notethat Theorem A has been already extended over the refined Sobolev scale inSection 4.1.1 (Theorem 4.1).

As has been mentioned in Section 4.2.1, Theorem A is not true if σ runsthrough the negative semiaxis. J.-L. Lions and E. Magenes suggested to replaceHσ+2q(Ω), as the domain of (L,B), with the narrower space

Dσ+2qL,X (Ω) := u ∈ Hσ+2q(Ω) : Lu ∈ Xσ(Ω), (4.178)

where Xσ(Ω) is a certain Hilbert space continuously embedded in Hσ(Ω). Hereand below, the image Lu of u ∈ D′(Ω) is understood in the sense of the theoryof distributions. In space (4.178), we introduce the graph inner product

(u1, u2)Dσ+2qL,X (Ω)

:= (u1, u2)Hσ+2q(Ω) + (Lu1, Lu2)Xσ(Ω) (4.179)

and the corresponding norm.The space Dσ+2q

L,X (Ω) with inner product (4.179) is complete. Indeed, if a se-quence (uk) is fundamental in Dσ+2q

L,X (Ω), then, since the sets Hσ+2q(Ω) andXσ(Ω) are complete, there exist limits u := limuk in Hσ+2q(Ω) → D′(Ω)and f := limLuk in Xσ(Ω) → D′(Ω) (embeddings are continuous). Since


the differential operator L is continuous on D′(Ω), we have Lu = limLuk inD′(Ω) by virtue of the first limit. Using the second limit, we obtain the equal-ity Lu = f ∈ Xσ(Ω). Therefore, u ∈ Dσ+2q

L,X (Ω) and limuk = u in the spaceDσ+2qL,X (Ω), i.e., this space is complete.J.-L. Lions and E. Magenes [119, 120, 121, 126] give some important examples

of spaces Xσ(Ω) such that the mapping (4.176) extends by continuity to theFredholm bounded operator

(L,B) : Dσ+2qL,X (Ω)→ Xσ(Ω)⊕

q⊕j=1

Hσ+2q−mj−1/2(Γ)

=: Xσ(Ω,Γ) (4.180)

if σ < 0. Unlike Theorem A, the domain of operator (4.180) depends, togetherwith topology, on coefficients of the elliptic expression L. Therefore, theoremson properties of operator (4.180) are individual theorems on solvability of theboundary-value problem (4.1).

We formulate two individual theorems established by J.-L. Lions and E. Ma-genes.

Theorem LM1 [119, 120]. Let σ < 0 and let Xσ(Ω) := L2(Ω). Thenmapping (4.176) extends uniquely (by continuity) to the Fredholm bounded op-erator (4.180). Its kernel coincides with N and the range consists of all vectors(f, g1, . . . , gq) ∈ Xσ(Ω,Γ) that satisfy condition (4.4). The index of operator(4.180) is equal to dimN − dimN+ and does not depend on σ.

Here, we should select the case of σ = −2q, which is important in the spectraltheory of elliptic operators [67, 68, 69, 137, 138]. In this case, the space

D0L,L2

(Ω) = u ∈ L2(Ω) : Lu ∈ L2(Ω) (4.181)

is the domain of the maximum operator corresponding to the differential ex-pression L acting in the space L2(Ω). Note that even if all coefficients of theexpression L are constant, then space (4.181) essentially depends on each ofthem. This follows from L. Hörmander’s result [80, Theorem 3.1] cited below.

Let L andM be two linear differential expressions with constant coefficients.Then if D0

L,L2(Ω) ⊆ D0

M,L2(Ω), then either M = αL + β for certain α, β ∈ C

or L and M are polynomials in the operator of differentiation along a certainvector e and, furthermore, ordM ≤ ordL. Note that the second possibility iseliminated for elliptic operators.

To formulate the second Lions–Magenes theorem, we consider the weightspace

%Hσ(Ω) := f = %v : v ∈ Hσ(Ω), (4.182)


where σ < 0, while the function % ∈ C∞(Ω) is positive. We endow this spacewith the inner product

(f1, f2)%Hσ(Ω) := (%−1f1, %−1f2)Hσ(Ω) (4.183)

and the corresponding norm. The space %Hσ(Ω) is complete and embeddedcontinuously in D′(Ω). This follows from the fact that the operator of multi-plication by the function % is continuous on D′(Ω) and sets an isomorphism ofthe complete space Hσ(Ω) onto %Hσ(Ω).

We consider weight functions of the form % := %−σ1 , where

%1 ∈ C∞(Ω ), %1 > 0 in Ω,

%1(x) = dist(x,Γ) in a neighborhood of Γ.(4.184)

Theorem LM2 [121, Chap. 2, Sec. 2.3]. Let σ < 0 and let

Xσ(Ω) :=

%−σ1 Hσ(Ω) if σ + 1/2 /∈ Z,

[ %−σ+1/21 Hσ−1/2(Ω), %

−σ−1/21 Hσ+1/2(Ω)]t1/2

if σ + 1/2 ∈ Z.

(4.185)

Then the conclusion of Theorem LM1 remains true.

Remark 4.9. J.-L. Lions and E. Magenes [121, Chap. 2, Sec. 6.3] use thefollowing Hilbert space Ξσ(Ω) as Xσ(Ω). For integer σ ≥ 0, the space Ξσ(Ω)is defined by (3.53). For fractional σ > 0, it is defined by interpolation withpower parameter

Ξσ(Ω) :=

[Ξ[σ](Ω),Ξ[σ]+1(Ω)

]tσ

.

Finally, for negative σ < 0, it is defined by passing to the dual space (withrespect to the inner product in L2(Ω)), namely, Ξσ(Ω) := (Ξ−σ(Ω))′. For anarbitrary σ < 0, the space Ξσ(Ω) coincides (up to equivalence of norms) withthe right-hand side of relation (4.185). This follows from J.-L. Lions and E. Ma-genes’ result [121, Chap. 2, Corollary 7.4].

4.4.2 Key individual theorem

Here, we prove the key individual theorem on solvability of the boundary-valueproblem (4.1). According to this theorem, operator (4.180) is well-defined,bounded, and Fredholm for every σ < 0 if the Hilbert space Xσ(Ω) → D′(Ω)satisfies condition Iσ given below. This theorem is a key for proving otherindividual theorems.


Condition Iσ. The set X∞(Ω) := Xσ(Ω)∩C∞(Ω ) is dense in Xσ(Ω), andthere exists a number c > 0 such that

‖Of‖Hσ(Rn) ≤ c ‖f‖Xσ(Ω) for any f ∈ X∞(Ω). (4.186)

Here, Of(x) = f(x) for every x ∈ Ω, and Of(x) = 0 for every x ∈ Rn \Ω.

Note that the smaller σ is the weaker condition Iσ will be, for the same spaceXσ(Ω).

Remark 4.10. Ya. A. Roitberg [207, Sec. 2.4] considers a condition for thespace Xσ(Ω) which is somewhat stronger than our condition Iσ. In addition,he assumes that C∞(Ω ) ⊂ Xσ(Ω). Under this condition, Ya. A. Roitbergproves the boundedness of operator (4.180) for all σ < 0 (see [207, Sec. 2.4] and[209, Remark 6.2.2]). Note that this condition does not include the importantcase Xσ(Ω) = 0 and several weight spaces Xσ(Ω) = %Hσ(Ω) consideredbelow.

Let us formulate the key individual theorem.

Theorem 4.25. Let σ < 0 and let Xσ(Ω) be an arbitrary Hilbert space that iscontinuously embedded in D′(Ω) and satisfies condition Iσ. Then the followingassertions are true:

(i) The setD∞L,X(Ω) := u ∈ C∞(Ω ) : Lu ∈ Xσ(Ω)

is dense in the space Dσ+2qL,X (Ω).

(ii) The mapping u → (Lu,Bu), where u ∈ D∞L,X(Ω), extends uniquely (bycontinuity) to the bounded linear operator (4.180).

(iii) Operator (4.180) is Fredholm. Its kernel coincides with N , and the rangeconsists of all vectors (f, g1, . . . , gq) ∈ Xσ(Ω,Γ) that satisfy condition (4.4).

(iv) If the set O(X∞(Ω)) is dense in the space HσΩ(Rn), then the index of

operator (4.180) is equal to dimN − dimN+.

Proof. The proof is based on Theorems 4.14, 4.18, 4.23, and 4.24 for theSobolev case of ϕ ≡ 1, we taking s = σ + 2q. Recall that, in this case, theyare proved by Ya. A. Roitberg [209], Theorem 4.24 being established for anys < −1/2.

It follows from the condition of the theorem that the mapping f 7→ Of,where f ∈ X∞(Ω), is extended by continuity to the bounded linear operator

O : Xσ(Ω)→ HσΩ(Rn) = Hσ,(0)(Ω). (4.187)


This operator is injective. Indeed, let Of = 0 for a certain distribution f ∈Xσ(Ω). We choose a sequence (fk) ⊂ X∞(Ω) such that fk → f in Xσ(Ω) →D′(Ω). Then Ofk → 0 in Hσ

Ω(Rn) → S ′(Rn), which yields

(f, v)Ω = limk→∞

(fk, v)Ω = limk→∞

(Ofk, v)Ω = 0 for all v ∈ C∞0 (Ω).

Thus, f = 0 as a distribution from the space Xσ(Ω) → D′(Ω), i.e., the operator(4.187) is injective. It defines the continuous embedding Xσ(Ω) → Hσ,(0)(Ω).

According to Theorem 4.13, for every element u ∈ Hσ+2q,(2q)(Ω), we defineLu ∈ Hσ,(0)(Ω) passing to the limit. We set

Dσ+2q,(2q)L,X (Ω) :=

u ∈ Hσ+2q,(2q)(Ω) : Lu ∈ Xσ(Ω)

.

In the space Dσ+2q,(2q)L,X (Ω), we introduce the graph inner product

(u1, u2)Dσ+2q,(2q)L,X (Ω)

:= (u1, u2)Hσ+2q,(2q)(Ω) + (Lu1, Lu2)Xσ(Ω).

The space Dσ+2q,(2q)L,X (Ω) is complete with respect to this inner product. In-

deed, let a sequence (uk) be fundamental in Dσ+2q,(2q)L,X (Ω). Since both the

spaces Hσ+2q,(2q)(Ω) and Xσ(Ω) are complete, there exist limits u := limukin Hσ+2q,(2q)(Ω) and f := limLuk in Xσ(Ω). The first limit yields the con-vergence limLuk = Lu in Hσ,(0)(Ω). Hence, using the second limit and con-tinuous embedding (4.187), we conclude that Lu = f ∈ Xσ(Ω). Therefore,u ∈ Dσ+2q,(2q)

L,X (Ω) and limuk = u in Dσ+2q,(2q)L,X (Ω), i.e., the space Dσ+2q,(2q)

L,X (Ω)is complete.

Based on Theorem 4.14, we conclude that the restriction of the operator

(L,B) : Hσ+2q,(2q)(Ω)→ Hσ,(0)(Ω)⊕q⊕j=1

Hσ+2q−mj−1/2(Γ)

=: Hσ,(0)(Ω,Γ) (4.188)

onto the space Dσ+2q,(2q)L,X (Ω) is the bounded operator

(L,B) : Dσ+2q,(2q)L,X (Ω)→ Xσ(Ω,Γ). (4.189)

The kernel of operator (4.189) is equal toN , and the range consists of all vectors(f, g1, . . . , gq) ∈ Xσ(Ω,Γ) that satisfy condition (4.4). Therefore, operator(4.189) is Fredholm, its kernel being of a dimension β ≤ dimN+.

In addition, if the set O(X∞(Ω) is dense in the space HσΩ(Rn), then β =

dimN+. Indeed, let Λ denote operator (4.188) and let Λ0 denote the narrower


operator (4.189); consider the adjoint operators Λ∗ and Λ∗0. Since the contin-uous embedding Xσ(Ω,Γ) → Hσ,(0)(Ω,Γ) is dense, we have ker Λ∗0 ⊇ ker Λ∗.Therefore,

β = dim coker Λ0 = dim ker Λ∗0 ≥ dim ker Λ

∗ = dim coker Λ = dimN+.

Hence, β = dimN+, and the index of operator (4.189) is equal to dimN −dimN+ in the considered case.

Let us show that the set D∞L,X(Ω) is dense in the space Dσ+2q,(2q)L,X (Ω). Using

the density of X∞(Ω)×(C∞(Γ))q in Xσ(Ω,Γ) and applying the Gohberg–Kreinlemma [65, Lemma 2.1], we can write

Xσ(Ω,Γ) = (L,B)(Dσ+2q,(2q)L,X (Ω)

)uQ(Ω,Γ), (4.190)

where Q(Ω,Γ) is a certain finite-dimensional space that satisfies the condition

Q(Ω,Γ) ⊂ X∞(Ω)× (C∞(Γ))q. (4.191)

Let Π denote the projection operator of the space Xσ(Ω,Γ) onto the first sum-mand in (4.190) parallel to the second.

Let u ∈ Dσ+2q,(2q)L,X (Ω). We approximate F := (L,B)u by a certain sequence

(Fk) ⊂ X∞(Ω) × (C∞(Γ))q with respect to the norm in the space Xσ(Ω,Γ).Then

limk→∞

ΠFk = ΠF = F = (L,B)u in Xσ(Ω,Γ), (4.192)

where, by virtue of (4.191),

(ΠFk) ⊂ X∞(Ω)× (C∞(Γ))q (4.193)

The Fredholm operator (4.189) naturally generates the isomorphism

Λ0 := (L,B) : Dσ+2q,(2q)L,X (Ω)/N ↔ Π(Xσ(Ω,Γ)).

By virtue of (4.192), we have

limk→∞

Λ−10 ΠFk = u+ w : w ∈ N in D

σ+2q,(2q)L,X (Ω)/N.

Therefore, there exists a sequence of representatives uk ∈ Dσ+2q,(2q)L,X (Ω) of the

cosets Λ−10 ΠFk such that

limk→∞

uk = u in Dσ+2q,(2q)L,X (Ω). (4.194)

Moreover, by virtue of (4.193), we have

(L,B)uk = ΠFk ∈ C∞(Ω )× (C∞(Γ))q.


Hence, using Theorem 4.18 and the Sobolev embedding theorem, we get

uk ∈⋂s>2q

Hs,(2q)(Ω) =⋂s>2q

Hs(Ω) = C∞(Ω ).

Thus, (uk) ⊂ D∞L,X(Ω) in (4.194). This proves that D∞L,X(Ω) is dense in

Dσ+2q,(2q)L,X (Ω).

Further, we separately consider the case of −2q − 1/2 ≤ σ < 0 and the caseof σ < −2q − 1/2.

The first case, namely, −2q − 1/2 ≤ σ < 0. Then Hσ+2q,(0)(Ω) = Hσ+2q(Ω).Indeed, since Hλ

0 (Ω) = Hλ(Ω) for 0 ≤ λ ≤ 1/2 (see [121, Chap. 1, Theo-rem 11.1]), we have

Hs,(0)(Ω) = (H−s(Ω))′ = (H−s0 (Ω))′ = Hs(Ω) for − 1/2 ≤ s < 0.

This yieldsHs,(0)(Ω) = Hs(Ω) for s ≥ −1/2 (4.195)

with equality of norms.We use Theorem 4.23 and consider the mapping I0 : u 7→ u0, where u ∈

Dσ+2q,(2q)L,X (Ω) and (u0, f) := ILu. The mapping defines the isomorphism

I0 : Dσ+2q,(2q)L,X (Ω)↔ Dσ+2q

L,X (Ω). (4.196)

Indeed, for arbitrary distributions

u0 ∈ Hσ+2q,(0)(Ω) = Hσ+2q(Ω) and f ∈ Xσ(Ω) → Hσ,(0)(Ω),

conditions (4.115) and (4.116) are equivalent by virtue of Lemma 4.5. Con-dition (4.116) means that Lu0 = f in the sense of the equality of distri-butions given in the domain Ω. Hence, using Theorem 4.23, we obtain theequality I0(D

σ+2q,(2q)L,X (Ω)) = Dσ+2q

L,X (Ω). In addition, for u ∈ Dσ+2q,(2q)L,X (Ω) and

(u0, f) = ILu, we have the following equivalence of norms:

‖u‖2Dσ+2q,(2q)L,X (Ω)

= ‖u‖2Hσ+2q,(2q)(Ω)

+ ‖f‖2Xσ(Ω)

‖u0‖2Hσ+2q,(0)(Ω)+ ‖f‖2

Hσ,(0)(Ω)+ ‖f‖2Xσ(Ω)

‖u0‖2Hσ+2q(Ω) + ‖f‖2Xσ(Ω) = ‖u0‖

2Dσ+2qL,X (Ω)

.

