SINGLE AND DOUBLE LAYER POTENTIALS ON DOMAINS

WITH CONICAL POINTS I: STRAIGHT CONES

VICTOR NISTOR AND YU QIAO

Abstract. Let Ω “ R`ω be an open straight cone in Rn, n ě 3, whereω Ă Sn´1 is a smooth subdomain of the unit sphere. Denote by K and

S the double and single layer potential operators associated to Ω and the

Laplace operator ∆. Let r be the distance to the origin. We consider anatural class of dilation invariant operators on BΩ, called Mellin convolution

operators and show that Ka :“ raKr´a and Sb :“ rb´12 Sr´b´ 1

2 are Mellin

convolution operators for a P p´1, n´1q and b P p 12, n´ 3

2q. It is known that a

Mellin convolution operator T is invertible if, and only if, its Mellin transform

T pλq is invertible for any real λ. We establish a reduction procedure thatrelates the Mellin transforms of Ka and Sb to the single and, respectively,

double layer potential operators of some other operators on ω, which can be

shown to be invertible using the classical theory of layer potential operators onsmooth domains. This reduction procedure thus allows us to prove that 1

2˘K

and S are invertible between suitable weighted Sobolev spaces. A classical

consequence of the invertibility of these operators is a solvability result inweighted Sobolev spaces for the Dirichlet problem on Ω.

Contents

Introduction 11. Preliminaries 51.1. Layer potentials 51.2. Mellin convolution operators 61.3. Convolution and the Mellin Transform 102. Norm closure and invertibility 133. The double layer potential operator K on Straight Cones 144. The Single Layer Potential Operator S on Straight Cones 175. Mapping properties of K and S 186. Invertibility of layer potentials 21References 23

Introduction

The Dirichlet problem for Laplace’s equation in a sufficiently regular boundeddomain Ω Ă Rn can be solved for continuous boundary values using the double

Date: November 19, 2011.Nistor was partially supported by the NSF Grants DMS-0713743, OCI-0749202, and DMS-

1016556. Manuscripts available from http://www.math.psu.edu/nistor/.

1

2 VICTOR NISTOR AND YU QIAO

layer potential operator

upxq “ Kϕpxq “ ´1

ωn

ż

BΩ

px´ yq ¨ νpyq

|x´ y|nϕpyqdSn´1pyq,

or the single layer potential operator

vpxq “ Sϕpxq “1

pn´ 2qωn

ż

BΩ

1

|x´ y|n´2ϕpyqdSn´1pyq.

Potential theory can be traced back to the works of Lagrange, Laplace, Pois-son, and Gauss [15]. One of the main motivations to study the layer potentialoperators in the setting of potential theory is that the single and double layer po-tentials represent harmonic functions in terms of its boundary data. Moreover, thelayer potentials play a fundamental role in many real-world problems, especially inphysics. For instance, Gauss used single layer potential to find, for an arbitraryconductor Ω, the equilibrium charge distribution with the total charge M . (See[17, 38], and [57] for a more detailed historical account and a more complete listof applications.) Many other mathematicians made important contributions to po-tential theory, such as Liouville, Neumann, Poincare, Fredholm, Hilbert, and manyothers.

Recently, the method of potential theory has attracted some renewed attentionboth in the theoretical and the applied mathematics. For instance, it plays an im-portant role in solving boundary value problems of elliptic equations, the Helmholtzequation, the equation of linear elasticity, and others. It has applications to elec-tromagnetic scattering [6, 34, 45]. In applied mathematics, the so-called BoundaryElement Methods is often used for domains with singularities. The advantage ofthis method is that it decreases the dimension of the discretization space. For in-stance, in this setting, Bremer and Rokhin [5] develop a numerical procedure forthe construction of quadrature formulae suitable for the efficient discretization ofboundary integral equations on two-dimensional polygonal domains. Although inthis paper we do not address applications to numerical methods, these applicationsare our main motivation. See [52] for more on the Boundary Element Method.

A great many number of papers and books have been written on the methodof layer potentials. It is not possible to mention them all here, but we do wantto name here a few, beginning with the books by Courant and Hilber [8], Foland[14], Hsiao and Wendland [17], Kress [29], McLean [38], and Taylor [58], whichgive a rather complete account of the theory on smooth domains, which is by nowfairly well understood. We are more interested in non-smooth domains. Thereare also many papers devoted to the method of layer potentials on non-smoothdomains. These works can be roughly divided into two categories: works devotedto Lipschitz domains and works devoted to polyhedral domains (mostly polygonal).The case of Lipschitz domains is by far the most studied among the class of non-smooth domains, and is also fairly well understood. We would like to mention herethe papers of Jerrison and Kenig [18, 19, 20], Kenig [23], Kenig and Pipher [24],Medkova [40], and Verchota [60] that give rather complete results for domains inthe euclidean space. In the works of D. Mitrea and I. Mitrea [44], I. Mitrea andM. Mitrea [47], M. Mitrea and Taylor [49, 50], Kohr, Pintea and Wendland [25],and Taylor [59], the method of layer potentials is applied to Lipschitz domains onmanifolds. See also Costabel’s paper [7] for a nice introduction to the method oflayer potentials using more elementary methods. The paper [13] contains some

STRAIGHT CONES 3

earlier results on C1-domains. Our long term interest is in polyhedral domains.By comparison, much fewer works were devoted to this case. We want to mentionhowever the papers of Lewis [31], Lewis and Parenti [32], and I. Mitrea [46] forwork on polygonal domains. The works of Elschner [10], Fabes, Jodeit, Lewis [12],Kral [27, 3], Kral and Medkova [28], Medkova [39], and Verchota and Vogel [61]deal with the case of polyhedral domains in three and four dimensions. See [22] forthe related case of interface problems.

The simplest type of a piecewise smooth domain is that of a domain with conicalsingularities. Earlier work on layer potentials on domains with conical points canbe found, for example, in [35, 48]. The survey [35] is especially important becauseit underscores the importance of understanding algebras of pseudodifferential oper-ators on singular spaces (a step in the solution of this problem on general domainsis in [2]).

Domains with conical points were studied by many authors. We want to men-tion in this regard the work of Kondratiev on boundary value problems on domainswith conical points, the papers of Mazzeo and Melrose [37] and Melrose [41], Ka-panadze and Schulze [21], and Schrohe and Schulze [53, 54]. See also the books ofEgorov and Schulze [9], Mazya and Rossmann [36], Melrose [42, 43], Schulze [55],Schulze, Sternin, and Shatalov [56], and Sauter and Schwab [52]. Most of theseworks are devoted to constructing suitable algebras of pseudodifferential operatorson conical manifolds. These algebras then allow a reduction of the study of bound-ary value problems on domains with conical points to boundary value problemson straight (infinite) cones and boundary value problems on smooth, bounded do-mains. This is done, for instance, in the works of Schulze (some with collaborators)mentioned above. See also the recent paper [1] using groupoids to construct alge-bras of pseudodifferential operators on singular spaces (see also [51] for some relatedconstructions).

It is important to understand the relation between our work and the papersand books mentioned above. The algebras of pseudodifferential operators Ψ˚ con-structed in the works above (and in all works on the subject that we know) reducesome questions about specific operators T P Ψ˚ on domains with conical points torelated questions for the indicial operators of T on some associated infinite cones.Usually (when proving the Fredholm property, for instance) one has to prove theinvertibility of these indicial operators. The issue of invertibility is, however, seldomaddressed in the above works, partly because one has to use different techniquesto answer it. It is the purpose of this paper to study the case of straight (infinite)cones and to prove invertibility results. These are the main results of this paper.

For straight cones, we can use suitable dilation invariant pseudodifferential op-erators that can be defined elementary (i.e. without using the theory of conepseudodifferential operators developed in the above references). These operatorsare often, but not always, simple particular cases of the operators considered in theworks mentioned above. Let us also mention that although we do not use the the-ory of pseudodifferential operators on conical manifolds, we do expect this theoryto play a role in the study of layer potential on domains with conical points. Thisstudy cannot be carried out, however, without understanding the case of straightcones. Also, in order to use directly the results on conical pseudodifferential opera-tors in the works above to study the layer potential on domains with conical points,one would have to first prove that these operators are indeed in these algebras of

4 VICTOR NISTOR AND YU QIAO

cone pseudodifferential operators, which does not seem to be trivial (and possiblyis not even true in general).