Therefore, the mapping I0 defines isomorphism (4.196).It follows from properties of operators (4.196) and (4.189) (recall that the

latter is denoted by Λ0) that the operator

Λ0I−10 : Dσ+2q

L,X (Ω)→ Xσ(Ω,Γ) (4.197)


is bounded and Fredholm. Moreover, its kernel, the range and index arethe same as for operator (4.189). Since the operator I0 establishes a one-to-one mapping of the set D∞L,X(Ω) onto itself, this set is dense in the spaceDσ+2qL,X (Ω) and operator (4.197) is an extension by continuity of the mapping

u→ (Lu,Bu), where u ∈ D∞L,X(Ω). Theorem 4.25 is proved in the first case.The second case, namely, σ < −2q − 1/2. Then Hσ+2q,(0)(Ω) = Hσ+2q

Ω(Rn).

Set Rw := w Ω for w ∈ D′(Rn). Let us prove that the mapping I0 : u 7→ Ru0,

where u ∈ Dσ+2q,(2q)L,X (Ω) and (u0, f) := ILu, defines isomorphism (4.196) in

the considered case (in the first case, Ru0 = u0). We use Theorem 4.23 andnote that (4.115) ⇒ (4.116). For an arbitrary u ∈ D

σ+2q,(2q)L,X (Ω), we have

Ru0 ∈ Hσ+2q(Ω) [see (4.173)], f = Lu ∈ Xσ(Ω) and

(Ru0, L+v)Ω = (u0, L

+v)Ω = (f, v)Ω for all v ∈ C∞0 (Ω),

i.e., LRu0 = f in the sense of the equality of distributions given in the do-main Ω. Therefore, I0u = Ru0 ∈ Dσ+2q

L,X (Ω). In addition, by virtue of (4.174)and the definition of the space Hσ+2q,(2q)(Ω), we have the estimate

‖I0u‖2Dσ+2qL,X (Ω)

= ‖Ru0‖2Hσ+2q(Ω) + ‖f‖2Xσ(Ω)

≤ ‖u0‖2Hσ+2q(Rn) + ‖f‖2Xσ(Ω) ≤ ‖u‖

2

Dσ+2q,(2q)L,X (Ω)

.

Therefore, the operator I0 : Dσ+2q,(2q)L,X (Ω)→ Dσ+2q

L,X (Ω) is bounded.We show that this operator is bijective. Let ω ∈ Dσ+2q

L,X (Ω) and let f := Lω ∈Xσ(Ω). By virtue of (4.173), there exists a distribution u0 ∈ Hσ+2q

Ω(Rn) such

that ω = Ru0. In this case, equality (4.116) is true, namely,

(u0, L+v)Ω = (ω,L+v)Ω = (f, v)Ω for all v ∈ C∞0 (Ω).

According to Theorem 4.24, for the distributions u0 ∈ Hσ+2q,(0)(Ω) and f ∈Xσ(Ω) → Hσ,(0)(Ω), there exists a unique pair (u∗0, f) ∈ Kσ+2q,L(Ω) thatsatisfies condition (4.118). (In the Sobolev case of ϕ ≡ 1, this theorem is truefor every s ∈ R as was shown by Ya. A. Roitberg [209, Theorem 6.2.1].) UsingTheorem 4.23, we can write

u∗ := I−1L (u∗0, f) ∈ Dσ+2q,(2q)L,X (Ω) and I0u

∗ = Ru∗0 = Ru0 = ω.

The element u∗ is a unique preimage of the distribution ω in the mapping I0.Indeed, if I0u′ = ω for a certain u′ ∈ Dσ+2q,(2q)

L,X (Ω), then the pair (u′0, f′) :=

ILu′ ∈ Kσ+2q,L(Ω) satisfies the conditions

f ′ = LRu′0 = Lω = f and

(u′0, v)Ω = (ω, v)Ω = (u0, v)Ω for all v ∈ C∞0 (Ω).


Therefore, by virtue of Theorem 4.24, the pairs (u′0, f′) = (u′0, f) and (u∗0, f)

are equal and, hence, their preimages u′ and u∗ in the mapping IL are equal.Thus, the linear bounded operator (4.196) is bijective in the considered case.

Therefore, by the Banach theorem on inverse operator, it is an isomorphismNow, using the Fredholm property of operator (4.189) and reasoning as in thefirst case, we complete the proof in the second case.


Remark 4.11. A statement analogous to Theorem 4.25 is proved in [126,Theorem 6.16] in the case of half-integer σ ≤ −2q and the Dirichlet boundary-value problem. Moreover, certain other conditions depending on the consideredboundary-value problem are imposed on the space Xσ(Ω). Our condition Iσ isindependent of this problem.

Remark 4.12. In Theorem 4.25, as in the individual theorems LM1 and LM2,the solution and the right-hand side of the elliptic equation Lu = f are dis-tributions given in Ω. Other individual theorems are proved in the papers byYa. A. Roitberg [204] and Yu. V. Kostarchuk, Ya. A. Roitberg [97]. In thesetheorems, u and/or f are not distributions in the domain Ω.

It is obvious that the space Xσ(Ω) := 0 satisfies condition Iσ. In thiscase, Theorem 4.25 describes properties of the semihomogeneous boundary-value problem (4.1), where f = 0, and is true for any σ ∈ R in view of Theo-rem 3.11. Despite the fact that Hs(Ω) 6= Hs(Ω) for half-integer s = σ+2q < 0,we have

u ∈ Hs(Ω) : Lu = 0 in Ω = u ∈ Hs(Ω) : Lu = 0 in Ω. (4.198)

In addition, the norms in the spaces Hs(Ω) and Hs(Ω) are equivalent on thedistributions u indicated in (4.198).

Further, we consider various applications of Theorem 4.25 caused by a specificchoice of the space Xσ(Ω). We separately study the cases where inner productSobolev spaces or their weight analogs are taken as Xσ(Ω).

4.4.3 Individual theorem for Sobolev spaces

The theorem given below describes all inner product (nonweight) Sobolev spacesthat satisfy condition Iσ.

Theorem 4.26. Let σ < 0 and λ ∈ R. The space Xσ(Ω) := Hλ(Ω) satisfiescondition Iσ if and only if

λ ≥ maxσ,−1/2. (4.199)


Proof. Note that, since Xσ(Ω) = Hλ(Ω), the set X∞(Ω) = C∞(Ω ) is densein the space Xσ(Ω).

Sufficiency. Assume that inequality (4.199) is true. Then, in view of (4.195),

Hλ(Ω) = Hλ,(0)(Ω) → Hσ,(0)(Ω)

together with topology. Recall that each function f ∈ C∞(Ω ) is identifiedwith the functional (f, · )Ω, which, in turn, is identified with the function Ofconsidered as an element of the space Hσ

Ω(Rn) = Hσ,(0)(Ω). Therefore,

‖Of‖Hσ(Rn) ≤ c ‖f‖Hλ(Ω) for any f ∈ C∞(Ω ),

where the number c > 0 is independent of f. Sufficiency is proved.Necessity. Assume that the space Xσ(Ω) := Hλ(Ω) satisfies condition Iσ. If

λ ≥ 0, then inequality (4.199) is true. For this reason, we restrict ourselves tothe condition λ < 0. Operator (4.187) defines the dense continuous embeddingHλ(Ω) → Hσ,(0)(Ω). This yields

H−σ(Ω) = (Hσ,(0)(Ω))′ ⊆ (Hλ(Ω))′ = H−λ0 (Ω).

Therefore, −σ ≥ −λ. In addition, −λ ≤ 1/2 because if −λ > 1/2, then thefunction f ≡ 1 ∈ H−σ(Ω) does not belong to the space H−λ0 (Ω) by virtue ofTheorem 3.20(i). Thus, σ satisfies inequality (4.199). Necessity is proved.


The following individual theorem on solvability of the boundary-value prob-lem (4.1) in Sobolev spaces follows from Theorems 4.25 and 4.26.

Theorem 4.27. Let σ < 0 and let λ ≥ maxσ,−1/2. Then mapping (4.176)extends uniquely (by continuity) to the Fredholm bounded operator

(L,B) : u ∈ Hσ+2q(Ω) : Lu ∈ Hλ(Ω)

→ Hλ(Ω)⊕q⊕j=1

Hσ+2q−mj−1/2(Γ) (4.200)

whose domain is a Hilbert space with respect to the graph norm(‖u‖2Hσ+2q(Ω) + ‖Lu‖

2Hλ(Ω)

)1/2.

The index of operator (4.200) is equal to dim N −dimN+ and does not dependon σ and λ.


Proof. The boundedness and the Fredholm property of operator (4.200)immediately follow from Theorems 4.25 and 4.26, where Xσ(Ω) := Hλ(Ω)and D∞L,X(Ω) = C∞(Ω ). In addition, since the set O(C∞(Ω )) identified withC∞(Ω ) is dense in the space Hσ

Ω(Rn) = Hσ,(0)(Ω), it implies from Theo-

rem 4.25(iv) that the index of operator (4.200) is equal to dimN − dimN+

and does not depend on σ and λ.Theorem 4.27 is proved.

Theorem LM1 is the special case of Theorem 4.27 where σ = 0, i.e., Xσ(Ω) =L2(Ω).

In Theorem 4.27, certain spaces Xσ(Ω) containing L2(Ω) are admitted. Thespace Xσ(Ω) = H−1/2(Ω) with σ ≤ −1/2 is the widest among them.

Note that, in the case of −1/2 ≤ σ = λ < 0, the domain of operator (4.200)is independent of L. Indeed, if −1/2 < σ = λ < 0, then, since the number σis not half-integer, we have the bounded operator L : Hσ+2q(Ω) → Hσ(Ω) (see[121, Chap. 1, Proposition 12.1]). This implies that the domain of operator(4.200) coincides with Hσ+2q(Ω) and is independent of L. In this case, Theo-rem 4.27 coincides with J.-L. Lions and E. Magenes’ theorem [121, Chap. 2,Theorem 7.5], which is proved under assumption that N = N+ = 0.

If σ = λ = −1/2, then the previous arguments are not correct becausethe space H−1/2(Ω) is narrower than L(H2q−1/2(Ω)). However, it follows fromTheorem 4.25(i), equality (4.195), and Theorem 4.23 that the domain of oper-ator (4.200) is the completion of the set of functions u ∈ C∞(Ω ) with respectto the norm

‖u‖2H2q−1/2(Ω)+ ‖Lu‖2H−1/2(Ω)

= ‖u‖2H2q−1/2,(0)(Ω)

+ ‖Lu‖2H−1/2,(0)(Ω)

‖u‖H2q−1/2,(2q)(Ω).

Therefore, it coincides with the space H2q−1/2,(2q)(Ω), which is independentof L.

In the conclusion of this section, note that the Hörmander spaces Hλ,ϕ(Ω),where λ > maxσ,−1/2 and ϕ ∈M, also satisfy condition Iσ for σ < 0. Indi-vidual theorems involving Hörmander spaces will be considered in Section 4.5.

4.4.4 Individual theorem for weight spaces

In Theorem 4.26, always Xσ(Ω) ⊆ H−1/2(Ω). The space Xσ(Ω) that containsa wide class of distributions f /∈ H−1/2(Ω) and satisfies condition Iσ can beobtained using certain weight spaces %Hσ(Ω).

The function % : Ω → C is called a multiplier in the space Hλ(Ω), whereλ ≥ 0, if the operator of multiplication by % continuously maps this spaceinto itself. We denote this operator by M%. The description of the class of all


multipliers in the space Hλ(Ω) = Hλ(Ω) for λ ≥ 0 is given in the monographby V. G. Maz’ya and T. O. Shaposhnikova [131, Sec. 6.3.3].

Let σ < −1/2. Consider the following condition on the function %.

Condition IIσ. The function % is a multiplier in the space H−σ(Ω), and

Djν % = 0 on Γ for all j ∈ Z with 0 ≤ j < −σ − 1/2. (4.201)

Note that if % is a multiplier in the space H−σ(Ω), then obviously % ∈ H−σ(Ω).Therefore, by virtue of Theorem 3.5, there exists a trace

(Djν%)Γ ∈ H−σ−j−1/2(Γ)

for any integer j ≥ 0 such that −σ − j − 1/2 > 0. Therefore, condition IIσ iswell-formulated.

Theorem 4.28. Let σ < −1/2 and let a function % ∈ C∞(Ω) be positive.The space Xσ(Ω) := %Hσ(Ω) satisfies condition Iσ if and only if the function %satisfies condition IIσ.

To prove this theorem, we need the following lemma.

Lemma 4.8. Let σ < −1/2. Multiplication by a function % is a bounded oper-ator

M% : H−σ(Ω)→ H−σ0 (Ω) (4.202)

if and only if condition IIσ is satisfied.

Proof. Necessity. If the multiplication by % defines the bounded operator(4.202), then % is a multiplier in the space H−σ(Ω) and, in addition, % ∈H−σ0 (Ω). As known, [258, Theorem 4.7.1 (a)],

% ∈ H−σ0 (Ω) ⇔ (% ∈ H−σ(Ω) and (4.201) is true) (4.203)

This yields condition IIσ. Necessity is proved.Sufficiency. Let the function % satisfy condition IIσ. It is only necessary

to prove that %u ∈ H−σ0 (Ω) for any u ∈ H−σ(Ω). By virtue of (4.203), con-dition IIσ yields inclusion % ∈ H−σ0 (Ω). Let sequences (uk) ⊂ C∞(Ω ) and(%j) ⊂ C∞0 (Ω) be such that uk → u and %j → % in H−σ(Ω). Since the functions% and uk are multipliers in the space H−σ(Ω), we have

limk→∞

(%uk) = %u and limj→∞

(%juk) = %uk for all k in H−σ(Ω).

In view of %juk ∈ C∞0 (Ω), this implies that %u ∈ H−σ0 (Ω). Sufficiency is proved.Lemma 4.8 is proved.


Proof of Theorem 4.28. By condition, σ < −1/2 and the function % ∈C∞(Ω) is positive. LetM% andM%−1 denote the operators of multiplication by %and %−1, respectively. We have the isomorphism M% : Hσ(Ω)↔ %Hσ(Ω). Usingthis result and the density of C∞0 (Ω) in Hσ(Ω), we conclude that C∞0 (Ω) is densein Xσ(Ω) := %Hσ(Ω). Therefore, the wider set X∞(Ω) is dense in Xσ(Ω).

Let us define the inner product space

%−1H−σ0 (Ω) := f = %−1v : v ∈ H−σ0 (Ω),

(f1, f2)%−1H−σ0 (Ω) := (%f1, %f2)H−σ(Ω).

We have the isomorphism

M%−1 : H−σ0 (Ω)↔ %−1H−σ0 (Ω). (4.204)

Therefore, the space %−1H−σ0 (Ω) is complete, and the set C∞0 (Ω) is dense inthis space.

Note that

(%−1H−σ0 (Ω))′ = %Hσ(Ω) with equality of norms. (4.205)

Indeed, passing in (4.204) to the adjoint operator, we obtain the isomorphism

M%−1 : (%−1H−σ0 (Ω))′ ↔ (H−σ0 (Ω))′ = Hσ(Ω).

Using this result and the definition of the space %Hσ(Ω), we obtain one moreisomorphism

I =M%M%−1 : (%−1H−σ0 (Ω))′ ↔ %Hσ(Ω),

where I is the identity operator. Thus, we prove relation (4.205) because theused isomorphisms are isometric.

Now we can complete the proof by using the following arguments. By virtueof Lemma 4.8, condition IIσ is equivalent to the boundedness of operator(4.202), which, in view of (4.204), is equivalent to the continuous embeddingH−σ(Ω) → %−1H−σ0 (Ω). This embedding is dense. By virtue of (4.205), it isequivalent to the dense continuous embedding

%Hσ(Ω) = (%−1H−σ0 (Ω))′ → (H−σ(Ω))′ = HσΩ(Rn).

Finally, the continuous embedding %Hσ(Ω) → HσΩ(Rn) is equivalent to condi-

tion Iσ. Note that the last embedding is dense because the set C∞0 (Ω) is densein the space Hσ

Ω(Rn). Thus, the equivalence of conditions IIσ and Iσ is proved

for the space Xσ(Ω) = %Hσ(Ω).Theorem 4.28 is proved.

Theorems 4.25 and 4.28 lead to the following individual theorem on solvabil-ity of the boundary-value problem (4.1) in weight Sobolev spaces.


Theorem 4.29. Let σ < −1/2 and let a positive function % ∈ C∞(Ω) satisfycondition IIσ. Then the mapping u → (Lu,Bu), where u ∈ C∞(Ω ) and Lu ∈%Hσ(Ω), extends uniquely (by continuity) to the Fredholm bounded operator

(L,B) :u ∈ Hσ+2q(Ω) : Lu ∈ %Hσ(Ω)

→ %Hσ(Ω)⊕

q⊕j=1

Hσ+2q−mj−1/2(Γ) (4.206)

whose domain is a Hilbert space with respect to the graph norm(‖u‖2Hσ+2q(Ω) + ‖%

−1Lu‖2Hσ(Ω)

)1/2.

The index of operator (4.206) is equal to dimN − dimN+ and does not dependon σ and %.

Proof. The boundedness and the Fredholm property of operator (4.206)result immediately from Theorems 4.25 and 4.28, where Xσ(Ω) := %Hσ(Ω).In addition, since the set O(X∞(Ω)) containing C∞0 (Ω) is dense in the spaceHσ

Ω(Rn), it follows by Theorem 4.25(iv) that the index of operator (4.206) is

equal to dimN − dimN+ and does not depend on σ and %.Theorem 4.29 is proved.

The following theorem gives an important example of a positive function% ∈ C∞(Ω) that satisfies condition IIσ.

Theorem 4.30. Let a number σ < −1/2 and a function %1 that satisfies condi-tion (4.184) be given. Assume that either δ ≥ −σ−1/2 ∈ Z or δ > −σ−1/2 /∈ Z.Then the function % := %δ1 satisfies condition IIσ.

This implies that Theorem LM2 for half-integer σ < −1/2 is a special case ofthe individual theorem 4.29. Recall that, in Theorem LM2, the weight function% := %−σ1 is used.

Proof of Theorem 4.30. The function % = %δ1 satisfies condition (4.201)because %1 = 0 on Γ and δ ≥ −σ−1/2. Therefore, it remains to prove that %δ1 isa multiplier in the space H−σ(Ω) = H−σ(Ω). If the positive number δ is integer,then the function %δ1 belongs to the space C∞(Ω) and, hence, is a multiplier inH−σ(Ω). Further, assume that δ /∈ Z. Then, by condition, δ > −σ − 1/2.

It is easy to verify that the function ηδ(t) := tδ, 0 ≤ t ≤ 1, belongs to theSobolev space H−σ((0, 1)) (this will be shown in the next paragraph). Thenthis function has a certain extension over R belonging to H−σ(R). We preservethe notation ηδ for the extension. By R. Strichartz’s theorem [250] (see also[131, Sec. 2.2.9]) any function from the space H−σ(R) is a multiplier in thisspace if −σ > 1/2. Therefore, ηδ is a multiplier in H−σ(R). Then the function


ηδ,n(t′, tn) := ηδ(tn) of arguments t′ ∈ Rn−1 and tn ∈ R is a multiplier in

H−σ(Rn) [131, Sec. 2.4, Proposition 5]. It coincides with %δ1 in the special localcoordinates (x′, tn) near the boundary Γ. Here, x′ is the coordinate of a pointin the local chart of the surface Γ, and tn is the distance to Γ. This impliesthat %δ1 is a multiplier in each space H−σ(Ω ∩ Vj), where Vj : j = 1, . . . , ris a finite system of balls in Rn, of a sufficiently small radius ε, covering theboundary Γ [131, Sec. 6.4.1, Lemma 3]. We complement this system with the setV0 := x ∈ Ω : dist(x,Γ) > ε/2 and obtain the finite open covering of the closeddomain Ω. Let functions χj ∈ C∞0 (Vj), j = 0, 1, . . . , r, form a partition of unityon Ω subordinated to this covering. Since the multiplication by a function inthe class C∞0 (Vj) is a bounded operator on the space H−σ(Ω∩Vj), the functionχj%

δ1 is a multiplier in this space. Therefore, the function %δ1 =

∑rj=0 χj%

δ1 is

a multiplier in H−σ(Ω).It remains to show that ηδ ∈ H−σ((0, 1)). We use the internal description

of the space H−σ((0, 1)). If −σ ∈ Z, then the inclusion ηδ ∈ H−σ((0, 1))

is equivalent to the pair of inclusions ηδ ∈ L2((0, 1)) and η(−σ)δ ∈ L2((0, 1)).

It is obvious that the last two inclusions are true for δ > −σ − 1/2. There-fore, ηδ ∈ H−σ((0, 1)) in the considered case. If −σ /∈ Z, then the inclusionηδ ∈ H−σ((0, 1)) is equivalent to the following: ηδ ∈ H [−σ]((0, 1)) and

1∫0

1∫0

|D[−σ]t tδ −D[−σ]

τ τ δ|2

|t− τ |1+2−σ dt dτ <∞ (4.207)

(see, e.g., [1, Theorem 7.48]). As usual, [−σ] and −σ are, respectively, theintegral and fractional parts of the number −σ. Since δ > [−σ]− 1/2, we con-clude that ηδ ∈ H [−σ]((0, 1)) by the result proved above. In addition, inequality(4.207) is true by virtue of the following elementary lemma, which will be provedat the end of this subsection.

Lemma 4.9. Let α, β, γ ∈ R and, furthermore, α 6= 0 and γ > 0. Then

I(α, β, γ) :=

1∫0

1∫0

|tα − τα|γ

|t− τ |βdt dτ <∞ (4.208)

if and only if all three inequalities

αγ − β > −2, γ − β > −1, αγ > −1 (4.209)

are true.

Indeed, the double integral in (4.207) is equal to c I(α, β, γ), where c is a cer-tain positive number and α = δ − [−σ], β = 1 + 2−σ, and γ = 2. For the

Section 4.5 Hörmander spaces and individual theorems on solvability 243

numbers α, β, and γ, inequalities (4.209) are true, namely,

αγ − β = 2(δ + σ)− 1 > −2,

γ − β = 1− 2−σ > −1,

αγ = 2(δ − [−σ]) > −1.

In the first and third inequalities, we use the condition δ > −σ − 1/2. Thus,ηδ ∈ H−σ((0, 1)) in the case of noninteger σ − 1/2 as well.


It remains to prove Lemma 4.9.

Proof of Lemma 4.9. Changing variable λ := τ/t in the inner integral, weconclude after obvious transformations that

I(α, β, γ) = 2

1∫0

dt

t∫0

|tα − τα|γ

|t− τ |βdτ = 2

1∫0

tαγ−β+1dt

1∫0

|1− λα|γ

|1− λ|βdλ.

Here, the integral with respect to t is finite if and only if αγ − β > −2, andthe integral with respect to λ is finite if and only if αγ > −1 and γ − β > −1.Therefore, (4.208)⇔ (4.209).


4.5 Hörmander spaces and individual theoremson solvability

We discuss analogs of individual theorems 4.25, 4.27, and 4.29 for Hörmanderspaces that form the refined scale.

4.5.1 Key individual theorem for the refined scale

Let σ < 0 and let ϕ ∈ M. Assume that a Hilbert space Xσ,ϕ(Ω) embeddedcontinuously in D′(Ω) be given. Consider the following analog of condition Iσ:

Condition Iσ,ϕ. The set X∞(Ω) = Xσ,ϕ(Ω) ∩C∞(Ω ) is dense in Xσ,ϕ(Ω),and there exists a number c > 0 such that

‖Of‖Hσ,ϕ(Rn) ≤ c ‖f‖Xσ,ϕ(Ω) for any f ∈ X∞(Ω). (4.210)

We setDσ+2q,ϕL,X (Ω) := u ∈ Hσ+2q,ϕ(Ω) : Lu ∈ Xσ,ϕ(Ω).


The space Dσ+2q,ϕL,X (Ω) is endowed with the graph inner product

(u1, u2)Dσ+2q,ϕL,X (Ω)

:= (u1, u2)Hσ+2q,ϕ(Ω) + (Lu1, Lu2)Xσ,ϕ(Ω).

This space is complete, which is proved by analogy with the case of ϕ ≡ 1 (seeSubsection 4.4.1).

Let us formulate the key individual theorem.

Theorem 4.31. Let ϕ ∈M and let a number σ < 0 be such that

σ + 2q 6= 1/2− k for any k ∈ N. (4.211)

Let Xσ,ϕ(Ω) be an arbitrary Hilbert space that is continuously embedded inD′(Ω) and satisfies condition Iσ,ϕ. Then the following assertions are true:

(i) The setD∞L,X(Ω) := u ∈ C∞(Ω ) : Lu ∈ Xσ,ϕ(Ω)

is dense in the space Dσ+2q,ϕL,X (Ω).

(ii) The mapping u → (Lu,Bu), where u ∈ D∞L,X(Ω), extends uniquely (bycontinuity) to the bounded linear operator

(L,B) : Dσ+2q,ϕL,X (Ω)→ Xσ,ϕ(Ω)⊕

q⊕j=1

Hσ+2q−mj−1/2,ϕ(Γ)

=: Xσ,ϕ(Ω,Γ). (4.212)

(iii) Operator (4.212) is Fredholm. Its kernel coincides with N , and the rangeconsists of all vectors (f, g1, . . . , gq) ∈ Xσ,ϕ(Ω,Γ) that satisfy condi-tion (4.4).

(iv) If the set O(X∞(Ω)) is dense in the space Hσ,ϕ

Ω(Rn), then the index of

operator (4.212) is equal to dimN − dimN+.

This theorem can be proved by analogy with Theorem 4.25 if we use Theo-rems 4.14, 4.18, 4.23, and 4.24 for an arbitrary ϕ ∈ M and, moreover, applyTheorem 3.21 instead of relations (4.173) and (4.174). The additional condition(4.211) being absent in Theorem 4.25 is caused by the circumstance that theparameter s = σ + 2q is not half-integer in Theorems 3.21 and 4.24.

4.5.2 Other individual theorems

We give important applications of Theorem 4.31 caused by a specific choice ofthe space Xσ,ϕ(Ω). Let σ < 0 and ϕ ∈M.

Section 4.5 Hörmander spaces and individual theorems on solvability 245

The space Xσ,ϕ(Ω) := 0 satisfies condition Iσ,ϕ. In this case, Theorem 4.31describes properties of the semihomogeneous boundary-value problem (4.1),where f = 0, and is true for any s ∈ R, which was established in Theorem 3.11.

The space Xσ,ϕ(Ω) := L2(Ω) satisfies condition Iσ,ϕ as well. As has beenmentioned in Subsection 4.4.1, this choice of the space Xσ,ϕ(Ω) is important inthe spectral theory of elliptic operators

As Xσ,ϕ(Ω), we admit the choice of several Hörmander spaces. In view ofTheorem 4.17, the case σ < −1/2 is interesting here. Every Hörmander spaceXσ,ϕ(Ω) := Hλ,η(Ω), where λ > −1/2 and η ∈ M, satisfies condition Iσ,ϕ forσ < −1/2. Indeed, by virtue of Theorem 3.9(i) and (iv), the space Hλ,η(Ω) =Hλ,η,(0)(Ω) is continuously embedded in Hσ,ϕ,(0)(Ω) = Hσ,ϕ

Ω(Rn). This yields

inequality (4.210) in which X∞(Ω) = C∞(Ω ). Setting Xσ,ϕ(Ω) := Hλ,η(Ω) inthe key theorem 4.31, we obtain the following individual theorem on solvabilityof the boundary-value problem 4.1 in Hörmander spaces.

Theorem 4.32. Let a number σ < −1/2 satisfy condition (4.211) and letλ > −1/2 and ϕ, η ∈M. Then the mapping u 7→ (Lu,Bu), where u ∈ C∞(Ω ),extends uniquely (by continuity) to the Fredholm bounded operator

(L,B) :u ∈ Hσ+2q,ϕ(Ω) : Lu ∈ Hλ,η(Ω)

→ Hλ,η(Ω)⊕

q⊕j=1

Hσ+2q−mj−1/2,ϕ(Γ) (4.213)

whose domain is a Hilbert space with respect to the graph norm(‖u‖2Hσ+2q,ϕ(Ω) + ‖Lu‖

2Hλ,η(Ω)

)1/2.

The index of operator (4.213) is equal to dim N −dimN+ and does not dependon the parameters σ, ϕ and λ, η.

Note that, in Theorem 4.32, the solution and the right-hand side of the ellipticequation Lu = f can have different additional smoothness ϕ and η.

In Theorem 4.32, the space Xσ,ϕ(Ω) := Hλ,η(Ω) lies in H−1/2(Ω) becauseλ > −1/2. A space Xσ,ϕ(Ω) that contains a wide class of distributions f /∈H−1/2(Ω) and satisfies condition Iσ,ϕ can be obtained by using the weightHörmander spaces.

Let σ < −1/2 and ϕ ∈ M and let a function % ∈ C∞(Ω) be positive. Wedefine the inner product space

%Hσ,ϕ(Ω) := f = %v : v ∈ Hσ,ϕ(Ω) ,

(f1, f2)%Hσ,ϕ(Ω) := (%−1f1, %−1f2)Hσ,ϕ(Ω).


The multiplication by % defines the isomorphism

M% : Hσ,ϕ(Ω)↔ %Hσ,ϕ(Ω).

Therefore, the space %Hσ,ϕ(Ω) is complete (Hilbert) and continuously embed-ded in D′(Ω).

If the function % is a multiplier in the space H−σ,1/ϕ(Ω) and if (4.201) is true,then the space Xσ,ϕ(Ω) := %Hσ,ϕ(Ω) satisfies condition Iσ,ϕ. This is provedby analogy with Theorem 4.28 with the following difference: assertions (i)and (iii) of Theorem 3.20 are used instead of relations (4.203) and (4.172).Setting Xσ,ϕ(Ω) := %Hσ,ϕ(Ω) in the key theorem 4.31, we obtain the followingindividual theorem on solvability of the boundary-value problem (4.1) in spacesconnected with weight Hörmander spaces.

Theorem 4.33. Let a number σ < −1/2 satisfy condition (4.211), let ϕ ∈M,and let a function % ∈ C∞(Ω) be positive. Suppose that % is a multiplierin the space H−σ,1/ϕ(Ω) and satisfy condition (4.201). Then the mappingu → (Lu,Bu), where u ∈ C∞(Ω ) and Lu ∈ %Hσ,ϕ(Ω), extends uniquely (bycontinuity) to the Fredholm bounded operator

(L,B) :u ∈ Hσ+2q,ϕ(Ω) : Lu ∈ %Hσ,ϕ(Ω)

→ %Hσ,ϕ(Ω)⊕

q⊕j=1

Hσ+2q−mj−1/2,ϕ(Γ)

whose domain is a Hilbert space with respect to the the graph norm

(‖u‖2Hσ+2q,ϕ(Ω) + ‖%

−1Lu‖2Hσ,ϕ(Ω)

)1/2.

The index of this operator is equal to dimN − dimN+ and does not depend onσ, ϕ, and %.