The virtue of using layer potential operators is that they reduce the existenceproblem for the Laplace equation to the invertibility of either of the operators12 ` K or S. For instance, if the boundary is C2, then we can apply the classical

Fredholm theory to the operators 12 ` K and S to solve the Laplace’s equation

when ϕ P L2pBΩq or ϕ P CpBΩq [14, 29]. If the boundary is only C1, the operatorK is still compact [13] on L2pBΩq. However, if the boundary of our domain Ωis not C1, which is the case, for instance, if Ω is a polygon or, more generally,a domain with conical points, the double layer potential operator is no longercompact [10, 12, 26, 29, 31, 32, 44, 46, 48, 58]. A similar approach applies also tothe Neumann problem for the Laplace equation.

These results justify studying in detail domains with conical points. As explainedabove, for technical reasons, we found it necessary to first take a close look at thecase of straight cones, which is the main topic of this paper. We need a thoroughunderstanding of the case of straight cones to be able to handle successfully thecase of domains with conical points. In addition to using the theory of compactand Fredholm operators, to study straight cones, we found it necessary to usetechniques of harmonic analysis such as the Mellin transform. Our main resultsare as follows. First, we provide a reduction, via Mellin transform, of the studyof layer potentials on cones to the study of a family layer potential operators onthe basis of the cone. This allows us to establish the invertibility of the relevantlayer potential operators on cones by reducing to the pointwise invertibility of afamily of operators on the boundary of the basis of the cone. (It is known thatfor the type of operators that we consider, the pointwise invertibility is enough forglobal invertibility [9, 42, 53], but since this may be a more subtle point, we haveincluded a proof using Banach Algebras.) This invertibility turns out to hold for arange of weights of our Sobolev spaces. In the process, we also determine in detailthe structure of the layer potentials in terms of operator valued Mellin convolutionoperators.

This paper is organized as follows. In Section 1 we include some preliminarydefinitions and results, which are mainly the adaptation of some known resultsto our setting. We provide some general references, but we include some of thesimple proofs, for the benefit of the reader. More precisely, in Subsection 1.1, weintroduce the single and double layer potentials for sufficiently regular domains.In Subsection 1.2, we define Mellin convolution operators and the filtered algebraR :“ YR´k of Mellin convolution pseudodifferential operators on BΩ, with Rk

denoting the operators of order k. We then prove mapping properties for operatorsin Rk. In Subsection 1.3, we introduce the Mellin transform for operators in R andstudy its properties. In Section 3 and Section 4, we define the double and singlelayer potential operators K and S associated to the Laplace operator on a straightcone, and related operator S0 “ M

r´12S0M

r´12

, which is unlike S, turns out to be

a Mellin convolution operator. For the purpose of working with a dilation invariant

measure on the cone, we introduce the modified operators rK :“ Mr

n´12KM

r´n´12

and rS0 :“Mr

n´22S0M

r´n´22

. In Section 5, we define weighted Sobolev spaces on Ω

or on BΩ. Then we prove that rK and rS0 belong to R´1. Using general results fromSubsection 1.2, we establish some mapping properties of K and S between some

STRAIGHT CONES 5

weighted Sobolev spaces. Lastly, Section 6 contains the proofs of our main results:that the operators 1

2`K and S0 are isomorphisms between weighted Sobolev spacesfor suitable weights.

We thank Irina Mitrea for some useful comments.

1. Preliminaries

In this section, we include some background results and definitions that arefor the most part well known. Some general references include Kapanadze andSchulze [21], Schrohe and Schulze [53, 54], Egorov and Schulze [9], Melrose [42, 43],and Schulze [55]. Our definitions and results in this section are very similar tosome results in these references, but are more elementary. The main differencebetween our results and the above references is that we use a slightly differentcompletion of the algebra of regularizing operators. For this reason, we include theeasy proofs, for the convenience of the reader. The reason we need to include aslightly different completion of the algebra of regularizing operators is to be ableto prove that our layer potentials are in the resulting algebras, while preservingthe important property of these algebras that they be closed under holomorphicfunctional calculus.

1.1. Layer potentials. Let us fix a Riemannian manifold M without boundary.The single and double layer potentials are defined for any domain Ω Ă M whoseboundary is sufficiently regular, but not necessarily smooth. These layer potentialsare associated to a positive, strongly elliptic operator P “

ř

aijBiBj`ř

bjBj`c anda fundamental solution Epx, yq of P . In this paper, we just focus on the case whereP is the positive Laplacian: P “ ´∆ and E is its usual fundamental solution.

We shall denote by ΨkpMq the space of pseudodifferential operators of order kon M [14, 16, 59]. Suppose we are given a domain Ω ĂM on which there is a givenfundamental solution Epx, yq of the Laplacian. Let ωn denote the area of Sn´1 andlet cn “ ´rpn ´ 2qωns

´1 for n > 3, and c2 “ 12π. More precisely, we assumethat we are given a smooth function E : M ˆM r diagM Ñ R, where diagM isthe diagonal of M , such that for any fixed y, Epx, yq defines a distribution in xsatisfying

´∆xEpx, yq “ δypxq,

where Epx, yq is the Schwartz kernel of an operator Epx,Dq P Ψ´2pMq and δy isthe Dirac distribution at y. We then have

Epx, yq „ cndistpx, yq2´n ` ¨ ¨ ¨

as xÑ y, if n > 3, while

Epx, yq „ c2 log distpx, yq ` ¨ ¨ ¨

if n “ 2. See [11, 59].Assume Ω is a polyhedral domain, so that the surface measure dSn´1 on Ω is

defined and that the unit normal vector νpyq to BΩ is defined almost everywherewith respect to dSn´1. We introduce the operators S and D, mapping functions onthe boundary BΩ to functions on the whole space Rn. Let ν denote the unit outernormal vector field on BΩ. For any integrable function f : BΩ Ñ C and for anyx PM r BΩ, we define

(1) Sfpxq “ż

BΩ

Epx, yqfpyqdSn´1pyq

6 VICTOR NISTOR AND YU QIAO

and

(2) Dfpxq “ż

BΩ

BE

Bνypx, yqfpyqdSn´1pyq.

Similarly, for x P BΩ, we define

(3) Sfpxq “ Sp∆, E; Ωqfpxq “

ż

BΩ

Epx, yqfpyqdSn´1pyq

and

(4) Kfpxq “ Kp∆, E; Ωq “

ż

BΩ

BE

Bνypx, yqfpyqdSn´1pyq.

In the cases of interest, we will check that these integrals converge.The operators K “ Kp∆, E; Ωq and S “ Sp∆, E; Ωq define the double and single

layer potential operators for ∆, E, and Ω. They map functions on BΩ to functionson BΩ. As indicated in the notation, the operators S and K depend on the choiceof E. In this paper we shall study the case when Ω is an open cone (5). In thiscase, we will show that K and S are related to some specific Mellin convolutionoperators, which will be used to study the properties of these operators.

1.2. Mellin convolution operators . Let ω Ă Sn´1 (n > 3) be an open subsetwith smooth boundary. We allow ω to be disconnected. Denote by

(5) Ω :“ R`ω :“ tty1 : y1 P ω, t P p0,8qu

the open cone with base ω. Since any point x P Ω can uniquely be written as rx1,where r P R` and x1 P ω, we shall sometimes identify a point x “ rx1 P Ω withthe pair pr, x1q P R` ˆ ω. Denote the surface measure on ω by dSn´1px

1q and themeasure on Bω by dSn´2px

1q, and set

(6) dµnpr, x1q “ r´1drdSn´1px

1q, and dµn´1pr, x1q “ r´1drdSn´2px

1q,

which give rise to Hilbert spaces L2pΩ, dµnq and L2pBΩ, dµn´1q.

Definition 1.1. Let p “ ppr, x1, y1q P C8pR` ˆ ω ˆ ωq and u P CcpΩq. Define thefunction Pu on Ω by

Pupr, x1q “ p ˚ upr, x1q “

ż

ω

ż 8

0

pprs, x1, y1qups, y1qds

sdSn´1py

1q.

The operator Cp :“ P will be called the smoothing Mellin convolution operator onΩ with (operator valued) convolution kernel p. If p is only continuous, the opera-tor Cp :“ P will be called the Mellin convolution operator on Ω with continuous(operator valued) convolution kernel p.

If p “ ppr, x1, y1q P C8pR` ˆ Bω ˆ Bωq, we define smoothing Mellin convolutionoperators on BΩ in the same way. Any extension of P contained in the closure ofP will also be called a Mellin convolution operator.

The operator valued convolution kernel of a Mellin convolution operator P willbe denoted by kP . Thus kP “ p in the above definition.

Definition 1.2. For any t P p0,8q, we define the dilation αt (by t) by the formula

αtfpxq “ fptxq.

It is clear that for each t P R`, αt is unitary on L2pΩ, dµnq or on L2pBΩ, dµn´1q.The following lemma is well known and its proof is a simple computation.

STRAIGHT CONES 7

Lemma 1.3. Let P be a bounded integral operator on L2pΩ, dµnq with continu-ous integral kernel p1pr, s, x

1, y1q. Then P is a Mellin convolution operator (withcontinuous kernel) if, and only if, it satisfies

Pαt “ αtP,

for any t P R`. The same is true if P is a (possibly unbounded) operator onL2pBΩ, dµn´1q.

For a Mellin convolution operator P on Ω “ R`ω with operator valued kernelppr, x1, y1q, we shall denote by P prq the integral operator on ω with kernel ppr, x1, y1q.By differentiating with respect to t in Lemma 1.3 and taking into account thatrBr “ pαtq

1|t“0, we obtain the following corollary.

Corollary 1.4. Let P be a Mellin convolution operator on Ω with operator-valuedconvolution kernel kP pr, x

1, y1q “ ppr, x1, y1q. Then

prBrqP “ P prBrq.

Moreover, the operator prBrqP is the Mellin convolution operator with operator-valued convolution kernel prBrqp. The same is true if P is a Mellin convolutionoperator on BΩ.

We now introduce suitable algebras of Mellin convolution operators. Let ∆BΩ bethe Laplace operator on R`ˆBω associated to the metric pr´1drq2`pdx1q2, wherepdx1q2 is the metric on Bω induced from Sn´1. Then

∆BΩ “ prBrq2 `∆Bω.

Proposition 1.5. Let ϕ P C8pR`ˆBωˆBωq and Cϕ denote the Mellin convolutionoperator on BΩ with kernel ϕ. Then we have, for j, k P NY t0u,

∆jBωCϕ∆k

Bω “ Cψ,

where ψpr, x1, y1q “ ∆jBω,x1∆

kBω,y1ϕpr, x

1, y1q. Also

prBrqjCϕ “ CϕprBrq

j “ CprBrqjϕ.

Proof. Integration by parts implies the first statement. To prove the second state-ment, it is enough to show the case j “ 1. The result follows from Corollary 1.4,using also a short calculation.

Definition 1.6. Define R´8 to be the space of Mellin convolution operators P onBΩ with kernel ppr, x1, y1q P C8pR` ˆ Bω ˆ Bωq for which there exists ε ą 0 suchthat, for all pr, x1, y1q P R` ˆ Bω ˆ Bω,

´

r `1

r

¯εˇˇ

ˇprBrq

i∆jBω,x1∆

kBω,y1 ppr, x

1, y1qˇ

ˇ

ˇď Cpi, j, kq,

where Cpi, j, kq is a constant depending on i, j, and k.

Remark 1.7. By Proposition 1.5, the above definition implies that p and its allpartial derivatives have rapid decay as r Ñ 0 and r Ñ8.

We endow R´8 with the convolution product:

(7) p1 ˚ p2pr, x1, y1q “

ż

Bω

ż 8

0

p1prs, x1, z1qp2ps, z

1, y1qds

sdSn´1pz

1q.

8 VICTOR NISTOR AND YU QIAO

We define an involution on R´8 by

(8) p˚pr, x1, y1q “ ppr´1, y1, x1q.

Lemma 1.8. The space R´8 is a ˚-algebra.

Proof. If p P R´8, it is easy to see that p˚ P R´8. We need to show that ifp1, p2 P R´8, then p1 ˚ p2 P R´8. Let ε ą 0 be the constant defining p1 and p2. Itis enough to check that

´

r `1

r

¯ε1 ˇˇ

ˇprBrq

i∆jBω,x1∆

kBω,y1 p1 ˚ p2pr, x

1, y1qˇ

ˇ

ˇď Cpε1, i, j, kq,

for any 0 ă ε1 ă ε. Indeed, we have

´

r `1

r

¯ε1 ˇˇ

ˇprBrq

i∆jBω,x1∆

kBω,y1 p1 ˚ p2pr, x

1, y1qˇ

ˇ

ˇď

´

r `1

r

¯ε1ż

Bω

ż 8

0

ˇ

ˇ

ˇprBrq

i∆jBω,x1p1prs, x

1, z1q∆kBω,y1p2ps, z

1, y1qˇ

ˇ

ˇ

ds

sdSn´1pz

1q

ď C´

r `1

r

¯ε1ż

Bω

ż 8

0

´r

s`s

r

¯´ε´

s`1

s

¯´ε ds

sdSn´1pz

1q

ď C

ż 8

0

´

s`1

s

¯ε1´ε ds

sď Cpε1, i, j, kq,

The proof is now complete.

Let us define now ΨmproppMq to be the set of properly supported pseudodifferential

operators on a manifold M . Then we let

Rm :“ R´8 `ΨmproppR` ˆ BωqR

`

,

where ΨmproppR` ˆ BωqR

`

denotes dilation invariant pseudodifferential operators

in ΨmproppR` ˆ Bωq. That is, we augment the spaces of properly supported, di-

lation invariant pseudodifferential operators on R` ˆ Bω with the subspace R´8of regularizing Mellin operators. Then RjRi Ă Ri`j and each Rk consists ofdilation invariant operators with suitable mapping properties between weightedSobolev spaces. The main difference between our algebra and the ones consideredin [42, 55] is in the completion of the ideal of regularizing operators. Our choice islarge enough to make it an algebra, to contain the layer potential operators (as weshow in this paper), but small enough to keep it tractable. Our choice leads to alarger algebra than the ones considered in [42, 55].

Recall that Ω Ă Rn is an open cone, so that BΩ has dimension n´ 1.

Lemma 1.9. If P P R´n and f P L2pR` ˆ Bωq, then

PfL2pR`ˆBωq ď CfL2pR`ˆBωq,

where C is a constant depending on P but not on f .

Proof. Let p P CpBΩ ˆ BΩq be the distribution kernel of P . Since P is dilationinvariant and has a continuous kernel, we can use Lemma 1.3 to conclude that it isa Mellin convolution operator.

Let us split P “ Q0 `Q1, where Q0 P Ψ´nproppBΩqR` and Q1 P R´8. Since Q0 is

properly supported, the decay properties of p will depend only on the decay prop-erties of the kernel of Q1, and hence there exists ε ą 0 such that pr`1rqεppr, x1, y1q

STRAIGHT CONES 9

is bounded. Let us consider the functions

Hpx1q “

ż

Bω

ż 8

0

|ppr, x1, y1q|dr

rdSn´2py

1q

and

Jpy1q “

ż

Bω

ż 8

0

|ppr, x1, y1q|dr

rdSn´2px

1q,

which will be bounded on Bω. We have

suppt,x1qPR`ˆBω

ż

R`ˆBω|ppts, x1, y1q|

ds

sdSn´2py

1q

“ supx1PBω

ż

R`ˆBω|pps, x1, y1q|

ds

sdSn´2py

1q “ supx1PBω

Hpx1q ă C.

Similarly,

supps,y1qPR`ˆBω

ż

R`ˆBω|ppts, x1, y1q|

dt

tdSn´2px

1q ă C.

A standard lemma (the generalized Young’s inequality [14]) then gives the result.

Next, we want to extend the definition of the operator valued convolution kernelof an operator in R´8 to operators in Rm. To do this, we choose an approximate

unit φj P R´8. Let P P ΨmpR` ˆ BωqR` . Define Pj “ CφjP . Hence Pj P R´8.

Denote the kernel of Pj by kPj . Then the convolution kernel of P is defined to bekp “ lim

jÑ8kPj . See [9, 42].