An important example of the function % that satisfies the condition of The-orem 4.33 is obtained by setting % := %δ1 for an arbitrarily chosen numberδ > −σ − 1/2. Recall that %1 is the function of distance, to the boundary Γ,subordinated to condition (4.184). Indeed, the function % ∈ C∞(Ω) is posi-tive and satisfies condition (4.201). By virtue of Theorem 4.30, this functionis a multiplier in the space H−σ∓ε(Ω), where the number ε > 0 is such thatδ > −σ∓ ε− 1/2. By virtue of the interpolation theorem 3.2, this implies that% is a multiplier in the space H−σ,1/ϕ(Ω). Thus, the function % = %δ1 satisfiesthe condition of Theorem 4.33.

Section 4.6 Remarks and Comments 247

4.6 Remarks and Comments

Section 4.1. In the fundamental paper [4], S. Agmon, A.Douglis, and L. Niren-berg proved a priori estimates for solutions to the elliptic boundary-value prob-lem in the corresponding pairs of Hölder spaces (of noninteger orders) andpositive Sobolev spaces under the assumption that the problem is given ina bounded Euclidean domain with smooth boundary. For positive Sobolevspaces, these estimates are proved independently by F.E.Browder [29], L.N. Slo-bodetskii [238, 239, 241], M. Schechter [222], and others (see references in theM. S. Agranovich’s review [11]). A priori estimates are equivalent to the Fred-holm property of the operator generated by the problem in the mentioned pairsof spaces. If the boundary-value problem is regular elliptic, then the defectsubspace and the range of the operator can be described with the use of differ-ential expressions connected with formally adjoint problem. It is also provedthat a priory estimates for solutions in pairs of Sobolev inner product spaceslead to the validity of the Lopatinskii complementing condition for a collectionof boundary expressions, i.e., to the ellipticity of the boundary-value prob-lem. These questions are systematically discussed, e.g., in the monographs byS. Agmon [3], Yu. M. Berezansky [21], L. Hörmander [81, 86], J.-L. Lions andE. Magenes [121], O. I. Panich [184], M. Schechter [227], H. Triebel [258], andthe review by M. S. Agranovich [11]. The index of the elliptic boundary-valueproblem was calculated by M. F. Atiyah and R. Bott in [17] using the funda-mental M. F. Atiyah and I. M. Singer’s [18] formula for the index of the ellipticmatrix pseudodifferential operator. For the detail description of the theory ofindex of elliptic boundary-value problems, see the monograph by S. Rempeland B.-W. Schulze [198].

Theorems on solvability of regular elliptic boundary-value problems are alsoproved for other scales of positive function spaces. G. Shlenzak proved theoremsfor a certain class of Hörmander inner product spaces [231]. H. Triebel [258, 256]and J. Franke [58] proved theorems for Lizorkin–Triebel and Nikol’skii–Besovboth Banach and non-Banach spaces.

Theorems on solvability of elliptic boundary-value problems find various andimportant applications. Among them, we mention theorems on increase insmoothness of the solutions of the elliptic equation up to the boundary ofthe domain, applications to the investigation of the Green function of theelliptic boundary-value problem, problems of optimal control, nonlocal ellip-tic boundary-value problems, some classes of nonlinear elliptic boundary-valueproblems, etc. See the monographs by Yu. M. Berezansky [21], Yu. M. Berezan-sky, G. F. Us, and Z. G. Sheftel [23], O. A. Ladyzhenskaya and N. N. Ural’tseva[111], J.-L. Lions [118, 117], J.-L. Lions and E. Magenes [121], Ya. A. Roitberg[209, 210], and I. V. Skrypnik [237], the review by M. S. Agranovich [11] andthe reference therein.


S. Agmon and L. Nirenberg [2, 6], M. S. Agranovich and M. I. Vishik [13]selected a subclass of elliptic boundary-value problems, depending on the com-plex-valued parameter, that have the following important property. For suffi-ciently large absolute values of the parameter, the operator corresponding tothe problem sets isomorphisms in appropriate pairs of Sobolev spaces. More-over, the norm of the operator admits a two-sided estimate with constants notdepending on parameter. Not that the index of this problem is equal to zero forall values of the parameter. Parameter-elliptic boundary-value problems haveimportant applications in the theory of parabolic problems and in the spectraltheory of differential operators. Various wider or different classes of ellipticoperators and elliptic boundary-value problems with parameter are studied byM. S. Agranovich [8, 9], G. Grubb [69, Chap. 2], R. Denk, R. Mennicken, andL. R. Volevich [38, 39, 42], A. N. Kozhevnikov [99, 100, 101], O. I. Panich[182, 183]; see also M. S. Agranovich’s surveys [10, Sec. 4] and [11, Sec. 3.6.4]and the reference therein.

All the theorems in Section 4.1 are proved, except for the last theorem 4.10,in our papers [145, 150]. Therein, we consider the more general case where theelliptic boundary-value problem is given on a smooth compact manifold withboundary. Theorem 4.10 is established in [166].

Section 4.2. J.-L. Lions and E. Magenes [119, 120, 126, 121] and Ya. A. Roit-berg [202, 203, 209] investigated the solvability of elliptic boundary-value prob-lems in various two-sided scales formed by positive order Sobolev spaces andtheir negative order analogs. These authors proposed essentially different meth-ods for the construction of the domain of the operator corresponding to theproblem, which leads to different types of theorems on solvability: generaland individual theorems. In the general Roitberg theorem, the domain doesnot depend on the coefficients of the elliptic equation and is general for allboundary-value problems of the same order. In the individual Lions–Magenestheorems, it depends on the coefficients. The terms “general” and “individual”theorems on solvability are proposed in [155, 171].

The general Roitberg theorem involves the two-sided scale of the modifiedSobolev spaces introduced by Ya. A. Roitberg in [202, 203] and the notion ofa generalized solution in these spaces. Ya. A. Roitberg calls them the spaces“of the Sobolev type.” We use the term “the Sobolev scale modified in the senseof Roitberg” for the class of these spaces.

Ya. A. Roitberg proved the general theorem on solvability for regular ellipticboundary-value problems [202, 203] at first and then extended it over nonregu-lar elliptic boundary-value problems [205, 206] and boundary-value problems forgeneral elliptic systems [208]. These results are known in literature as “theoremson complete collection of isomorphisms.” Ya. A. Roitberg’s monograph [209] isdevoted to them. They are also set forth in the monograph by Yu. M. Berezan-sky [21, Chap. 3, Sec. 6, Subsec. 8], handbook [107, Chap. 3, Sec. 6, Subsec. 5],

Section 4.6 Remarks and Comments 249

and survey by M. S. Agranovich [11, Sec. 7.9]. In the most general form, thetheorem on complete collection of isomorphism is proved by A. N. Kozhevnikovin [102] for general elliptic pseudodifferential boundary-value problems.

Ya. A. Roitberg, Z. G. Sheftel, and their disciples systematically appliedthe modified Sobolev scale and theorems on complete collection of isomor-phisms to the investigation of various classes of elliptic boundary-value prob-lems: the problems with power singularities on right-hand sides, problemswith strong degeneration at the boundary, transmission problems, nonlocalproblems, the Sobolev problem, etc; see the monographs by Ya. A. Roitberg[209, 210] and the reference therein. Ya. A. Roitberg [204] used the theorem oncomplete collection of isomorphisms to derive a number of other theorems aboutthe solvability of the regular elliptic boundary-value problem. These resultswere generalized by Ya. A. Roitberg and Yu. V. Kostarchuk [97] to nonregularelliptic boundary-value problems and by I. Ya. Roitberg and Ya. A. Roitberg[200] to elliptic boundary–value problems for mixed order systems (see alsothe monograph by Ya. A. Roitberg [210, Sec. 1.3]). In [163, 164], A. A. Mu-rach proved theorems on complete collection of isomorphisms for the two-sidedscales of the Lizorkin–Triebel and Nikol’skii–Besov spaces modified in the senseof Roitberg.

The concept of the Sobolev scale modified in the sense of Roitberg and theRoitberg generalized solution is used by V. A. Kozlov, V. G. Maz’ya, andJ. Rossmann [104] in the theory of elliptic boundary-value problems in domainswith nonsmooth boundary, by N. V. Zhitarashu and S. D. Eidel’man [53] in thetheory of parabolic equations, and by Ya. A. Roitberg [210] in the theory ofhyperbolic equations.

All theorems in Section 4.2 are proved in [151] except for the last theorem 4.22that has been published for the first time.

Section 4.3. The main result of this section consists of Theorems 4.6 and 4.7.The former endowed the spaces formed the refined Sobolev scale with an equiv-alent norm associated with the elliptic expression. The latter states that ev-ery generalized solution of the elliptic equation has traces on the boundary ofthe Euclidean domain. In the Sobolev case, these theorems were proved byYa. A. Roitberg (see [207] and [209, Theorems 6.1.1 and 6.2.1]). For the re-fined Sobolev scale, they have been established for the first time in the presentmonograph. Lemmas 4.6 and 4.7 on projectors in certain Sobolev spaces are ofinterest in their own right.

Section 4.4. J.-L. Lions and E. Magenes [119, 120, 126, 121] proved variousindividual theorems on the solvability of the regular elliptic boundary-valueproblem in scales of Sobolev spaces containing nonregular distributions. In thistheorems, certain spaces of restrictions of the elliptic differential operator servesas the domain of the operator corresponding to the boundary-value problem.


These spaces are determined by the given right-hand sides of the elliptic equa-tion. In particular, the domain can coincide with the domain of the maximumoperator corresponding to the elliptic differential expression. L. Hörmander[80, Theorem 3.1] proved that the letter domain depends essentially on all co-efficients of the elliptic expression even if they are constant. This individualtheorem is used in the spectral theory of elliptic operators; see the papers byG. Grubb [67, 68, 69] and V. A. Mikhailets [137, 138, 139].

Our condition Iσ for the space of the right-hand sides of the elliptic equationis general enough. This condition holds for the spaces used by J.-L. Lions andE. Magenes. Ya. A. Roitberg [207, Sec. 2.4] used a condition which is somewhatstronger than ours. Under this condition, Ya. A. Roitberg proved that theoperator corresponding to the elliptic boundary-value problem is bounded foreach σ < 0 (see [207, Sec. 2.4] and [209, Remark 6.2.4]).

All results in Subsections 4.4.2–4.4.4, specifically, the individual theorems4.25, 4.27, and 4.29, are proved in [171]. As a special case, they contain theindividual Lions–Magenes theorems formulated in Subsection 4.4.1. The keyindividual theorem 4.25 is close (as to the choice of a wide class of spaces of theright-hand sides) to the two individual theorems discussed in the surveys byE. Magenes [126, Theorem 6.16] and M. S. Agranovich [11, Sec. 7.9, p. 85]. Theformer is due to J.-L. Lions and E. Magenes, and the latter to Ya. A. Roitberg.

Other individual theorems are proved by Ya. A. Roitberg in [204, Sec. 5,Theorem 4] and by Yu. V. Kostarchuk and Ya. A. Roitberg in [97, Sec. 5, The-orem 4]. In these theorems, solutions and/or right-hand sides of the ellipticequation are not distributions given in a Euclidean domain, which differs themfrom the individual theorems in Section 4.4

Section 4.5. The results of this section have been announced in [155, Sec. 6].They generalize the individual theorems proved in Section 4.4 to wide classesof spaces related to Hörmander spaces.

Chapter 5

Elliptic systems

5.1 Uniformly elliptic systems in the refinedSobolev scale

In the present section, we study uniformly elliptic systems of pseudodifferentialequations given in the Euclidean space. We prove an a priori estimate for theirsolutions in the refined Sobolev scale and investigate the interior smoothnessof the solutions. It is useful to compare these results with the theorems inSection 1.4, where the case of one equation is investigated. The proofs of theresults are performed following the same scheme as in the scalar case. For thesake of completeness, we present their full versions.

5.1.1 Uniformly elliptic systems

In the space Rn, we consider a linear system of pseudodifferential equationsp∑k=1

Aj,k uk = fj , j = 1, . . . , p, (5.1)

where n, p ∈ N and Aj,k ∈ Ψ∞(Rn), j, k = 1, . . . , p, are scalar polyhomogeneousPsDOs in Rn. The solutions of equations (5.1) are considered in the class ofdistributions in Rn.

As an important example of system (5.1), we can mention a system of linearpartial differential equations with coefficients in C∞b (Rn).

We study the following class of systems of pseudodifferential equations [10,Sec. 3.2 b]:

Definition 5.1. System (5.1) is called Douglis–Nirenberg uniformly ellipticin Rn if there exist collections of real numbers l1, . . . , lp and m1, . . . ,mpsuch that

(i) ordAj,k ≤ lj +mk for all j, k ∈ 1, . . . , p;

(ii) there exists a number c > 0 such that∣∣det(a(0)j,k(x, ξ)

)pj,k=1

∣∣ ≥ c for all x, ξ ∈ Rn with |ξ| = 1,

where a(0)j,k(x, ξ) is the principal symbol of the PsDO Aj,k in the case where

ordAj,k = lj +mk or a(0)j,k(x, ξ) ≡ 0 in the case where ordAj,k < lj +mk.

252 Chapter 5 Elliptic systems

This definition specifies a fairly broad class of elliptic systems of pseudod-ifferential equations. For differential equations, this class was introduced byDouglis and Nirenberg in [47] who considered ellipticity in Euclidean domains.This class contains homogeneous elliptic systems, for which all lj = 0 andmk = m ∈ R, and Petrovskii’s elliptic systems [191], for which all lj = 0 butmk can be different for different k.

In what follows, we assume that system (5.1) satisfies Definition 5.1.We rewrite this system in the matrix form as follows:

Au = f.

Here,A := (Aj,k)

pj,k=1

is a matrix PsDO, whereas u = col (u1, . . . , up) and f = col (f1, . . . , fp) arefunction columns. As the system itself, the matrix PsDO A is called uniformlyelliptic in Rn. For this operator, there exists a parametrix B, i.e., the followingstatement is true [10, Sec. 3.2 b]:

Proposition 5.1. There exists a matrix classical PsDO B = (Bk,j)pk,j=1 such

that all Bk,j ∈ Ψ−mk−lj (Rn), as well as

BA = I + T1 and AB = I + T2, (5.2)

where T1 = (T j,k1 )pj,k=1 and T2 = (T k,j2 )pk,j=1 are matrix PsDOs, all elements ofwhich belong to the class Ψ−∞(Rn), and I is the identity operator on S ′(Rn).

5.1.2 A priori estimate for the solutions of the system

We establish an a priori estimate for the solutions of the system Au = f con-sidered in the refined Sobolev scale. In view of the inclusion Aj,k ∈ Ψlj+mk(Rn)and Lemma 1.6, we get the linear bounded operator

A :p⊕k=1

Hs+mk, ϕ(Rn)→p⊕j=1

Hs−lj , ϕ(Rn) (5.3)

for any s ∈ R and ϕ ∈M.

Theorem 5.1. Let s ∈ R, σ > 0, and ϕ ∈ M. There exists a number c =c(s, σ, ϕ) > 0 such that, for any vector-valued functions

u = col (u1, . . . , up) ∈p⊕k=1

Hs+mk, ϕ(Rn), (5.4)

f = col (f1, . . . , fp) ∈p⊕j=1

Hs−lj , ϕ(Rn) (5.5)

Section 5.1 Uniformly elliptic systems in the refined Sobolev scale 253

satisfying the equation Au = f in Rn, the following a priori estimate is true:( p∑k=1

‖uk‖2Hs+mk,ϕ(Rn)

)1/2

≤ c( p∑j=1

‖fj‖2Hs−lj , ϕ(Rn)

)1/2

+ c

( p∑k=1

‖uk‖2Hs+mk−σ, ϕ(Rn)

)1/2

. (5.6)

Proof. Let ‖ · ‖′s,ϕ, ‖ · ‖′′s,ϕ, and ‖ · ‖′s−σ,ϕ denote norms in the spaces

p⊕k=1

Hs+mk,ϕ(Rn),p⊕j=1

Hs−lj ,ϕ(Rn), andp⊕k=1

Hs+mk−σ,ϕ(Rn),

respectively. Let the vector-valued functions (5.4) and (5.5) satisfy the equationAu = f in Rn. Due to the first equality in (5.2), we write u = Bf − T1u. Thisyields estimate (5.6):

‖u‖′s,ϕ = ‖Bf − T1u‖′s,ϕ ≤ ‖Bf‖′s,ϕ + ‖T1u‖′s,ϕ ≤ c ‖f‖′′s,ϕ + c ‖u‖′s−σ,ϕ,

where c is the maximum of norms of the operators

B :p⊕j=1

Hs−lj , ϕ(Rn)→p⊕k=1

Hs+mk, ϕ(Rn), (5.7)

T1 :p⊕k=1

Hs+mk−σ, ϕ(Rn)→p⊕k=1

Hs+mk, ϕ(Rn). (5.8)

These operators are bounded by Lemma 1.6 and Proposition 5.1.Theorem 5.1 is proved.