We turn our attention to studying the mapping property for P P Rk. Denote byg the cylindrical metric on R` ˆ Bω, that is, g “ pr´1drq2 ` pdx1q2, where pdx1q2

is the metric on Bω induced from the Euclidean metric on Rn. Let L2pR` ˆ Bω, gqbe the L2 space defined by the volume form determined by the metric g and letHspR` ˆ Bω, gq be the Sobolev spaces defined by the same metric. Let ∆g :“prBrq

2 `∆Bω be the Laplacian associated to the metric g. Then HspR` ˆ Bω, gq isobtained as the domain of the powers of ∆g.

Theorem 1.10. Let P P R´8. Then for all k, l P Z, it defines a continuous map

P : H´2kpR` ˆ Bω, gq Ñ H2lpR` ˆ Bω, gq.

Proof. Let Q “ pI ´ ∆gqlP pI ´ ∆gq

k. By Proposition 1.5 and Definition 1.6, wehave Q P R´8. Then the desired result follows from Lemma 1.9 and the fact that

pI ´∆BΩqk : H2kpR` ˆ Bω, gq Ñ L2pR` ˆ Bω, gq

is an isomorphism.

Theorem 1.11. If P P ΨkproppR` ˆ BωqR

`

, then for all m P Z, we have that Pdefines a bounded operator

P : HmpR` ˆ Bω, gq Ñ Hm´kpR` ˆ Bω, gq.

Proof. First of all, suppose that P P Ψ´aproppR` ˆ BωqR`

with a ą 0 and f P

L2pR` ˆ Bωq. Since Pf2L2pR`ˆBωq “ pP˚Pf, fqL2pR`ˆBωq, it is enough to show

that Q :“ P˚P is bounded on L2pR`ˆBωq. But for i large enough, by Lemma 1.9,

Qi P Ψ´2iproppR`ˆBωq is bounded on L2pR`ˆBωq. Hence, if P P Ψ´aproppR`ˆBωqR

`

,

then P is bounded on L2pR` ˆ Bωq.

10 VICTOR NISTOR AND YU QIAO

Then we can use the standard Hormander trick (see for example, [59]) to show

that if P P Ψ0proppR` ˆ BωqR

`

, then P defines a bounded operator

P : L2pR` ˆ Bω, gq Ñ L2pR` ˆ Bω, gq.This completes the proof.

We therefore obtain

Corollary 1.12. Let P P Rk, then for all m P Z, we have that P defines a boundedoperator

P : HmpR` ˆ Bω, gq Ñ Hm´kpR` ˆ Bω, gq.

1.3. Convolution and the Mellin Transform. A standard tool in the study ofdomains with conical points is the Mellin Transform [9, 12, 21, 36]. Recall that theMellin transform M7f of f P CcpR`q is given by

M7fptq “

ż 8

0

r´itfprqdr

r.

We extend this definition to f P CcpΩq by

M7fpt, x1q “

ż 8

0

r´itfprx1qdr

r.

A similar formula defines M7fpt, x1q if f P CcpBΩq. As for the Fourier transform,the Mellin transform extends to L2pΩ, dµnq and to L2pBΩ, dµn´1q, where µn andµn´1 are the measures defined in Equation (6).

Proposition 1.13. The map

p2πq´12M7 : L2pΩ, dµnpx1qq Ñ L2pRˆ ω, dtdSn´1px

1qq

is unitary.

Proof. The Plancherel formula for Fourier transform implies the result since theMellin transform is the composition of

L2pΩ, dµnq Q fprx1q Ñ fpetx1q P L2pRˆ ω, dtdSn´1px

1qq

with the Fourier transform in the t variable.

Definition 1.14. Let p be the convolution kernel of a smoothing Mellin convolutionoperator P on Ω or on BΩ. The Mellin transform M7p of p is defined by

M7ppt, x1, y1q “ qpt, x1, y1q “

ż 8

0

s´itpps, x1, y1qds

s.

The Mellin transform M7P ptq of P is defined to be integral kernel operator withkernel M7ppt, x1, y1q. This definition extends to P P Rk by allowing p to be adistribution with singular support at 1.

We shall need the following.

Example 1.15. We have

M7prBrqptq “ prBrq˚r´it|r“1 “ ´prBrqr

´it|r“1 “ it,

where prBrq˚ is the adjoint of rBr with respect to the measure µn or µn´1. On the

other hand, if P is a differential operator that acts only on the x1 variable, thenM7pP q “ P .

STRAIGHT CONES 11

Remark 1.16. Let us denote by =pzq “ b the imaginary part of a complex numberz “ a ` bi. If P P R´8, then pM7P qptq extends to a holomorphic function int P t|=ptq| ă εu, with ε as in the Definition 1.6. On the other hand, if P P

ΨkproppR` ˆ BωqR

`

, then pM7P qptq is defined for all t P C.

Let us recall the following standard properties of the Mellin transform.

Proposition 1.17. Suppose P P Rk is a Mellin convolution operator on BΩ withoperator-valued convolution kernel p. Then M7P ptq P ΨkpBωq and for any u PCcpBΩq, we have

M7pp ˚ uqptq “M7pPuqptq “M7P ptqM7uptq.

The same is true if P is a smoothing Mellin convolution operator on Ω.

Proof. We have that M7P ptq P ΨkpBωq by [9, 42]. Let vpt, x1q “ pPuqpt, x1q. Letus assume first that u P CcpBΩq and P P R´8. Then we calculate using Fubini’stheorem to interchange the order of integration.

M7vpt, x1q “

ż 8

0

r´itvpr, x1qdr

r

“

ż 8

0

r´itˆż 8

0

ż

ω

pprs, x1, y1qups, y1qdSn´1py1qds

s

˙

dr

r

“

ż

ω

ˆż 8

0

ż 8

0

prsq´itpprs, x1, y1qs´itups, y1qdr

r

ds

s

˙

dSn´1py1q

“

ż

ω

ˆż 8

0

z´itppz, x1, y1qdz

z

˙ˆż 8

0

s´itups, y1qds

s

˙

dSn´1py1q

“

ż

ω

M7ppt, x1, y1qM7upt, y1qdSn´1py1q “ pM7PM7uqpt, x1q.

The general case P P Rk follows by considering an approximate unit φj P R´8,that is a sequence such that φj ˚ f Ñ f . This completes the proof.

Let us denote by Mf the multiplication operator by f .

Proposition 1.18. Let P P Rk be a Mellin convolution operator on BΩ withoperator-valued convolution kernel ppr, x1, y1q. Then Q :“ MraPMr´a is still aconvolution operator with kernel kQpr, x

1, y1q “ rappr, x1, y1q. The same is true if Pis a smoothing Mellin convolution operator on Ω.

Proof. If u P CcpBΩq, we have

pMraPMr´auq pr, x1q “

ż

ω

raż 8

0

pprs, x1, y1qs´aups, y1qds

sdSn´1py

1q

“

ż

ω

ż 8

0

prsqapprs, x1, y1qups, y1qds

sdSn´1py

1q,

where the integrals represent pairings of distributions and functions.

This gives the following.

Corollary 1.19. Let P P ΨkproppR` ˆ BωqR

`

. Then

rzPr´z P ΨkproppR` ˆ BωqR

`

for all z P C. Also, if P P R´8, then rzPr´z P R´8 if |=pzq| ă ε, where ε is as inDefinition 1.6.

12 VICTOR NISTOR AND YU QIAO

The above corollary allows us to extend the Mellin transform to the class ofoperators P such that raPr´a P Rk for some a and k. The Mellin transform ofP will then be defined and holomorphic in a set containing |=pz ´ aq| ă ε formsome small ε ą 0. Also, we shall agree that the Mellin transform is extendedholomorphically to the largest set of the form ta ă =pzq ă bu.

The Lebesgue measure on Rn is given in spherical coordinates by

dx “ rn´1drdSn´1px1q,

where dSn´1px1q is the surface measure on the unit sphere in Rn. Consider the

Hilbert spaces L2pΩ, dxq and L2pΩ, dµnpx1qq. Then the map

Mr

n2

: L2pΩ, dxq Ñ L2pΩ, dµnpx1qq

is unitary.

Definition 1.20. We define the shifted Mellin transform on Ω

M : L2pΩ, dxq Ñ L2pRˆ ω, dtdSn´1px1qq

by the formula

Mu “M7pMr

n2uq,

so Muptq “M7upt` in2q.

Since the measure on BΩ is rn´2drdSn´2px1q, for BΩ we adapt the above defini-

tion as follows.