Theorem5.1 refines the a priori estimate obtained by Hörmander [83, Sec. 1.0]for the Sobolev scale.

5.1.3 Smoothness of solutions

Assume that the right-hand side of the equation Au = f has a certain interiorsmoothness in the refined Sobolev scale on an open nonempty set V ⊆ Rn.We study the interior smoothness of the solutions u on this set. First, weconsider the case where V = Rn. Recall that H−∞(Rn) is the union of allspaces Hs,ϕ(Rn), where s ∈ R and ϕ ∈M.

Theorem 5.2. Suppose that u ∈ (H−∞(Rn))p is a solution of the equationAu = f in Rn under the condition

fj ∈ Hs−lj , ϕ(Rn) for every j ∈ 1, . . . , p


with certain parameters s ∈ R and ϕ ∈M. Then

uk ∈ Hs+mk, ϕ(Rn) for every k ∈ 1, . . . , p.

Proof. By Theorem 1.15(i), for the vector-valued function u ∈ (H−∞(Rn))p,there exists a number σ > 0 such that

u ∈p⊕k=1

Hs+mk−σ, ϕ(Rn). (5.9)

Using this result, the condition of the theorem, and relations (5.2), (5.7), and(5.8), we obtain the required property

u = BAu− T1u = Bf − T1u ∈p⊕k=1

Hs+mk, ϕ(Rn).


We now consider the general case where V is an arbitrary open nonemptysubset of the space Rn. Recall that the space Hσ,ϕ

int (V ), which consists of distri-butions with given interior smoothness on V, is defined in Section 1.4.3.

Theorem 5.3. Suppose that u ∈ (H−∞(Rn))p is a solution of the equationAu = f on the set V under the condition

fj ∈ Hs−lj , ϕint (V ) for every j ∈ 1, . . . , p (5.10)


uk ∈ Hs+mk, ϕint (V ) for every k ∈ 1, . . . , p. (5.11)

Proof. First, we prove that condition (5.10) yields the following property ofincrease in interior smoothness of the solution to Au = f :

The following implication is true for any r ≥ 1:

u ∈p⊕k=1

Hs−r+mk, ϕint (V ) ⇒ u ∈

p⊕k=1

Hs−r+1+mk, ϕint (V ). (5.12)

We arbitrarily choose a function χ ∈ C∞b (Rn) such that

suppχ ⊂ V and dist(suppχ, ∂V ) > 0. (5.13)

For this function, there exists a function η ∈ C∞b (Rn) such that

supp η ⊂ V, dist(supp η, ∂V ) > 0,

and η = 1 in a neighborhood of suppχ(5.14)

(this was shown in the proof of Theorem 1.18).

Section 5.1 Uniformly elliptic systems in the refined Sobolev scale 255

Changing the sequence order for the matrix PsDO A and the operator ofmultiplication by the function χ, we find

Aχu = Aχηu

= χAηu+A′ηu

= χAu+ χA(η − 1)u+A′ηu

= χf + χA(η − 1)u+A′ηu in Rn. (5.15)

Here, the matrix PsDO A′ = (A′j,k )pj,k=1 is the commutator of the PsDO A

and the operator of multiplication by the function χ. In view of the inclusionA′j,k ∈ Ψlj+mk−1(Rn) and Lemma 1.6, we get the bounded operator

A′ :p⊕k=1

Hs−r+mk, ϕ(Rn) →p⊕j=1

Hs−r+1−lj , ϕ(Rn).

Therefore,

u ∈p⊕k=1

Hs−r+mk, ϕint (V ) ⇒ A′ηu ∈

p⊕j=1

Hs−r+1−lj , ϕ(Rn). (5.16)

Further, according to condition (5.10) and in view of the inequality r ≥ 1, wehave

χf ∈p⊕j=1

Hs−lj ,ϕ(Rn) →p⊕j=1

Hs−r+1−lj ,ϕ(Rn). (5.17)

In addition, since supports of functions χ and η− 1 do not intersect, the PsDO

χAj,k(η − 1) ∈ Ψ−∞(Rn)

for all j, k ∈ 1, . . . , p. Since for the vector-valued function u ∈ (H−∞(Rn))pwe have inclusion (5.9) valid for some σ > 0, by virtue of Lemma 1.6 we obtainthe inclusion

χA(η − 1)u ∈p⊕j=1

Hs−r+1−lj ,ϕ(Rn). (5.18)

By using relations (5.15)–(5.18) and Theorem 5.2, we conclude that

u ∈p⊕k=1

Hs−r+mk, ϕint (V ) ⇒ Aχu ∈

p⊕j=1

Hs−r+1−lj ,ϕ(Rn)

⇒ χu ∈p⊕k=1

Hs−r+1+mk, ϕ(Rn).


Thus, implication (5.12) is proved since the function χ ∈ C∞b (Rn), which sat-isfies condition (5.13), is chosen arbitrarily.

It is now easy to derive property (5.11) from implication (5.12). We canassume that the number σ > 0 in relation (5.9) is integer. Hence,

u ∈p⊕k=1

Hs−σ+mk, ϕint (V ).

Applying (5.12) successively to r = σ, σ− 1, . . . , 1, we obtain property (5.11):

u ∈p⊕k=1

Hs−σ+mk, ϕint (V ) ⇒ u ∈

p⊕k=1

Hs−σ+1+mk, ϕint (V )

⇒ . . . ⇒ u ∈p⊕k=1

Hs+mk, ϕint (V ).


Theorem 5.3 enables us to establish the continuity of generalized derivativesfor the solutions of system (5.1). To this end, we also use Theorem 1.15(iii).

Theorem 5.4. Let integers k ∈ 1, . . . , p and r ≥ 0 and a function ϕ ∈ Msatisfying condition (1.37) be given. Suppose that u ∈ (H−∞(Rn))p is a solutionof the equation Au = f on an open set V ⊆ Rn under the condition

fj ∈ Hr−mk−lj+n/2, ϕint (V ) for every j ∈ 1, . . . , p. (5.19)

Then the component uk of the solution has continuous partial derivatives on theset V up to the order r inclusive, and moreover, these derivatives are boundedon each set V0 ⊂ V such that dist(V0, ∂V ) > 0.In particular, if V = Rn, then uk ∈ C r

b (Rn).

Proof. By virtue of Theorem 5.3, where we set s := r − mk + n/2, theinclusion uk ∈ H

r+n/2, ϕint (V ) is true. Let a function η ∈ C∞b (Rn) satisfy the

conditions

supp η ⊂ V, dist(supp η, ∂V ) > 0, and η = 1 in a neighborhood of V0.

For the distribution ηuk, by virtue of Theorem 1.15(iii) we have

ηuk ∈ Hr+n/2, ϕ(Rn) → C rb (Rn).

This implies that all partial derivatives of the function uk, up to the order rinclusive, are continuous and bounded in a certain neighborhood of the set V0.

Section 5.2 Elliptic systems on a closed manifold 257

Then these derivatives are continuous on the set V as well because we can takeV0 := x0 for any point x0 ∈ V.


As an application of Theorem 5.4, we give the following sufficient conditionfor a solution of the system Au = f to be classical in the case where all Aj,kare linear differential operators with coefficients from the class C∞b (Rn) andall lj = 0. In other words, we consider the case where Au = f is a Petrovskiiuniformly elliptic system of differential equations.

Corollary 5.1. Suppose that u ∈ (H−∞(Rn))p is a solution of the equationAu = f in an open set V ⊆ Rn under the condition that fj ∈ Hn/2,ϕ

int (V ) foreach number j ∈ 1, . . . , p and a parameter ϕ ∈ M satisfying (1.37). Thenthe solution u is classical in the set V, i.e., uk ∈ Cmk(V ) for all k ∈ 1, . . . , p.

This result immediately follows from Theorem 5.4 for r = mk. Note thatfor the classical solution u of system (5.1) the left-hand sides of the systemare determined by classical derivatives (i.e., derivatives which are not properlygeneralized in the sense of the theory of distributions) and these derivatives arecontinuous in the set V.

5.2 Elliptic systems on a closed manifold

In the present section, we study Douglis–Nirenberg systems of pseudodifferen-tial equations defined on an infinitely smooth closed (compact) oriented mani-fold Γ. We prove that the operator corresponding to these systems is boundedand Fredholm in related pairs of Hörmander spaces from the refined Sobolevscale. In addition, we investigate the class of elliptic systems (parameter-ellipticsystems) for which the indicated operator is an isomorphism for large absolutevalues of the complex parameter. It makes sense to compare the obtained re-sults with theorems presented in Section 2.2 where the case of a single equationis investigated. For the proofs of these results, we use the same scheme that isused for proofs of similar results in the scalar case.

5.2.1 Elliptic Systems

Consider a system of linear pseudodifferential equationsp∑k=1

Aj,k uk = fj on Γ, with j = 1, . . . , p. (5.20)

Here, p ∈ N and Aj,k ∈ Ψ∞(Γ), j, k = 1, . . . , p, are scalar classical PsDOsdefined on the manifold Γ. Equations (5.20) are understood in the sense of thetheory of distributions.


Definition 5.2. System (5.20) is called Douglis–Nirenberg elliptic on Γ if thereexist collections of real numbers l1, . . . , lp and m1, . . . ,mp such that

(i) ordAj,k ≤ lj +mk for all j, k ∈ 1, . . . , p;

(ii) det(a(0)j,k(x, ξ)

)pj,k=1

6= 0 for arbitrary point x ∈ Γ and covector ξ ∈ T ∗xΓ\0;here, a(0)j,k(x, ξ) is the principal symbol of the PsDO Aj,k in the case where

ordAj,k = lj +mk or a(0)j,k(x, ξ) ≡ 0 in the case where ordAj,k < lj +mk.

Further in Section 5.2, it is assumed that system (5.20) satisfies Definition 5.2.We rewrite this system in the matrix form as follows:

Au = f on Γ.

Here, A := (Aj,k)pj,k=1 is a matrix PsDO on Γ and u = col (u1, . . . , up) and f =

col (f1, . . . , fp) are function columns. As the system itself, the correspondingmatrix PsDO A is called elliptic on Γ.

By A+ = ((A+j,k)

pj,k=1)

t we denote the matrix PsDO which is formally adjointto the operator A relative to the C∞-density dx on Γ. In this case, everyPsDO A+

j,k is formally adjoint to Aj,k. The ellipticity of the system Au = f isequivalent to the (Douglis–Nirenberg) ellipticity of the system A+v = g.We set

N := u ∈ (C∞(Γ))p : Au = 0 on Γ,

N+ := v ∈ (C∞(Γ))p : A+v = 0 on Γ.

Since the systems Au = f and A+v = g are elliptic, the spaces N and N+ arefinite-dimensional [10, Sec. 3.2 b].

5.2.2 Operator of the elliptic system on the refined scale

We study properties of the matrix PsDO A on the refined Sobolev scale overthe manifold Γ. In view of the inclusion Aj,k ∈ Ψlj+mk(Γ) and Lemma 2.5, weget the linear bounded operator

A :p⊕k=1

Hs+mk,ϕ(Γ) →p⊕j=1

Hs−lj ,ϕ(Γ) (5.21)

for any s ∈ R and ϕ ∈M. Let us study its properties.

Theorem 5.5. For arbitrary parameters s ∈ R and ϕ ∈ M, the boundedoperator (5.21) is Fredholm. Its kernel coincides with the space N , and therange consists of all vector-valued functions

f = col (f1, . . . , fp) ∈p⊕j=1

Hs−lj ,ϕ(Γ) (5.22)


that satisfy the condition

p∑j=1

(fj , wj)Γ = 0 for any w = (w1, . . . , wp) ∈ N+. (5.23)


Proof. For ϕ ≡ 1 (the Sobolev scale), this theorem is known [10, Theo-rem 3.2.1]. In order to rove this theorem in the general case of ϕ ∈M, we usethe interpolation with a function parameter. Namely, let s ∈ R. We have theFredholm bounded operators

A :p⊕k=1

Hs∓1+mk(Γ) →p⊕j=1

Hs∓1−lj (Γ) (5.24)

with the common kernelN and the same index κ := dimN−dimN+.Moreover,

A

( p⊕k=1

Hs∓1+mk(Γ)

)

=

f = col (f1, . . . , fp) ∈

p⊕j=1

Hs∓1−lj (Γ) : (5.23) is true. (5.25)

By applying to (5.24) the interpolation with the function parameter ψ fromTheorem 2.2, where we set ε = δ = 1, we obtain the bounded operator

A :[ p⊕k=1

Hs−1+mk(Γ),

p⊕k=1

Hs+1+mk(Γ)

]ψ

→[ p⊕j=1

Hs−1−lj (Γ),

p⊕j=1

Hs+1−lj (Γ)

]ψ

,

which coincides with operator (5.21) by virtue of Theorems 1.5 and 2.2. There-fore, in accordance with Theorem 1.7, operator (5.21) is Fredholm with thekernel N and the index κ = dimN − dimN+. The range of this operator isequal to ( p⊕

j=1

Hs−lj ,ϕ(Γ)

)∩ A

( p⊕k=1

Hs−1+mk(Γ)

).

This implies, by virtue of (5.25), that this range coincides with the range indi-cated in the formulation of the theorem.



According to this theorem, N+ is a defect subspace of operator (5.21). Byvirtue of Theorem 2.3(v), the operator

A+ :p⊕j=1

H−s+lj ,1/ϕ(Γ) →p⊕k=1

H−s−mk,1/ϕ(Γ) (5.26)

is adjoint to operator (5.21). Since the adjoint system A+v = g is elliptic, byTheorem 5.5 we conclude that the bounded operator (5.26) is Fredholm andhas the kernel N+ and the defect subspace N .

If the spaces N and N+ are trivial, then Theorem 5.5 and the Banach theo-rem on inverse operator imply that operator (5.21) is an isomorphism. In thegeneral case, it is convenient to define the isomorphism in terms of the followingprojectors:

The spaces of action of operator (5.21) can be represented in the followingform of the direct sum of (closed) subspaces:

p⊕k=1

Hs+mk,ϕ(Γ)

= N uu ∈

p⊕k=1

Hs+mk,ϕ(Γ) :p∑k=1

(uk, vk)Γ = 0 for all v ∈ N,

p⊕j=1

Hs−lj ,ϕ(Γ)

= N+ u

f ∈

p⊕j=1

Hs−lj ,ϕ(Γ) :p∑j=1

(fj , wj)Γ = 0 for all w ∈ N+

.

Recall here that u = col (u1, . . . , up), f = col (f1, . . . , fp) and v = (v1, . . . , vp),w = (w1, . . . , wp).

These decompositions in direct sums do exist because their terms have thetrivial intersection and, in addition, the finite dimension of the first term isequal to the codimension of the second one. (Indeed, for example, in the firstsum, the factor space of the space

⊕pk=1 H

s+mk,ϕ(Γ) by the second term isequal to the space which is dual to the subspace N of

⊕pk=1 H

−s−mk,1/ϕ(Γ)).Let P and P+ denote, respectively, oblique projectors of the spaces

p⊕k=1

Hs+mk,ϕ(Γ) andp⊕j=1

Hs−lj ,ϕ(Γ),

onto the related second terms in the indicated sums taken to be parallel to theirfirst terms. These projectors are independent of s and ϕ.