Definition 1.21. We define the shifted Mellin transform on the boundary by BΩ,

MBΩ : L2pBΩ, rn´2drdSn´2px1qq Ñ L2pRˆ Bω, dtdSn´2px

1qq,

by the formula

pMBΩ fqpt, x1q “M7pM

rn´12fq,

so MBΩfptq “M7fpt` ipn´ 1q2q.

The shift in Definition 1.21 is necessary to make MBΩ unitary up to a multiple.We will drop BΩ if no confusion can arise.

Lemma 1.22. Let us use the same notation as above.

(i) Let P be a smoothing Mellin convolution operator on Ω with operator-valuedconvolution kernel ppr, x1, y1q. Then for any u P CcpΩq, we have

Mpp ˚ uqptq “MpPuqptq “MP ptqMuptq

and MpP q “M7pMr

n2PM

r´n2q.

(ii) Let P P Rk be a Mellin convolution operator on BΩ with operator-valuedconvolution kernel ppr, x1, y1q. Then for any f P CcpBΩq, we have

Mpp ˚ fqptq “MpPfqptq “MP ptqMfptq

and MpP q “M7pMr

n´12PM

r´n´12q.

Proof. The results immediately follow from Propositions 1.17 and 1.18

STRAIGHT CONES 13

2. Norm closure and invertibility

We now study the relation between the invertibility of the operators in R0 andthe Mellin transform. To this end, we need to consider the norm closure of thesealgebras. See [33] for some similar results.

Definition 2.1. We shall denote by A the completion of R´1 in the norm ofbounded operators acting on L2pBΩ, dµn´1q.

Denote by A` the unitalization of A and by C0pR,KpL2pBωqqq the C˚-algebraof all continuous functions from R to KpL2pBωqq that vanish at infinity.

Theorem 2.2. The Mellin transform M7 induces an isomorphism from A to theC˚-algebra C0pR,KpL2pBωqqq. Moreover, λ ` P is invertible in A` if, and only ifλ` pM7P qptq is invertible in LpL2pBωqq for all t P R.

We need some preparations to prove this theorem.

Lemma 2.3. Let n P N0 Y t8u be arbitrary and B be a Banach algebra. Supposethat κ P CnpR,Bq fulfills κptq´1 P B for all t P R. Then κ´1 P CnpR,Bq also. Inparticular, for κ P CpR,Bq, we have

κ´1 P CpR,Bq exists ðñ κptq´1 P B exists for all t P R.

Proof. First of all, recall that since B is a Banach algebra, the group of its invertibleelements B´1 is open and the inversion B´1 Q a ÞÝÑ a´1 is infinitely many timesdifferentiable. Let us first treat the case n “ 0: It is clear, that the existence ofκ´1 P CpR,Bq implies that κptq is invertible for all t P R. Conversely, assume thatκptq´1 P B exist for all t P R. We define κ´1ptq :“ κptq´1. Then we have to checkthat κ´1 P CpR,Bq holds. Let t0 P R be arbitrary:

κpt0q´1 ´ κptq´1 “ κpt0q

´1 pκptq ´ κpt0qqκ´1ptq

which gives κ´1 P CpR,Bq after passing to the limit t Ñ t0, since B has an opengroup of invertible elements and the inversion is continuous.

Now, if n “ 1, we use

κptq´1 ´ κpt0q´1

t´ t0“ ´κpt0q

´1κpt0q ´ κptq

t0 ´ tκptq´1,

to see the differentiability of κ´1. An iteration of this argument yields then theassertion.

Proposition 2.4. We have the following:

(i) R´8 Ă A;

(ii) If P is a Mellin convolution operator and P P Ψ´1proppR`ˆBωqR

`

, then M7P P

C0pR,KpL2pBωqqq.

Proof. The first statement is clear. By Theorem 8.8 in [14], if P P Ψ´nproppR` ˆBωqR

`

, then ppr, x1, y1q is continuous for all variables. We notice that if P P

Ψ´nproppR` ˆ BωqR`

, by Lemma 1.9, there exists a constant C such that for allt P R,

pM7P qptq ď C.

Now, if P P Ψ´n´1prop pR` ˆ BωqR

`

, then p1 ` rBrqP P Ψ´nproppR` ˆ BωqR`

. Also wehave

M7´

`

1` rBr˘

P¯

ptq “ p1` itqpM7P qptq.

14 VICTOR NISTOR AND YU QIAO

So we obtain

pM7P qptq ď Cp1` |t|q´1 Ñ 0 as |t| Ñ 8.

This implies M7P P C0pR,KpL2pBωqqq.We then use Hormander’s trick to complete the proof of the second statement.

Suppose P P Ψ´1proppR` ˆ BωqR

`

and f P L2pR` ˆ Bωq. Since Pf2L2pR`ˆBωq “

pP˚Pf, fqL2pR`ˆBωq, it suffices to prove that some power of Q :“ P˚P is in

Ψ´n´2prop pR` ˆ BωqR

`

. But Qj P Ψ´2jproppR` ˆ Bωq, so for j large enough, this fol-

lows from the above discussion.

We are ready now to complete the proof of Theorem 2.2.Proof of Theorem 2.2. By Proposition 2.4, we have if P P R´1, then M7P P

C0pR,KpL2pBωqqq. Since C0pR,KpL2pBωqqq – C0pRq bKpL2pBωqq and the set

spantpfT qptq :“ fptqT | f P C0pRq, T P KpL2pBωqqu

is dense in C0pRqbKpL2pBωqq, the image of R´1 under M7 is dense in C0pR,KpL2pBωqqq.This gives surjectivity. To prove injectivity, we notice that A “ KpL2pBωqq or R(reduced crossed-product), where R acts on KpL2pBωqq trivially. Since R is com-mutative and its action is trivial, we have KpL2pBωqq or R – KpL2pBωqq b C0pRq.Hence the Mellin transform M7 is an isomorphism. Let us also notice by the re-sults of [30], the algebra R´1 is closed under holomorphic functional calculus in A.Similarly, R0 is also closed under holomorphic functional calculus.

The second part follows from Lemma 2.3. l

3. The double layer potential operator K on Straight Cones

Let ω Ă Sn´1 (n > 3) be an open subset with smooth boundary. We allow ω tobe disconnected. Denote by Ω “ R`ω the cone with base ω, as before (Equation(5)). We denote by dSn´1pxq be the surface measure with respect to BΩ. In thesegeneralized spherical coordinates, the Laplacian ∆ “

ř

j B2j on Ω is given by

∆ “ r´2´

prBrq2` pn´ 2qrBr `∆Sn´1

¯

,

where ∆Sn´1 is the Laplacian on Sn´1. Let

(9) Epx, yq “1

p2´ nqωn

1

|x´ y|n´2“

cn|x´ y|n´2

be the standard fundamental solution for ∆ (so ωn is the area of the unit sphere inRn). We shall denote K “ Kp∆;E; Ωq, S “ Sp∆;E; Ωq.

Let ν denote the outward unit normal vector. A convenient way of writing theoperators K and S for the cone takes into account the dilation invariance of thecone. Let x “ rx1, y “ sy1, and denote by dSn´2py

1q the measure on Bω and νpyqthe unit outer normal vector to BΩ at the point y P BΩ. Then the double layer

STRAIGHT CONES 15

potential operator K can be written as

Kgpxq “

ż

BΩ

BE

Bνypx, yqgpyq dSn´1pyq(10)

“ ´

ż

BΩ

px´ yq ¨ νpyq

ωn|x´ y|ngpyq dSn´1pyq

“ ´1

ωn

ż 8

0

ż

Bω

prx1 ´ sy1q ¨ νpy1q

|rx1 ´ sy1|ngpsy1qsn´2 dSn´2py

1q ds

“ ´1

ωn

ż 8

0

ż

Bω

rsx1 ¨ νpy1q

| rsx1 ´ y1|n

gpsy1q dSn´2py1qds

s.

Thus Equation (10) shows that K is a Mellin convolution operator with theoperator-valued convolution kernel

(11) kpr, x1, y1q :“ ´1

ωn

rx1 ¨ νpy1q

|rx1 ´ y1|n“ ´

1

ωn

rx1 ¨ νpy1q

pr2 ´ 2rx1 ¨ y1 ` 1qn2.

To obtain similar results for the single layer potential operator S, we shall needthe operator Φ : C8c pΩq Ñ L2

locpΩq defined by

pΦuqpxq “

ż

Ω

cn|x´ y|n´2

upyqdy(12)

“

ż 8

0

ż

ω

cn|rx1 ´ sy1|n´2

upsy1qsn´1 dSn´1py1q ds

“

ż 8

0

ż

ω

cn s2

| rsx1 ´ y1|n´2

upsy1q dSn´1py1qds

s,

where x “ rx1, y “ sy1 and dSn´1py1q denotes the surface measure on the unit

sphere in Rn.Let Φ0 “Mr´1ΦMr´1 . Then for u P CcpΩq, we have

(13) Φ0upxq “

ż 8

0

ż

ω

cnp rs q|

rsx1 ´ y1|n´2

upsy1q dSn´1py1qds

s.

From Equation (13), we see that Φ0 is a Mellin convolution operator with kernel

(14) kΦ0pr, x1, y1q “ φ0pr, x

1, y1q “cn

r|rx1 ´ y1|n´2.

Lemma 3.1. For any t P R`, we have

(i) Kαt “ αtK;(ii) Φ0αt “ αtΦ0;

(iii) prBrqkαt “ αtprBrq

k, where r is the radial variable.

Proof. Since K and Φ0 are both Mellin convolution operators, the conclusions of(i) and (ii) follow from Lemma 1.3. To prove (iii), it is enough to notice thatrBr “ α1t|t“0.

Proposition 3.2. For u P C1c pΩq, we have

MprBruqptq “ pit´n

2qpMuqptq.

Proof. This follows from Example 1.15 and Lemma 1.17. Indeed, we have MprBrq “M7prn2prBrqr

´n2q “M7prBr ´ n2q “ it´ n2.

16 VICTOR NISTOR AND YU QIAO

We shall need the range for which MpΦ0q is defined.Let Diff2

pωq denote the set of differential operators of order two on ω, and alsolet ∆0 “Mr∆Mr.

Lemma 3.3. Let r∆0 :“Mr

n2p∆0qMr´

n2

and rΦ0 “Mr

n2

Φ0Mr´n2

.

(i) Mp∆0qptq “M7r∆0ptq P Diff2

pωq;

(ii) MΦ0 “M7rΦ0;

(iii) The operator rΦ0 has Mellin convolution kernel

rφ0pr, x1, y1q “

cn

|r12x1 ´ r´

12 y1|n´2

,

where x1, y1 P ω.

Proof. By Lemma 1.22, the first two statements are clear from the definitions. We

need to check that rΦ0 has convolution kernel rφ0. Indeed, by Proposition 1.18, the

operator rΦ0 has Mellin convolution kernel

rφ0pr, x1, y1q “ r

n2 φ0 “

cnrn2

r|rx1 ´ y1|n´2“

cn

|r12x1 ´ r´

12 y1|n´2

.

The proof is complete.

We can now prove the following.

Lemma 3.4. For u P C8c pΩq, we have

(15) MpΦ0uqptq “MΦ0ptqMuptq,

provided that =ptq P p´n´22 , n´2

2 q.

Proof. Equation (15) follows right away from Lemma 1.22, whenever both sidesare defined. To check the range for which MΦ0 is defined, we use Lemma 3.3 (ii)

and (iii) that identifies the convolution kernel φ0 of rΦ0. Then we notice that forα P p´n´2

2 , n´22 q

|rαφ0pr, x1, y1q| “

cnrα

|r12x1 ´ r´

12 y1|n´2

.

decays fast as r Ñ 0 or r Ñ8.

Denote by ∆ω the restriction of ∆Sn´1 to ω Ă Sn´1. Let Φ0 “ Mr´1ΦMr´1 , asbefore, and recall that ∆0 “Mr∆Mr.

Proposition 3.5. We have

Mp∆Φqptq “Mp∆0Φ0qptq “

ˆ

∆ω ´ t2 ´

pn´ 2q2

4

˙

pMΦ0qptq “ I.

In particular, the function Mφ0ptq is a fundamental solution for the operator ∆ω´

t2 ´pn´ 2q2

4and domain ω, provided that =ptq P p´n´2

2 , n´22 q.

Proof. First of all, since ∆Φ “ I, we have

∆0Φ0 “Mr∆ΦMr´1 “MrIMr´1 “ I.

We compute

Mr∆Mr “ prBrq2 ` nprBrq ` pn´ 1q `∆ω.

STRAIGHT CONES 17

By Lemma 3.1, both Mr∆Mr and Mr´1ΦMr´1 are invariant with respect to dila-tions. Proposition 3.2 and Lemma 3.4 give

Mp∆Φqptq “ Mp∆0Φ0qptq “M∆0ptqMΦ0ptq

“

´

pit´n

2q2 ` npit´

n

2q ` pn´ 1q `∆ω

¯

pMΦ0qptq

“

ˆ

∆ω ´ t2 ´

pn´ 2q2

4

˙

MΦ0ptq.

On the other hand, since ∆Φ “ I, we get

Mp∆Φqptq “ I.

From the calculation above, we have´

∆ω ´ t2 ´

pn´ 2q2

4

¯

MΦ0ptq “ I. Hence,

we have completed the proof.

Corollary 3.6. We have

M7pr∆ rΦ0qptq “´

∆ω ´ t2 ´

pn´ 2q2

4

¯

M7rΦ0ptq “ I.

In particular, the function M7rφ0ptq is a fundamental solution for the operator

∆ω ´ t2 ´

pn´ 2q2

4, provided that =ptq P p´n´2

2 , n´22 q.

Proof. This follows right away from Proposition 3.5 and Lemma 3.3.

We now turn our attention to the double layer potential operator K.

Lemma 3.7. The operator rK “Mr

n´12KM

r´n´12

has convolution kernel

(16) rkpr, x1, y1q “ ´1

ωn

r12x1 ¨ νpy1q

|r12x1 ´ r´

12 y1|n

.

Proof. Recall that the kernel k of double layer potential operator K is given in

Equation (11). By Proposition 1.18, the operator rK has Mellin convolution kernel

(17) rkpr, x1, y1q “ rn´12 kpr, x1, y1q “ ´

1

ωn

r12x1 ¨ νpy1q

|r12x1 ´ r´

12 y1|n

.

The proof is complete.

4. The Single Layer Potential Operator S on Straight Cones

Let us consider now the single layer potential operator S. With the notation inthe above section, the single layer potential S satisfies

Sgpxq “

ż

BΩ

cn|x´ y|n´2

gpyq dSn´1pyq(18)

“

ż 8

0

ż

Bω

cn|rx1 ´ sy1|n´2

gpsy1qsn´2 dSn´2py1q ds

“

ż 8

0

ż

Bω

cnrrs |rsx1 ´ y1|n´2

gpsy1q dSn´2py1qds

s,

where x “ rx1 and y “ sy1. The operator S is not dilation invariant, but behavesunder dilations like r. This suggests to introduce S0 :“M

r´12SM

r´12

, then, unlike

18 VICTOR NISTOR AND YU QIAO

S, S0 is a Mellin convolution operator, and its operator-valued convolution kernelis given by

(19) hpr, x1, y1q “cn

r12 |rx1 ´ y1|n´2

.

Let us also define rS0 “Mr

n´12S0M

r´n´12

.

Lemma 4.1. The operator rS0 has Mellin convolution kernel

rhpr, x1, y1q “cn

|r12x1 ´ r´

12 y1|n´2

,

where x1, y1 P Bω.

Proof. By Proposition 1.18, the operator ĂS0 has Mellin convolution kernel

rhpr, x1, y1q “ rn´12 hpr, x1, y1q “

cnrn´12

r12 |rx1 ´ y1|n´2

“cn

|r12x1 ´ r´

12 y1|n´2

.

Remark 4.2. So rh is obtained as a boundary value of rφ0.

Lemma 4.3. The operator rS0 is symmetric.

Proof. Let rS˚0 be the formal adjoint of rS0. Then the convolution kernel of rS˚0 is

rh˚pr, x1, y1q :“ rhpr´1, y1, x1q.

From the expression of rh in Lemma 4.1, we have rh˚ “ rh. Hence rS0 is symmetric.