Theorem 5.6. For any s ∈ R and ϕ ∈ M, the restriction of operator (5.21)to the subspace P

(⊕pk=1 H

s+mk,ϕ(Γ))is the isomorphism

A : P( p⊕k=1

Hs+mk,ϕ(Γ)

)↔ P+

( p⊕j=1

Hs−lj ,ϕ(Γ)

). (5.27)

Proof. By Theorem 5.5, N is the kernel and

P+( p⊕j=1

Hs−lj ,ϕ(Γ))

is the range of operator (5.21). Therefore, operator (5.27) is a bijection. In ad-dition, this operator is bounded. Therefore, it is an isomorphism by virtue ofthe Banach inverse operator theorem.

Theorem 5.6 is proved.Theorem 5.6 yields the following a priori estimate for the solutions of the

equation Au = f :

Theorem 5.7. Let s ∈ R and ϕ ∈M. Suppose that the vector-valued function

u = col (u1, . . . , up) ∈p⊕k=1

Hs+mk,ϕ(Γ) (5.28)

is a solution of the equation Au = f on Γ under condition (5.22) for the right-hand side of the equation. Then, for chosen parameters s and ϕ and an arbitrarynumber σ > 0, there exists a number c > 0 such that( p∑

k=1

‖uk‖2Hs+mk,ϕ(Γ)

)1/2

≤ c( p∑j=1

‖fj‖2Hs−lj ,ϕ(Γ)

)1/2

+ c

( p∑k=1

‖uk‖2Hs+mk−σ,ϕ(Γ)

)1/2

(5.29)

and c is independent of u and f.

Proof. For the sake of brevity, we denote norms in the spacesp⊕k=1

Hs+mk,ϕ(Γ),

p⊕j=1

Hs−lj ,ϕ(Γ), andp⊕k=1

Hs−σ+mk,ϕ(Γ),

which are used in (5.29), by ‖ · ‖′s,ϕ , ‖ · ‖′′s,ϕ , and ‖ · ‖′s−σ,ϕ respectively. SinceN is a finite-dimensional subspace of these spaces, these norms are equivalenton N . In particular, for the vector-valued function u− Pu ∈ N , we get

‖u− Pu‖′s,ϕ ≤ c1 ‖u− Pu‖′s−σ,ϕ


with a constant c1 > 0 which is independent of u. This yields

‖u‖′s,ϕ ≤ ‖u− Pu‖′s,ϕ + ‖Pu‖′s,ϕ

≤ c1 ‖u− Pu‖′s−σ,ϕ + ‖Pu‖′s,ϕ

≤ c1 c2 ‖u‖′s−σ,ϕ + ‖Pu‖′s,ϕ,

where c2 is the norm of the projector 1− P acting in the spacep⊕k=1

Hs−σ+mk,ϕ(Γ).

Thus,‖u‖′s,ϕ ≤ ‖Pu‖′s,ϕ + c1 c2 ‖u‖′s−σ,ϕ. (5.30)

We now use the condition Au = f. Since N is the kernel of operator (5.21)and u − Pu ∈ N , we have APu = f. Thus, Pu is the preimage of the vectorfunction f under the action of isomorphism (5.27). Therefore,

‖Pu‖′s,ϕ ≤ c3 ‖f‖′′s,ϕ,

where c3 is the norm of the operator inverse to (5.27). Using this result andinequality (5.30), we immediately obtain estimate (5.29).


Note that if N = 0, i.e., the equation Au = f has at most one solution,then the quantity

p∑k=1

‖uk‖Hs−σ+mk,ϕ(Γ)

on the right-hand side of estimate (5.29) can be omitted. If N 6= 0, then thisquantity can be made arbitrarily small for any fixed function u by choosinga sufficiently large number σ.

5.2.3 Local smoothness of solutions

Let Γ0 be an open nonempty subset of the manifold Γ. We investigate the localsmoothness of solutions of the elliptic equation Au = f on Γ0 in the refinedSobolev scale. First, we consider the case where Γ0 = Γ.

Theorem 5.8. Suppose that a vector-valued function u ∈ (D′(Γ))p is a solutionof the equation Au = f on the manifold Γ under the condition

fj ∈ Hs−lj ,ϕ(Γ) for every j ∈ 1, . . . , p (5.31)


uk ∈ Hs+mk,ϕ(Γ) for every k ∈ 1, . . . , p. (5.32)


Proof. Since the manifold Γ is compact, the space D′(Γ) is the union of theSobolev spaces Hσ(Γ), where σ ∈ R. Therefore, for the vector-valued functionu ∈ (D′(Γ))p, there exists a number σ < s such that

u ∈p⊕k=1

Hσ+mk(Γ).

By virtue of Theorems 5.5 and 2.3(iii), we arrive at the equality( p⊕j=1

Hs−lj ,ϕ(Γ)

)∩A

( p⊕k=1

Hσ+mk(Γ)

)= A

( p⊕k=1

Hs+mk,ϕ(Γ)

).

Hence, it follows from condition (5.31) that

f = Au ∈ A( p⊕k=1

Hs+mk,ϕ(Γ)

).

Thus, on Γ, in addition to the equality Au = f, we get the equality Av = f fora vector-valued function

v ∈p⊕k=1

Hs+mk,ϕ(Γ).

Therefore, A(u− v) = 0 on Γ, and we conclude that w := u− v ∈ N by virtueof Theorem 5.5. However,

N ⊂ (C∞(Γ))p ⊂p⊕k=1

Hs+mk,ϕ(Γ).

Thus,

u = v + w ∈p⊕k=1

Hs+mk,ϕ(Γ),

i.e., u satisfies property (5.32).Theorem 5.8 is proved.

We now consider the general case of arbitrary Γ0. Recall that the spaceHσ,ϕ

loc (Γ0) of distributions, which have a predetermined local smoothness on Γ0,is defined in Subsection 2.2.3.

Theorem 5.9. Suppose that u ∈ (D′(Γ))p is a solution of the equation Au = fon the set Γ0 under the condition

fj ∈ Hs−lj ,ϕloc (Γ0) for every j ∈ 1, . . . , p


uk ∈ Hs+mk,ϕloc (Γ0) for every k ∈ 1, . . . , p.


Theorem 5.9 is proved by analogy with Theorem 5.3. However, in this case,we have to apply Theorem 5.8 instead of Theorem 5.2.

Using Theorems 5.9 and 2.8, we immediately obtain the following sufficientcondition of continuity of derivatives for the chosen component uk of the solu-tion to the system Au = f :

Corollary 5.2. Let integers k ∈ 1, . . . , p and r ≥ 0 and a function parameterϕ ∈ M satisfying inequality (1.37) be given. Suppose that u ∈ (D′(Γ))p isa solution of the equation Au = f on the set Γ0 under the condition

fj ∈ Hr−mk−lj+n/2, ϕloc (Γ0) for every j ∈ 1, . . . , p.

Then uk ∈ C r(Γ0).

5.2.4 Parameter-elliptic systems

Following the survey [10, Sec. 4.3 d], we consider a sufficiently broad class ofparameter-elliptic systems on the manifold Γ. Fix arbitrarily numbers p, q ∈ N,m > 0, and m1, . . . ,mp ∈ R. Consider a matrix PsDO A(λ) that depends ona complex parameter λ as follows:

A(λ) :=q∑

r= 0

λq−r A(r). (5.33)

Here, A(r) :=(A

(r)j,k

)pj,k=1

is a square matrix formed by arbitrary scalar polyho-

mogeneous PsDOs A(r)j,k on Γ of order

ordA(r)j,k ≤ mr +mk −mj .

Assume that A(0) = −I, where I is the identity matrix.Consider a system of linear equations

A(λ)u = f on Γ (5.34)

depending on the parameter λ ∈ C. Here, as before, u = col (u1, . . . , up) andf = col (f1, . . . , fp) are function columns whose components are distributionson the manifold Γ.

Let K be a fixed closed angle on the complex plane with the vertex at theorigin (we do not exclude the case where K degenerates into a ray).

Definition 5.3. System (5.34) is called parameter-elliptic in the angle K if

detq∑

r= 0

λq−r ar,0(x, ξ) 6= 0 (5.35)


for any x ∈ Γ, ξ ∈ T ∗xΓ, and λ ∈ K satisfying the condition (ξ, λ) 6= 0. Here,

ar,0(x, ξ) :=(ar,0j,k(x, ξ)

)pj,k=1

is a square matrix whose arbitrary element ar,0j,k(x, ξ) is defined as follows: it is

either the principal symbol of the PsDO A(r)j,k in the case where ordA(r)

j,k =mr+mk−mj or the zero function, otherwise. In the case r ≥ 1, it is additionallyassumed that the function ar,0j,k(x, ξ) is equal to 0 at ξ = 0 (because of theprincipal symbol is not initially defined for ξ = 0).

In what follows in the present subsection, we assume that system (5.34)satisfies Definition 5.3.

By virtue of Definition 5.3, system (5.34) is a Douglis–Nirenberg ellipticsystem for every fixed λ ∈ C. Indeed, due to (5.33), the matrix A(λ) is formedby the elements

q∑r= 0

λq−rA(r)j,k, j, k = 1, . . . , p, (5.36)

and these elements are classical scalar PsDOs of orders not greater than lj +m′k,where lj := −mj and m′k := mq+mk. The principal symbol of the PsDO (5.36)is equal to aq,0j,k(x, ξ) for every fixed λ. According to condition (5.35) with λ := 0,the following inequality is true:

det(aq,0j,k(x, ξ)

)pj,k=1

6= 0 for all x ∈ Γ and ξ ∈ T ∗xΓ \ 0.

This means that system (5.34) is Douglis–Nirenberg elliptic for every fixedλ ∈ C.

Therefore, Theorem 5.5 is true for the elliptic system (5.34). According tothis theorem, the bounded operator

A(λ) :p⊕k=1

Hs+mq+mk,ϕ(Γ) →p⊕j=1

Hs+mj ,ϕ(Γ) (5.37)

is Fredholm for arbitrary λ ∈ C, s ∈ R, and ϕ ∈ M. Moreover, since sys-tem (5.34) is parameter-elliptic in the angle K, this operator has the followingadditional properties:


(i) There exists a number λ0 > 0 such that, for any parameter value λ ∈ Ksatisfying |λ| ≥ λ0, the matrix PsDO A(λ) generates the isomorphism

A(λ) :p⊕k=1

Hs+mq+mk,ϕ(Γ) ↔p⊕j=1

Hs+mj ,ϕ(Γ) (5.38)

for any s ∈ R and ϕ ∈M.


(ii) For any parameters s ∈ R and ϕ ∈ M chosen arbitrarily, there existsa number c ≥ 1 such that the two-sided estimate

c−1p∑j=1

‖fj‖Hs+mj,ϕ(Γ)

≤p∑k=1

‖uk‖Hs+mq+mk,ϕ(Γ) + |λ|qp∑k=1

‖uk‖Hs+mk,ϕ(Γ)

≤ cp∑j=1

‖fj‖Hs+mj,ϕ(Γ) (5.39)

is valid for any λ ∈ K satisfying the condition |λ| ≥ λ0 and any vector-valued functions

u = col (u1, . . . , up) ∈p⊕k=1

Hs+mq+mk,ϕ(Γ), (5.40)

f = col (f1, . . . , fp) ∈p⊕j=1

Hs+mj ,ϕ(Γ) (5.41)

satisfying equation (5.34). Here, the number c is independent of λ andvector-valued functions u and f.

In the case of ϕ ≡ 1 (the Sobolev scale), this theorem is known [10, Sec. 4.3 d].Note that the left inequality in the two-sided estimate (5.39) is true even if weomit the assumption that system (5.34) is parameter-elliptic (cf. [13, Sec. 2,Subsec. 1]). In order to avoid awkward expressions in (5.39), we used equivalentnon-Hilbert norms in spaces (5.40) and (5.41).

We prove statements (i) and (ii) of Theorem 5.10 separately. The generalcase of ϕ ∈M is derived from the Sobolev case of ϕ ≡ 1.

Proof of Theorem 5.10. As already indicated, the statement of the the-orem is true in the Sobolev case where ϕ ≡ 1. Hence, there exists a numberλ0 > 0 such that the isomorphisms

A(λ) :p⊕k=1

Hs∓1+mq+mk(Γ, |λ|q,mq) ↔p⊕j=1

Hs∓1+mj (Γ) (5.42)

take place for any λ ∈ K with |λ| ≥ λ0 and arbitrary s ∈ R and ϕ ∈ M.Furthermore, the norm of operator (5.42) and the norm of its inverse operatorare uniformly bounded in λ. Here, we use the Hilbert space Hσ,ϕ(Γ, %, θ) withσ ∈ R, % = |λ|q, and θ = mq, defined in Subsection 2.2.4. This space is equalto Hσ,ϕ(Γ) up to equivalence of norms.


We choose arbitrary s ∈ R and ϕ ∈M. Let ψ be the interpolation parameterfrom Proposition 2.2, where we set ε = δ = 1. Applying the interpolation withthis parameter to (5.42), we obtain the isomorphism

A(λ) :[ p⊕k=1

Hs−1+mq+mk(Γ, |λ|q,mq),p⊕k=1

Hs+1+mq+mk(Γ, |λ|q,mq)]ψ

↔[ p⊕j=1

Hs−1+mj (Γ),

p⊕j=1

Hs+1+mj (Γ)

]ψ

. (5.43)

By virtue of Theorem 1.8, the norm of operator (5.43) and the norm of itsinverse operator are uniformly bounded in the parameter λ. [In (5.43), theadmissible pairs of spaces are normal.] Hence, by using Theorem 1.5 on inter-polation of direct sums of spaces, we obtain the isomorphism

A(λ) :p⊕k=1

[Hs−1+mq+mk(Γ, |λ|q,mq), Hs+1+mq+mk(Γ, |λ|q,mq)

]ψ

↔p⊕j=1

[Hs−1+mj (Γ), Hs+1+mj (Γ)

]ψ. (5.44)

Moreover, the norms of operators (5.43) and (5.44) are equal and the norms oftheir inverse operators are equal too.

We now use Lemma 2.6 with

σ := s+mq +mk, % := |λ|q, θ := mq, and ε = δ = 1,

and apply Theorem 2.2. Due to these results, (5.44) implies the isomorphism

A(λ) :p⊕k=1

Hs+mq+mk,ϕ(Γ, |λ|q,mq) ↔p⊕j=1

Hs+mj ,ϕ(Γ) (5.45)

such that the norm of operator (5.45) and the norm of its inverse operator areuniformly bounded in λ. This yields the required isomorphism (5.38) and thetwo-sided estimate (5.39).


Theorem 5.10(i) yields the following assertion on the index of the operatorcorresponding to the parameter-elliptic system.

Corollary 5.3. Suppose that system (5.34) is parameter-elliptic on a certainclosed ray

K := λ ∈ C : argλ = const.

Then, for any λ ∈ C, operator (5.37) has zero index.


Proof. For any fixed λ ∈ C, (5.34) is a Douglis–Nirenberg elliptic system.Therefore, by virtue of Theorem 5.5, the index of operator (5.37) is finite andindependent of s ∈ R and ϕ ∈ M. Moreover, this index is also independent ofthe parameter λ. Indeed, by virtue of (5.33), the parameter λ affects only thelowest terms of every element of the matrix PsDO A(λ) :

A(λ)−A(0) =q−1∑r=0

λq−rA(r) =

( q−1∑r=0

λq−rA(r)j,k

)pj,k=1

,

where

ordq−1∑r=0

λq−rA(r)j,k ≤ m(q − 1) +mk −mj .

Hence, in view of Lemma 2.5, we get the bounded operator

A(λ)−A(0) :p⊕k=1


Hs+m+mj ,ϕ(Γ).

However, by virtue of Theorem 2.3(iii) and the conditionm > 0, the embeddingHs+m+mj ,ϕ(Γ) → Hs+mj ,ϕ(Γ) is compact. Therefore, the operator

A(λ)−A(0) :p⊕k=1


Hs+mj ,ϕ(Γ)

is compact. This implies (see, e.g., [86, Corollary 19.1.8]) that the operatorsA(λ) and A(0) have the same index, i.e., the index does not depend on theparameter λ. According to Theorem 5.10(i), isomorphism (5.38) holds for suffi-ciently large absolute values of the parameter λ ∈ K. Consequently, the indexof the operator A(λ) is equal to zero for λ ∈ K satisfying the condition |λ| 1,and hence, for any λ ∈ C.

Corollary 5.3 is proved.

5.3 Elliptic boundary-value problems for systemsof equations

In the present section, we study elliptic boundary-value problems for systemsof partial differential equations in the refined scale of spaces. As in the caseof a single equation, these problems generate bounded and Fredholm operatorsin the related pairs of positive Sobolev spaces; see monograph [270, Sec. 9.4],the survey [11, Sec. 6], and the references therein. We extend this result to therefined Sobolev scale applied to Petrovskii elliptic systems.

Section 5.3 Elliptic boundary-value problems for systems of equations 269

5.3.1 Statement of the problem

Consider the linear system of p ≥ 2 partial differential equations

p∑k=1

Lj,k uk = fj in Ω, j = 1, . . . , p, (5.46)

where Lj,k = Lj,k(x,D), j, k = 1, . . . , p, are scalar linear partial differentialexpressions with complex-valued coefficients from C∞(Ω ). For each numberk = 1, . . . , p, we set

mk := maxordLj,k : j = 1, . . . , p.

Thus, mk is the maximal differentiation order for the required function uk. It isassumed that all mk ≥ 1.

We associate system (5.46) with the p× p square matrix

L(0)(x, ξ) :=(L(0)j,k(x, ξ)

)pj,k=1

of x ∈ Ω and ξ ∈ Cn.

Here, L(0)j,k(x, ξ) is the principal symbol of the partial differential equation Lj,k in

the case where ordLj,k = mk or L(0)j,k(x, ξ) ≡ 0 in the case where ordLj,k < mk.

Definition 5.4. System (5.46) is called Petrovskii elliptic on Ω if

detL(0)(x, ξ) 6= 0 for all x ∈ Ω and ξ ∈ Rn \ 0.

If system (5.46) is Petrovskii elliptic and n ≥ 3, then the homogeneous polyno-mial detL(0)(x, ξ) has even degree [11, Sec. 6.1 a]:

p∑k=1

mk = 2q for a certain q ∈ N.

We assume that this condition is satisfied for any integer n ≥ 2.Consider the solutions of system (5.46) that satisfy the boundary conditions

p∑k=1

Bj,k uk = gj on Γ, j = 1, . . . , q. (5.47)

Here, Bj,k = Bj,k(x,D), where j = 1, . . . , q and k = 1, . . . , p, are scalar linearboundary differential expressions defined on Γ. The expression Bj,k has theorder ordBj,k ≤ mk − 1 and infinitely smooth complex-valued coefficients. Foreach number j ∈ 1, . . . , q, we set

rj := minmk − ordBj,k : k = 1, . . . , p ≥ 1.


Here, we define ordBj,k := −∞ if Bj,k ≡ 0. Thus, ordBj,k ≤ mk − rj for allj ∈ 1, . . . , q and k ∈ 1, . . . , p.

We associate the boundary conditions (5.47) with the q × p matrix

B(0)(x, ξ) :=(B

(0)j,k (x, ξ)

)j=1,...,qk=1,...,p

of x ∈ Γ and ξ ∈ Cn.

Here, B(0)j,k (x, ξ) is the principal symbol of the differential expression Bj,k in the

case where ordBj,k = mk − rj or B(0)j,k (x, ξ) ≡ 0 in the case where ordBj,k <

mk − rj .

Definition 5.5. The boundary-value problem (5.46), (5.47) is called Petrovskiielliptic in the domain Ω if the following conditions are satisfied:

(i) System (5.46) is properly elliptic on Ω, i.e., for any point x ∈ Γ and anylinearly independent vectors ξ′, ξ′′ ∈ Rn, the polynomial

detL(0)(x, ξ′ + τξ′′)

in the variable τ has exactly q roots τ+j (x; ξ′, ξ′′), j = 1, . . . , q, with positiveimaginary parts and the same number of roots with negative imaginaryparts (taken with regard for multiplicities of the roots).

(ii) The boundary conditions (5.47) satisfy the complementing condition withrespect to system (5.46) on Γ, i.e., for any point x ∈ Γ and any vectorξ 6= 0 tangent to Γ at the point x, the rows of the matrix

B(0)(x, ξ + τν(x)) · L(0)c (x, ξ + τν(x)),

whose elements are treated as polynomials in τ, are linearly independentmodulo the polynomial

∏qj=1(τ − τ+j (x; ξ, ν(x))). Here, L

(0)c (x, ξ) is the

transposed matrix of the algebraic complements of elements of the matrixL(0)(x, ξ).

Remark 5.1. If system (5.46) satisfies condition (i) of Definition 5.5, then it isPetrovskii elliptic on Ω. The converse statement is true for n ≥ 3 [11, Sec. 6.1 a].

We give examples of elliptic boundary-value problems for systems of partialdifferential equations.

Example 5.1. The elliptic boundary-value problem for the Cauchy–Riemannsystem

∂u1∂x1− ∂u2∂x2

= f1,∂u1∂x2

+∂u2∂x1

= f2 in Ω,

u1 + u2 = g on Γ.

Section 5.3 Elliptic boundary-value problems for systems of equations 271

Here, n = p = 2 and m1 = m2 = 1, and hence, q = 1. The Cauchy–Riemannsystem is an example of a homogeneous elliptic system. These systems satisfyDefinition 5.4, with m1 = . . . = mp.

Example 5.2. The Petrovskii elliptic boundary-value problem

∂u1∂x1− ∂3u2

∂x32= f1,

∂u1∂x2

+∂3u2∂x31

= f2 in Ω,

u1 = g1, u2

(or∂u2∂ν

, or∂2u2∂ν2

)= g2 on Γ.

Here, n = p = 2, m1 = 1, andm2 = 3, and hence, q = 2. Note that the analyzedsystem is not homogeneous elliptic [268, Sec. 1, Subsec. 2 b].

5.3.2 Theorem on solvability

We rewrite the boundary-value problem (5.46), (5.47) in the matrix form

Lu = f in Ω, Bu = g on Γ,

where L := (Lj,k)pj,k=1 andB := (Bj,k)j=1,...,q

k=1,...,pare matrix differential expressions

and u := col (u1, . . . , up), f := col (f1, . . . , fp), and g := col (g1, . . . , gq) arefunction columns.

Theorem 5.11. Suppose that the boundary-value problem (5.46), (5.47) isPetrovskii elliptic in the domain Ω. Let s > 0 and ϕ ∈ M. Then the mappingu 7→ (Lu,Bu), where u ∈ (C∞(Ω ))p, can be uniquely extended (by continuity)to the bounded Fredholm operator

(L,B) :p⊕k=1

Hs+mk,ϕ(Ω)→ (Hs,ϕ(Ω))p ⊕q⊕j=1

Hs+rj−1/2,ϕ(Γ)

=: Hs,ϕ(Ω,Γ). (5.48)

The kernel N of operator (5.48) belongs to (C∞(Ω ))p and does not depend ons and ϕ. The range of this operator consists of all vector-valued functions

(f1, . . . , fp; g1, . . . , gq) ∈ Hs,ϕ(Ω,Γ)

such thatp∑j=1

(fj , wj)Ω +

q∑j=1

(gj , hj)Γ = 0


for any vector-valued function

(w1, . . . , wp; h1, . . . , hq) ∈W.

Here, W is a finite-dimensional subspace of (C∞(Ω ))p × (C∞(Γ))q, indepen-dent of s and ϕ. The index of operator (5.48) is equal to dimN− dimW and,hence, does not depend on s and ϕ.

In the Sobolev case of ϕ ≡ 1, this theorem is a special case of the well-knowntheorem on solvability of elliptic boundary-value problems for general systemsof mixed order; see, e.g., the monograph [270, Sec. 9.4] and the survey [11,Sec. 6.3]. The general case ϕ ∈ M is deduced from the Sobolev case with thehelp of interpolation by analogy with the proof of Theorem 4.1.


Section 5.1. Broad classes of elliptic systems of partial differential equationswere introduced and studied by I. G. Petrovskii [191] (see also [192, p. 328]) andA. Douglis and L. Nirenberg [47]. For these systems, the theorems on smooth-ness of solutions are proved for the Hölder spaces (with noninteger indices) andSobolev spaces. Moreover, a priori estimates of solutions are obtained. Seealso the monograph by L. Hörmander [81, Sec. 10.6], where the case of Sobolevspaces is considered. L. Hörmander also established a priori estimates for thesolutions of elliptic systems of pseudodifferential equations [83, Sec. 1.0].

Elliptic systems are encountered in continuum mechanics (e.g., the matrixLamé equation), in hydrodynamics (the linearized Navier–Stokes system) [11,Sec. 6.2], and in acoustics (the system of Biot equations) [95]. The last twocases are examples of Douglis–Nirenberg elliptic systems. Note that, as a resultof reduction of an arbitrary elliptic partial differential equation to the systemof first-order equations, we arrive at a Douglis–Nirenberg elliptic system.

It is worth noting that there are classes of Douglis–Nirenberg uniformly ellip-tic systems generating Fredholm operators on the Sobolev scale over Rn [196].

All theorems in this section were proved in [169]. They generalize the resultspresented in Section 1.4 to the case of Douglis–Nirenberg uniformly ellipticsystems of pseudodifferential equations defined in the Euclidean space. The caseof Petrovskii uniformly elliptic systems was separately considered in [172, 173].For the extended Sobolev scale, various classes of uniformly elliptic systemswere investigated in [173, 272].

Section 5.2. The theory of elliptic systems on closed smooth manifolds ispresented, e.g., in the monograph by L. Hörmander [86, Chap. 19] and in thesurvey by M. S. Agranovich [10, Sec. 3.2]. The a priori estimates for thesolutions of these systems are equivalent to the assertion that the boundedoperator corresponding to the system is a Fredholm operator in appropriate


pairs of Sobolev spaces. The index of the operator was found by M. F. Atiyahand I. M. Singer in [18].

For the parameter-elliptic systems, the index is equal to zero and, more-over, the corresponding operator realizes isomorphisms in appropriate pairs ofSobolev spaces provided that the absolute values of the parameter are suffi-ciently large; see, e.g., the survey by M. S. Agranovich [10, Sec. 4.3]. Variousclasses of elliptic systems with parameter were studied by A. N. Kozhevnikov[99, 100, 101, 103], M. S. Agranovich [8, 9], R. Denk, R. Mennicken, andL. R. Volevich [38], R. Denk and L. R. Volevich [40, 41], and R. Denk andM. Fairman [36, 37].

There exists a close connection between the elliptic boundary-value prob-lems and matrix elliptic pseudodifferential operators, namely, every ellipticboundary-value problem is equivalent to a certain elliptic system of pseudodif-ferential equations given on the boundary of the domain. This enables one touse the theory of elliptic systems in proving theorems on solvability of ellipticboundary-value problems; see, e.g., the monographs by J. T. Wloka, B. Row-ley, and B. Lawruk [270, Chap. IV], Yu. V. Egorov [52, Chap. III, Sec. 3], andL. Hörmander [86, Chap. 20].

Systems whose properties are close to the properties of elliptic systems werestudied by B. R. Vainberg and V. V. Grushin [263] and R. S. Saks [214, 215,216, 219, 220, 221].

All theorems presented in this section were proved in [170]. They generalizethe results presented in Section 2.2 to the case of Douglis–Nirenberg ellipticsystems of pseudodifferential equations. The case of Petrovskii elliptic systemswas separately studied in [167, 154]. In the extended Sobolev scale, Douglis–Nirenberg elliptic systems were investigated by T. N. Zinchenko in [271].

Section 5.3. The elliptic boundary-value problems for various systems ofdifferential equations were investigated by S. Agmon, A. Douglis, and L. Niren-berg [5], M. S. Agranovich and A. S. Dynin [12], L. R. Volevich [267, 268],L. N. Slobodetskii [238, 241], V. A. Solonnikov [244, 245, 246, 247], and L. Hör-mander [81, Sec. 10.6], [86, Sec. 19.5]. See also the monograph by J. T. Wloka,B. Rowley, and B. Lawruk [270] devoted to elliptic boundary-value problemsfor systems and a survey by M. S. Agranovich [11, Sec. 6] and the referencestherein. It is well known that the operator corresponding to the problem isbounded and Fredholm in appropriate pairs of positive Sobolev spaces.

In the two-sided modified Sobolev scale, these problems were studied byYa. A. Roitberg and Z. G. Sheftel [211, 212], I. A. Kovalenko [98], Ya. A. Roit-berg [208], and I. Ya. Roitberg and Ya. A. Roitberg [200]. They establishedthe theorems on complete collection of isomorphisms generated by the operatorcorresponding to the problem. These results are presented in the monographsby Ya. A. Roitberg [209, Chap. 10] and [210, Sec. 1.3] in connection with theDouglis–Nirenberg elliptic systems.


The boundary-value problems for systems whose properties are close to theproperties of elliptic systems were studied by B. R. Vainberg and V. V. Grushin[264] and R. S. Saks [213, 217, 218]. These problems are investigated in pairsof positive Sobolev spaces.

The main result of this section (Theorem 5.11) was established in [168]. Thisresult extends Theorem 4.7 (on solvability of scalar elliptic boundary-valueproblems) to the case of Petrovskii elliptic systems.