5. Mapping properties of K and S

Let m P Z`, α P Zn` be a multi-index. We define the m-th Sobolev space on Ωwith weight r and index a by

Kma pΩq “ tu P L2locpΩ, dxq, r

|α|´aBαu P L2pΩ, dxq, for all |α| ď mu.

The norm on Kma pΩq is u2Kma pΩq

:“ř

|α|ďm r|α|´aBαu2L2pΩ,dxq. By Theorem

5.6 in [4], this norm is equivalent to u2m,a :“ř

|α|ďm r´aprBqαu2L2pΩ,dxq, where

prBqα “ prB1qα1prB2q

α2 ¨ ¨ ¨ prBnqαn . Clearly, we have that rtKma pΩq – Kma`tpΩq. In

general, this isomorphism may not be an isometry.

Proposition 5.1. We have, for all m P Z,

Kmn2pΩq – HmpΩ, gq,

where the metric g is pr´1drq2 ` pdx1q2.

Proof. The result follows from Proposition 5.7 in [4], and can also be easily checkedby direct computations.

The identification given above allows us to define weighted Sobolev spaces onthe boundary Kma pBΩq. For more details, see [4].

Proposition 5.2. For m P Z`, we have the following identification:

Kmn´12

pBΩq – HmpBΩ, gq,

STRAIGHT CONES 19

Proof. The result follows from Definition 5.8 in [4], and can also be easily checkedby direct computations.

To study the mapping properties of operators in Rk, we need to isolate theirsingularity near 1. For this purpose, let us now choose a smooth cutoff function

(20) χpr, x1, y1q “ χprq “

"

1 if |r ´ 1| 6 δ,0 if |r ´ 1| > 2δ.

First, let us recall the explicit form of some of the operators that we will needbelow.

By Lemma 3.7 and Proposition 1.18, the operator MrarKMr´a has convolution

kernel

(21) rkapr, x1, y1q “ rarkpr, x1, y1q “ ´

1

ωn

ra`12x1 ¨ νpy1q

|r12x1 ´ r´

12 y1|n

.

By Lemma 4.1 and Proposition 1.18 MrbrS0Mr´b is also a Mellin convolution

operator with kernel

rhbpr, x1, y1q “ rbrhpr, x1, y1q “

cnrb

|r12x1 ´ r´

12 y1|n´2

.(22)

Proposition 5.3. Let rka and rhb be given by Equations (21) and (22) above.

(i) If ´n`12 ă a ă n´1

2 , then p1´ χqrka P C8pr0,8s ˆ Bω ˆ Bωq.

(ii) If ´n´22 ă b ă n´2

2 , then p1´ χqrhb P C8pr0,8s ˆ Bω ˆ Bωq.

Proof. Let rpzq “1` z

1´ z. Then r : r´1, 1s Ñ r0,8s is a map with rp´1q “ 0 and

rp1q “ 8. Under this changes of variables, the differential operator rBr (acting onfunctions on R`) corresponds to the differential operator Dz “

12 p1 ` zqp1 ´ zqBz

(acting on functions on r´1, 1s). The pullback of the differential structure on p0,8qdefines a differential structure on r´1, 1s with smooth functions C8b pr´1, 1sq. Moreprecisely, C8b pr´1, 1sq consists of continuous functions ξ : r´1, 1s Ñ C such thatDkz ξ is also continuous on r´1, 1s for all k. Clearly, we have

Dz

ˆ

1` z

1´ z

˙m

“1

2p1` zqp1´ zqBz

ˆ

1` z

1´ z

˙m

“ m

ˆ

1` z

1´ z

˙m

.

Moreover, we calculate

gapz, x1, y1q :“

´

1´ χprpzqq¯

rka

´

rpzq, x1, y1¯

“ ´1

ωn

´1` z

1´ z

¯

n`12 `a x

1 ¨ νpy1q´

1´ χp 1`z1´z q

¯

ˇ

ˇ

ˇ

´

1`z1´z

¯

x1 ´ y1ˇ

ˇ

ˇ

n .

If ´n`12 ă a ă n´1

2 , the function ga can be extended to ´1 and 1. Applying theoperator Dz to the function ga, we obtain a function of the same type as ga. By

iterating differentiation, we find that ga P C8b pr´1, 1sq ˆ Bω ˆ Bω, so p1 ´ χqrka P

20 VICTOR NISTOR AND YU QIAO

C8pr0,8s ˆ Bω ˆ Bωq. Similarly, we compute

´

1´ χprpzqq¯

rhb

´

rpzq, x1, y1¯

“

cnp1`z1´z q

b´

1´ χp 1`z1´z q

¯

|p 1`z1´z q

12x1 ´ p 1`z

1´z q´ 1

2 y1|n´2

“ cn

´1` z

1´ z

¯

n´22 `b 1´ χp 1`z

1´z qˇ

ˇ

ˇ

´

1`z1´z

¯

x1 ´ y1ˇ

ˇ

ˇ

n´2 .

For the same reason, p1´ χqrhb P C8pr0,8s ˆ Bω ˆ Bωq, provided that ´n´2

2 ă

b ă n´22 .

Lemma 5.4. Let rka and rhb be given by Equations (21) and (22) above.

(i) If ´n`12 ă a ă n´1

2 , choose ε1 ą 0 so that a ` ε1 ă n ´ 1 and a ´ ε1 ą ´1,then there exists some constant C1 “ C1pε1, i, j, lq, such that

suppr,x1,y1qPR`ˆBωˆBω

´

r `1

r

¯ε1 ˇˇ

ˇprBrq

i∆jBω,x1∆

lBω,y1p1´ χq

rkapr, x1, y1q

ˇ

ˇ

ˇď C1;

As a consequence, p1´ χqrka P R´8.(ii) If ´n´2

2 ă b ă n´22 , choose ε2 ą 0 so that b ` ε2 ă

n´22 , then there exists

some constant C2 “ C2pε2, i, j, lq, such that

suppr,x1,y1qPR`ˆBωˆBω

´

r `1

r

¯ε2 ˇˇ

ˇprBrq

i∆jBω,x1∆

lBω,y1p1´ χq

rhbpr, x1, y1q

ˇ

ˇ

ˇď C2.

As a consequence, p1´ χqrhb P R´8.

Proof. This is a consequence of Proposition 5.3.

We are ready to prove one of our main technical results.

Theorem 5.5. Let rK “Mr

n´12KM

r´n´12

and rS0 “Mr

n´12S0M

r´n´12

, then

(i) If ´n`12 ă a ă n´1

2 , then MrarKMr´a P R´1;

(ii) If ´n´22 ă b ă n´2

2 , then MrbrS0Mr´b P R´1.

Proof. Using the cutoff function χ in Equation (20), we write

rka “ p1´ χqrka ` χrka.

By Lemma 5.4, we have p1´χqrka P R´8. Since χrk is properly supported, dilation

invariant, and of order ´1, by Corollary 1.19, we have χrka P Ψ´1proppR` ˆ BωqR

`

.

Therefore, we have MraKMr´a P R´8 ` Ψ´1proppR` ˆ BωqR

`

“ R´1. The same

argument can be applied to MrbrS0Mr´b .

Similarly, we have the following.

Theorem 5.6. The following operators are well-defined and continuous for allm P Z:

(i) MrarKMr´a : Kmn´1

2

pBΩq Ñ Km`1n´12

pBΩq, for ´n`12 ă a ă n´1

2 ;

(ii) MrbrS0Mr´b : Kmn´1

2

pBΩq Ñ Km`1n´12

pBΩq for ´n´22 ă b ă n´2

2 .

Proof. The results follow immediately from Theorem 5.5, Theorem 5.2, and Corol-lary 1.12.

STRAIGHT CONES 21

For the operators of interest, we obtain the following.

Corollary 5.7. We have the following mapping properties:

(i) if ´1 ă a ă n´ 1 and m P Z, then

K : Kmn´12 ´a

pBΩq Ñ Km`1n´12 ´a

pBΩq,

(ii) if 12 ă b ă n´ 3

2 and m P Z, then

S0 : Kmn´12 ´b

pBΩq Ñ Km`1n´12 ´b

pBΩq.

(iii) if 12 ă b ă n´ 3

2 and m P Z, then

S : Kmn´22 ´b

pBΩq Ñ Km`1n2´b

pBΩq.

Proof. The conclusions follow from the relations rK “ Mr

n´12KM

r´n´12

, rS0 “

Mr

n´12S0M

r´n´12

, and S0 “Mr´

12SM

r´12

.