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Index

A 252, 258A(λ) 86, 264A+ 80, 258Aj,k 251B 111B

(0)j,r(x, ξ) 176

B(0)j (x; ξ, λ) 176

B+j 113

B2,µ(Rn) 3, 42Bj,k 189Bj 111Bj(λ) 175Bj(x,D) 111B

(0)j (x, ξ) 112

Bk,j 186Bp,µ(Rn) 41B 9B 271B(0)(x, ξ) 270C+j 113

C∞(Rn) 40C∞(b.c.) 142C∞(b.c.)+ 142C∞0 (Ω) 117C∞0 (Rn) 38C∞b (Rn) 40C∞ν,k(Ω ) 159Ck(Rn) 40Ckb (Rn) 40Cj 113coker 21D∞L,X(Ω) 231, 244Dµ 48, 111, 112Dσ+2q,(2q)L,X (Ω) 232

Dσ+2q,ϕL,X (Ω) 243

Dσ+2qL,X (Ω) 228

DsL(Ω) 132

Dν 159Dk 111, 112D′(Γ) 59D′(Ω) 115D′(Rn) 39D(Rn) 38Er 190Fu 39H−∞,(r)(Ω) 204H−∞ 156H−∞(Rn) 51Hµp (Rn) 41

Hσ,ϕloc (Ω0,Γ0) 169

Hσ,ϕ,(r)loc (Ω0,Γ0) 205

Hσ,ϕ(G, %) 179Hσ,ϕ(Γ, %, θ) 88Hσ,ϕ

int (V ) 52Hσ,ϕ

loc (V ) 54Hσ,ϕ

loc (Γ0) 85Hσ

0 (Ω) 132HσD(Ω) 133

Hϕ(Γ) 105Hϕ(Rn) 100Hs+(Rn) 43Hs,(0)(Ω) 123Hs,(r)(Ω) 190Hs,ϕ,(0)(Ω) 123Hs,ϕ,(r)(Ω) 190Hs,ϕ 142Hs,ϕ(Γ) 59, 60Hs,ϕ(Ω) 115Hs,ϕ(Rn) 5, 42Hs,ϕ(b.c.) 142Hs,ϕ(b.c.)+ 142Hs,ϕ

0 (Ω) 158Hs,ϕQ (Rn) 120

Hs,ϕν,k (Ω) 159

Hs−(Rn) 43

292 Index

Hs(Rn) 4, 43Hs(Γ) 60Hs(Ω) 115Hs(Ω) 227Hs,ϕ

loc (Ω0, b.c.,Γ0) 156Hs,ϕ,(0)loc (Ω0) 156Hs,ϕ 187H−∞,(0)(Ω,Γ) 204Hs,ϕ,(0)(Ω,Γ) 199Hs,ϕ(Γ) 126Hs,ϕ(Ω,Γ) 165Hs(Γ) 131Hσ,(0)(Ω,Γ) 232Hσ(Ω,Γ) 228IL 211ind 21K 176, 264K∞L (Ω) 126Ks,ϕL (Ω) 126

K0σ,ϕ,L(Ω) 220

Ks,ϕ,(r)(Ω,Γ) 191Ks,ϕ,L(Ω) 211ker 21L 111L(λ) 175L(x,D) 111L(0)(x, ξ) 112L(0)(x; ξ, λ) 176L(0)r (x, ξ) 176

L(k) 189L+ 113L2(Γ) 59L∞(Rn) 40Lj,k 269Lk 189Lp(Rn) 40L 271L(0)(x, ξ) 269lj 251M−∞ 156M+−s, 1/ϕ 145

M% 238Ms−2q, ϕ 143M 42m+j 214

mk 111, 251, 264, 269N 114

N+ 114NΛ 187N 81, 258N+ 81, 258O 231P 167P2q 201Q+ 167Q+

0 201RΓ 118Rs,ϕ(Γ) 138RO 98rj 269Smh (R2n) 48SΓ 118S ′(Rn) 39SV 30Spec 10T ∗xΓ 80T2q 201Tr 191u 39X∞(Ω) 231, 243Xψ 10Xσ,ϕ(Ω,Γ) 244∆Γ 80Γ 59, 111γj 133Λ 187ν 111ν(x) 111Ξσ(Ω) 132, 230ξµ 48Π

(r)σ 220

Πσ 214Πs,ϕ,(r)(Ω,Γ) 191%Hσ(Ω) 229σ0(ϕ) 99σ1(ϕ) 99Ψ−∞(Γ) 79Ψ−∞(Rn) 48Ψ∞(Γ) 79Ψ∞(Rn) 48Ψm(Γ) 79Ψm(Rn) 48Ψm

ph(Γ) 79Ψm

ph(Rn) 49

Index 293

Ω 111Ω 111Ω 1112q 111(H)T,Φ 128(·, ·)Γ 59, 113(·, ·)Ω 113(b.c.) 142(b.c.)+ 142(u, v)Rn 39[X0, X1]ψ 10〈ξ〉 42 10

Adams, R. A. 56, 242, 275Adjoint system of boundary expres-

sions 114Admissible pair 9Agmon, S. 1, 86, 109, 162, 175, 247,

248, 273, 275Agranovich, M. S. 1, 2, 47, 49–51, 53,

58, 69, 72, 79, 81–83, 86, 87,92, 109, 113, 127, 131, 144,162, 163, 170, 171, 174, 175,177, 178, 203, 247–252, 258,259, 266, 268–270, 272, 273,275

Alexits, G. 94, 109, 275Anop, A. V. 170, 276Aronszajn, N. 113, 162, 276Atiyah, M. F. 83, 247, 273, 276Avakumović, V. G. 98, 99, 110, 276

Beltrami–Laplace operator 80Bennet, K. 55, 276Berezansky, Yu. M. 1, 2, 51, 53, 58,

74, 95, 123, 124, 131, 144,157, 162, 163, 166, 196, 203,247, 248, 276

Bergh, J. 10, 24–28, 55, 276Besov, O. V. 57, 276Bingham, N. H. 5, 30, 31, 56, 99,

100, 110, 276Bott, R. 247, 276Boundary-value problem

Dirichlet 113formally adjoint 113parameter-elliptic 176

Petrovskii elliptic 270regular elliptic 112semihomogeneous 114

Boyd indices 100Boyd, D. W. 100, 276Browder, F. E. 1, 247, 276Brudnyi, Yu. A. 55, 276Burenkov, V. I. 57, 276

Caetano, A. M. 4, 57, 276Calderon, A. P. 55, 277Carro, M. J. 56, 277Cauchy–Riemann system 270Cerdà, J. 56, 277Classical PsDO 49, 79Cobos, F. 4, 57, 277Cokernel 21Complementing condition 112, 270Condition

complementing 112, 270Iσ,ϕ 243Iσ 231IIσ. 239

Definition of Hs,ϕ(Γ)local 59via interpolation 60via the operator 60

Denk, R. 248, 273, 277Differential expression

elliptic 113formally adjoint 114properly elliptic 112

Dines, N. 186, 277Dirichlet boundary-value prob-

lem 113Distribution

regular 39tempered 3, 39

Djakov, P. B. 109, 277Donoghue, W. F. 55, 277Douglis, A. 1, 247, 252, 272, 273,

275, 277Douglis–Nirenberg system

elliptic 258uniformly elliptic 251

Dual space 15Dynin, A. S. 1, 273, 275

294 Index

Edmunds, D. E. 4, 57, 277, 278Egorov, Yu. V. 53, 85, 273, 278Eidel’man, S. D. 249, 278Elliptic

differential expression 113principal symbol 80PsDO 80

Eskin, G. I. 186, 289Extended Sobolev scale 105, 107

Fairman, M. 273, 277Farkas, W. 4, 57, 278Fernandez, D. L. 4, 57, 277Foiaş, C. 55, 278Formally adjoint

boundary-value problem 113differential expression 114PsDO 80

Formally self-adjoint PsDO 80Fourier transform 39

inverse 39Franke, J. 247, 278Fredholm operator 21Function

pseudoconcave 25quasiregularly varying 32quasislowly varying 32regularly varying 29RO-varying 98slowly varying 30normalized 31

Gagliardo, E. 55, 278Gel’fand, I. M. 56, 278Geluk, J. L. 56, 278General theorem on solvability 188Generating operator 9Geymonat, G. 23, 56, 278Gilbarg, D. 1, 172, 278Gohberg, I. Ts. 22, 233, 278Goldie, C. M. 5, 30, 31, 56, 99, 100,

110, 276Green formula 113, 189, 214Grisvard, P. 146, 163, 278Grubb, G. 229, 248, 250, 278, 279Grushin, V. V. 273, 274, 289Gurka, P. 4, 57, 277, 279Gustavsson, J. 56, 279

Hörmander, L. 1, 3, 4, 21, 40, 41, 46,47, 50, 53, 57, 58, 60, 63,65, 66, 82, 92, 101, 109, 119,127, 141, 162, 174, 178, 229,247, 250, 253, 268, 272, 273,279

Hörmanderinner product space 42space 4, 41

Haan, de, L. 56, 278, 279Haroske, D. D. 4, 57, 58, 277, 279Harutyunyan, G. 186, 277, 279Hegland, M. 6, 56, 279Homogeneous elliptic system 252

Il’in, V. P. 57, 276Index of Fredholm operator 21Indices

Boyd 100Matuszewska 100

Individual theorem on solvability 188Inner product space

Hörmander 42Sobolev 43

Interpolationparameter 10space 29with a function parameter 10

Inverse Fourier transform 39

Jacob, N. 4, 57, 278, 279Janson, S. 56, 280

Kalugina, T. F. 56, 280Kalyabin, G. A. 4, 57, 280Karamata, J. 5, 30, 56, 280Kashin, B. S. 94, 109, 280Khashanah, K. 272, 280Kohn, J. J. 58, 280Kostarchuk, Yu. V. 163, 236, 249,

250, 280Kovalenko, I. A. 273, 280Kozhevnikov, A. N. 86, 248, 249, 273,

280Kozlov, V. A. 249, 280Krein, M. G. 22, 233, 278Krein, S. G. 1, 9, 24, 55, 100, 144,

157, 163, 276, 280, 281Krugljak, N. Ya. 55, 56, 281

Index 295

Löfström, J. 10, 24–28, 55, 163, 276,282

Ladyzhenskaya, O. A. 2, 247, 281Lawruk, B. 268, 272, 273, 290Leopold, H.-G. 4, 57, 276, 278, 281Lewy, H. 1, 281Lions, J.-L. 1, 2, 9, 55, 56, 108, 112–

114, 125, 127, 131–134, 138,161–163, 166, 188, 189, 214,226–230, 238, 247–250, 278,281

Lions–Magenes theorem 229, 230Lizorkin, P. I. 4, 57, 280Local definition of Hs,ϕ(Γ) 59Logarithmic multiscale 30Lopatinskii, Ya. B. 113, 162, 281

Magenes, E. 1, 2, 9, 55, 56, 108, 112–114, 125, 127, 131–134, 138,161–163, 166, 188, 189, 214,226–230, 236, 238, 247–250,281, 282

Malgrange, B. 57, 282Maric, V. 30, 56, 282Mathé, P. 6, 56, 282Matuszewska indices 100Matuszewska, W. 100, 282Maz’ya, V. G. 4, 239, 241, 242, 249,

280, 282Meaney, C. 96, 110, 282Men’shov, D. E. 94, 109, 282Men’shov–Rademacher theorem 93Mennicken, R. 248, 273, 277Merucci, C. 4, 56, 57, 282Mikhailets, V. A. 4, 7, 56, 57, 109,

110, 162, 163, 229, 248–250,272, 273, 282–284

Mikhlin, S. G. 56, 127, 284Milgram, A. N. 113, 162, 276Mityagin, B. S. 109, 277Modified refined scale 190Molyboga, V. 4, 109, 282Moura, S. D. 4, 57, 58, 279, 284Multiplier 238Multiscale 30Murach, A. A. 2, 7, 56–58, 109, 110,

162, 163, 170, 188, 248–250,272–274, 276, 283, 284, 290

Nicola, F. 4, 57, 284Nikol’skii, S. M. 57, 276, 285Nirenberg, L. 1, 58, 86, 109, 175, 247,

248, 252, 272, 273, 275, 277,280

Normalpair 24system of boundary expres-

sions 112Normalized slowly varying func-

tion 31

OperatorBeltrami–Laplace 80Fredholm 21generating 9

Opic, B. 4, 57, 277, 279, 285Order of

modification 190variation 100

Orlicz, W. 95, 109, 285Orlicz–Ul’yanov theorem 94Ovchinnikov, V. I. 29, 55, 56, 285

Pöschel, J. 109, 285Pair

admissible 9normal 24

Paneah, B. P. 4, 41, 57, 114, 117–119, 122, 162, 285, 290

Panich, O. I. 162, 247, 248, 285Parameter of interpolation 10Parameter-elliptic

boundary-value problem 176PsDO 86system 264

Parametrix 49, 252Parseval equality 40Paszkiewicz, A. 110, 285Peetre, J. 25–28, 55, 186, 281, 285Persson, L.-E. 56, 285Petrovskii elliptic

boundary-value problem 270system 252, 269

Petrovskii, I. G. 252, 272, 285Petunin, Yu. I. 24, 55, 100, 281Pliś, A. 1, 285Polyhomogeneous PsDO 49, 79

296 Index

Principal symbol 49, 80Properly elliptic

differential expression 112system 270

PsDO 48, 79classical 49, 79elliptic 80formally adjoint 80formally self-adjoint 80parameter-elliptic 86polyhomogeneous 49, 79uniformly elliptic 49

PsDO symbol 48principal 49

Pseudoconcave function 25Pseudodifferential operator 48Pustyl’nik, E. I. 56, 286

Quasiregularly varying function 32Quasislowly varying function 32

Rabier, P. 272, 286Rademacher, H. 94, 109, 286Refined

scale 5, 43, 60, 115Sobolev scale 5, 43, 60, 115modified in the sense of Roit-

berg 190Regular

distribution 39elliptic boundary-value prob-

lem 112Regularly varying function 29Rempel, S. 247, 286Representation theorem 30Reshnick, S. I. 30, 56, 286Rigging 123RO-varying function 98Rodino, L. 4, 57, 284Roitberg generalized solution 190Roitberg, I. Ya. 163, 249, 273, 286Roitberg, Ya. A. 1, 2, 131, 144, 148,

152, 156–158, 162, 163, 175,188–190, 192, 199, 203, 206,212, 214, 217, 219–221, 223,225, 231, 235, 236, 247–250,273, 276, 280, 286, 287

Rossmann, J. 249, 280

Rowley, B. 268, 272, 273, 290

Saakyan, A. A. 94, 109, 280Saks, R. S. 273, 274, 287Scale

refined 5, 43, 60Sobolev refined 5

Schechter, M. 1, 4, 55, 57, 113, 114,162, 163, 186, 247, 287

Schilling, R. L. 4, 57, 278Schulze, B.-W. 186, 247, 277, 279,

286Schwartz space 38Schwartz, L. 56, 288Seeley, R. T. 127, 146, 163, 288Semenov, E. M. 24, 55, 100, 281Semihomogeneous boundary-value

problem 114Seneta, E. 5, 30, 31, 34, 56, 99, 110,

288Shapiro, Z. Ya. 113, 162, 288Shaposhnikova, T. O. 4, 239, 241,

242, 282Sharpley, R. 55, 276Sheftel, Z. G. 2, 51, 74, 95, 123, 124,

162, 163, 196, 247, 249, 273,276, 286, 287

Shilov, G. E. 56, 278Shlenzak, G. 17, 18, 55–57, 109, 162,

163, 247, 288Shubin, M. A. 47, 58, 60, 61, 64–66,

69, 72, 85, 109, 288Simanca, S. R. 186, 288Singer, I. M. 83, 247, 273, 276Skrypnik, I. V. 2, 247, 288Slobodetskii, L. N. 1, 247, 273, 288Slowly varying function 30Smoothed modulus 42Sobolev

extended scale 105, 107inner product space 43refined scale 43, 60space 4

Sobolev, S. L. 56, 288Solonnikov, V. A. 1, 273, 288, 289Space

dual 15Hörmander 4, 41

Index 297

inner product 42interpolation 29of generalized smoothness 4Schwartz 38Sobolev 4inner product 43

Stepanets, A. I. 4, 77, 109, 289Strichartz, R. S. 241, 289Symbol

principal 49, 80elliptic 80uniformly elliptic 49

Symbol of PsDO 48System

Cauchy–Riemann 270Douglis–Nirenberg uniformly ellip-

tic 251ellipticDouglis–Nirenberg 258homogeneous 252Petrovskii 252, 269properly 270

parameter-elliptic 264System of boundary expressions

adjoint 114normal 112

Tandori, K. 95, 109, 289Tartar, L. 55, 56, 289Tautenhahn, U. 6, 56, 282Taylor, M. 47, 53, 58, 72, 108, 109,

289Teugels, J. L. 5, 30, 31, 56, 99, 100,

110, 276Theorem

A 228Lions–Magenes 229, 230

LM1 229LM2 230Men’shov–Rademacher 93Orlicz–Ul’yanov 94Representation 30Uniform Convergence 30

Trebels, W. 4, 57, 285Treves, F. 47, 58, 60, 109, 289Triebel, H. 2, 4, 10, 55–58, 112, 113,

115–117, 119, 120, 122, 132,134, 146, 147, 159, 162, 182,198, 214, 227, 239, 247, 277,278, 289

Trudinger, N. S. 1, 172, 278

Ul’yanov, P. L. 95, 109, 289Uniform Convergence theorem 30Uniformly elliptic

principal symbol 49PsDO 49

Ural’tseva, N. N. 2, 247, 281Us, G. F. 2, 51, 74, 95, 123, 124, 162,

163, 196, 247, 276

Vainberg, B. R. 273, 274, 289Vishik, M. I. 86, 92, 109, 175, 178,

186, 248, 266, 275, 289Vladimirov, V. S. 56, 289Volevich, L. R. 1, 4, 41, 57, 114, 117–

119, 122, 162, 248, 271, 273,277, 290

Weight function 40Wloka, J. T. 268, 272, 273, 290

Zhitarashu, N. V. 249, 278Zinchenko, T. N. 110, 272, 273, 284,

290

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Hoermander Spaces, Interpolation, and Elliptic Problems