We shall also need the following in order to prove that the pointwise invertibilityof the Mellin transform implies the global invertibility.

Lemma 5.8. We have rK, rS0 P A, where A is defined in Subsection 1.3. As aresult, K, S0 P A.

Proof. This is a consequence of Theorem 5.5 and Definition 2.1.

6. Invertibility of layer potentials

In this section, we will use functional analytic argument to reduce global invert-ibility of 1

2 ` K and S, acting on suitable Sobolev spaces, to the invertibility of

their Mellin transform for each t. To this end, we will use that rS0 and rK are inR´1.

We now come to one of our main results, which identifies the Mellin transformof the double layer potential operator Kp∆, E; Ωq of ∆ on the cone Ω :“ R`ω in

terms of the double layer potential of ∆ω ´ t2 ´

pn´2q2

4 on ω. Recall that E is thestandard fundamental solution of ∆, that φ0 is the Mellin convolution kernel of Φ0,

and that rφ0 is the Mellin convolution kernel of rΦ0 :“Mr

n2

Φ0Mr´n2

.

Theorem 6.1. Let ´n´22 ă =ptq ă n´2

2 . We have

pMKqpt´ i2q “M7rKpt´ i2q “ K

´

∆ω ´ t2 ´

pn´ 2q2

4,M7

rφ0ptq; ω¯

.

Let us notice also that M7rφ0 “Mφ0, and hence

pMKqpt´ i2q “ K´

∆ω ´ t2 ´

pn´ 2q2

4,Mφ0ptq; ω

¯

.

In the following, we shall denote

cptq :“ ´t2 ´pn´ 2q2

4.

22 VICTOR NISTOR AND YU QIAO

Proof. It is clear from the definition and Lemma 1.22 that MK “M7rK. Let and

Lt “ ∆ω`cptq, cptq :“ ´t2´ pn´2q2

4 , as above. By Corollary 3.6, for each fixed t sat-

isfying ´n´22 ă =ptq ă n´2

2 , the function M7rφ0pt, x

1, y1q is a fundamental solution

for the elliptic operator Lt “ ∆ω ` cptq on ω Ă Sn´1. Furthermore, by Corollaries

3.3(iii) and 3.6, the double layer potential associated to Lt, M7rφ0pt, x

1, y1q, and ωis

K´

Lt,M7rφ0ptq; ω

¯

“ Bνy1M7rφ0pt, x

1, y1q “

ż 8

0

r´itBνy1rφ0pr, x

1, y1qdr

r

“

ż 8

0

r´itBνy1cn

|r12x1 ´ r´

12 y1|n´2

dr

r“ ´

1

ωn

ż 8

0

r´itx1 ¨ νpy1q

|r12x1 ´ r´

12 y1|n

dr

r,

since ´1ωn “ pn´ 2qcn.On the other hand, Lemma 3.7 gives

M7rKpt´ i2, x1, y1q “

ż 8

0

r´it´12rkpr, x1, y1q

dr

r

“ ´1

ωn

ż 8

0

r´it´12 r12x1 ¨ νpy1q

|r12x1 ´ r´

12 y1|n

dr

r“ K

´

Lt,M7rφ0ptq; ω

¯

.

This gives the desired result.

Theorem 6.2. Let ´n´22 ă =ptq ă n´2

2 . We have

MS0ptq “M7rS0ptq “ S

´

∆ω ´ t2 ´

pn´ 2q2

4,M7

rφ0ptq;ω¯

.

Let us notice also that M7rφ0 “Mφ0, and hence

MSptq “ S´

∆ω ´ t2 ´

pn´ 2q2

4,M7

rφ0ptq; ω¯

“ S´

∆ω ´ t2 ´

pn´ 2q2

4,Mφ0ptq; ω

¯

.

Proof. The relation MS0 “ M7rS0 follows from definitions. Let Lt “ ∆ω ` cptq,

with cptq “ ´t2 ´ pn´2q2

4 as before . By Proposition 3.5, for each fixed t satisfying

´n´22 ă =ptq ă n´2

2 , the function M7rφ0pt, x

1, y1q, x1, y1 P ω, is a fundamental

solution for the elliptic operator Lt “ ∆ω ` cptq on ω Ă Sn´1. Hence, the singlelayer potential associated to Lt, Mφ0pt, x

1, y1q, and ω is simply the operator with

kernel M7rφ0pt, x

1, y1q, x1, y1 P Bω, and hence

SpLt,M7rφ0ptq;ωq “M7

rφ0pt, x1, y1q “

ż 8

0

r´itcn

|r12x1 ´ r

12 y1|n´2

dr

r.

Similarly, recall that rhpt, x1, y1q is the convolution kernel of rS0, and hence

M7rS0pt, x

1, y1q “

ż 8

0

r´itrφ0pt, x1, yq

dr

r

“

ż 8

0

r´itcn

|r12x1 ´ r

12 y1|n´2

dr

r,

so the proof is complete.

STRAIGHT CONES 23

In the next two theorems, in order to apply theorems in [58] to obtain invertibilityof 1

2 `K and S, we need to assume that ω Ă Sn´1 is connected.

Theorem 6.3. Assume ω is connected. Let ´n´22 ă =ptq ă n´2

2 . Then the oper-

ator 12 `MKptq “ 1

2 `M7rKptq : HmpBωq Ñ HmpBωq is invertible. In particular,

if ´n´22 ă a ă n´2

2 , then

1

2` rK : Kmn´1

2 ´apBΩq Ñ Kmn´1

2 ´apBΩq

is invertible for all m P Z.

Proof. By Theorem 6.1, we can apply the double layer potential theory to the

operator ∆ω ` cptq, cptq “ ´t2 ´ pn´2q2

4 , with domain ω. First, we notice that

∆ω ` cptq : H10 pωq Ñ H´1pωq is invertible for all t satisfying ´n´2

2 ă =ptq ă n´22 .

Since ω has smooth boundary, by Proposition 6.3 in Chapter 3 in [58], we obtain

that1

2`MKptq : HmpBωq Ñ HmpBωq is invertible for each t satisfying ´n´2

2 ă

=ptq ă n´22 . By Lemma 5.8 and Theorem 2.2, the operator 1

2 `rK : Kmn´1

2 ´apBΩq Ñ

Kmn´12 ´a

pBΩq is invertible.

We now prove analogous results for the single layer potential operator.

Theorem 6.4. Assume ω is connected. Let ´n´22 ă =ptq ă n´2

2 . Then the

operator pMS0qptq “M7rS0ptq : HmpBωq Ñ Hm`1pBωq is invertible. In particular,

if ´n´22 ă b ă n´2

2 then

rS0 : Kmn´12 ´b

pBΩq Ñ Km`1n´12 ´b

pBΩq

is invertible for each m P Z.

Proof. By Theorem 6.2, we can apply the single layer potential theory to ∆ω` cptqwith domain ω. Since ω has smooth boundary, by Proposition 6.5 in Chapter 3 in

[58], we know that pMS0qptq “ M7rS0ptq : HmpBωq Ñ Hm`1pBωq is invertible for

each t satisfying ´n´22 ă =ptq ă n´2

2 . By Lemma 5.8 and Theorem 2.2 again, the

operator rS0 : Kmn´12 ´b

pBΩq Ñ Km`1n´12 ´b

pBΩq is invertible.

Corollary 6.5. Let m P Z be arbitrary.

(1) 12 `K : Kmn´1

2 ´apBΩq Ñ Kmn´1

2 ´apBΩq is invertible for 1

2 ă a ă n´ 32 .

(2) S0 : Kmn´12 ´b

pBΩq Ñ Km`1n´12 ´b

pBΩq is invertible for 12 ă b ă n´ 3

2 .

(3) S : Kmn´22 ´b

pBΩq Ñ Km`1n2´b

pBΩq is invertible for 12 ă b ă n´ 3

2 .

Proof. Since rK “ Mr

n´12KM

r´n´12

and rS0 “ Mr

n´12S0M

r´n´12

, the results are

direct consequences of the preceding two theorems.

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Pennsylvania State University, Math. Dept., University Park, PA 16802E-mail address: nistor@math.psu.edu

Pennsylvania State University, Math. Dept., University Park, PA 16802E-mail address: qiao@math.psu.edu