Interpolation of Linear Operators

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T R A N S L A T I O N S O F

MATHEMATICAL M O N O G R A P H S

V O L U M E 5 4

S. G. Kreln Ju. I. Petunin E. M. Semenov

Interpolatio n of Linea r Operator s

S|f IHl̂ H JJB American Mathematical Society

10.1090/mmono/054

HHTEPnOJIflUHfl JIHHEftHM X OnEPATOPO B

C. T. KPEMH, K). HL nETYHHH H E. M, CEMEHOB

M34ATEJIBCTBO «HAYKA » TJIABHAH PE^AKLJM H

d>M3MKOMATEMATHHECKOfi[ JIMTEPATYPb l MOCKBA 197 8

Translated fro m th e Russia n by J . Szuc s Translation edite d b y Le v J. Leifma n

2000 Mathematics Subject Classification. Primar y 46-XX .

ABSTRACT. Th e boo k i s devote d t o an importan t directio n i n functiona l analysis : interpolatio n theory fo r linea r operators . Th e mai n method s fo r constructin g interpolatio n space s ar e ex -pounded an d thei r propertie s ar e studied . Thes e method s allo w on e t o loo k a t a numbe r o f theorems an d inequalitie s o f classica l analysi s fro m a ne w standpoint . Interpolatio n theor y fo r operators ha s numerous applications i n Fourie r series , approximation theory , partia l differentia l equations, etc . Some of the m are developed i n the book.

Library o f Congres s Cataloging in Publicatio n Dat a

Krein, S. G. (Seli m Grigor'evich) , 1917— Interpolation o f linea r operators . (Translations o f mathematica l monographs ; 54) Translation of : Interpoliatsii a linelnyk h operatorov . Bibliography: p . Includes indexes . 1. Linea r operators . I . Petunin , IUri i Ivanovich . II . Semenov , E . M. III .

Title. IV . Series . QA329.2.K7313 515.7*24 6 81-2063 7 ISBN 0-8218-4504- 7 (Har d cover ) AACR 2 ISBN 0-8218-3176-3 (Sof t cover ) ISSN 0065-928 2

© Copyrigh t 198 2 b y th e America n Mathematica l Society . Printed i n the Unite d State s o f America .

The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s within th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http://www.ams.org / Copying an d reprintin g informatio n ca n b e foun d a t th e bac k o f thi s volum e

10 9 8 7 6 5 4 3 2 0 7 0 6 0 5 04 0 3 0 2

TABLE OF CONTENT S

FOREWORD T O THE AMERICAN EDITIO N vi i

FOREWORD i x

CHAPTER I . IMBEDDED, INTERMEDIATE, AN D INTERPOLA -TION BANAC H SPACE S 1

1. Imbeddin g of Banach spaces 1 2. Dua l space s of imbedded Banac h spaces 6 3. Intermediat e Banach space s 9 4. Interpolatio n space s and interpolation triple s 1 8

CHAPTER II . INTERPOLATION I N SPACE S OF MEASURABL E FUNCTIONS 3 9

Introduction 3 9 1. Positiv e functions o n a semiaxis and thei r dilation function s 4 6 2. Rearrangement s o f measurable functions 5 8 3. Operator s in th e Banach couple L,(0, oc), £^(0, °°) 7 7 4. Symmetri c spaces. Interpolation betwee n L x an d L^ 9 0 5. Lorent z and Marcinkiewic z spaces 10 7 6. Operator s of weakened an d weak type 12 4 7. Th e singular Hilber t operato r 15 0 8. Interpolatio n theorem s for space s with different measure s 15 6 9. Application s to the theory of orthogonal serie s 17 3

CHAPTER III . SCALES OF BANACH SPACE S 18 7 1. Scales of Banac h spaces . Related space s 18 7 2. Maxima l and minima l normal scale s 19 2 3. Th e scale of Holde r spaces 20 0

v

VI CONTENTS

CHAPTER IV . INTERPOLATION METHOD S 20 9 Introduction 20 9 1. Th e complex method o f interpolation 21 4 2. Th e methods of constants and mean s (%- an d ^-methods) 24 6

CHAPTER V (B Y S . G . KREIN) . INTERPOLATIO N I N SPACE S OF SMOOT H FUNCTION S 28 3

1. Interpolatio n space s constructed fro m a n unbounde d operato r and a smoothing approximation proces s 28 3

2. Trac e theory 30 9 3. Space s of smooth function s o f n variables 32 2

NOTES ON THE LITERATUR E 34 9

BIBLIOGRAPHY 35 5

SUBJECT INDE X 37 3

NOTATION INDE X 375

FOREWORD TO THE AMERICAN EDITIO N

This editio n include s Chapte r V , "Interpolatio n i n space s o f smoot h func -tions", written a t th e sam e tim e a s the preceding chapter s bu t no t include d i n the Sovie t editio n fo r technica l reasons . Thi s chapte r expound s th e abstrac t scheme fo r constructin g interpolatio n space s b y mean s o f a n unbounde d operator i n a Banac h space , an d th e correspondin g approximatio n process . Starting points o f thi s theory wer e papers by J . L . Lions , Lions an d J . Peetre , and P . Grisvard , whic h the n wer e extende d b y othe r authors . As a n applica -tion we consider onl y Sobolev an d Beso v spaces.

The reade r ca n ge t acquainte d wit h othe r familie s o f space s i n th e book s referred t o in the foreword t o the Sovie t edition .

The author s expres s thei r sincer e gratitud e t o th e edito r o f th e translation , Dr. L . J . Leifman , fo r hi s penetratin g remark s tha t helpe d i n eliminatin g a number o f shortcomings .

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FOREWORD

The presen t boo k i s devote d t o th e systemati c expositio n o f a chapte r i n functional analysi s tha t has appeared an d develope d i n th e past two decade s and has found applications in various fields.

The basi c object s o f classica l functiona l analysi s wer e operator s actin g from one Banach space (or later from a topological linear space) into another. The space s themselve s wer e considere d a s give n i n advance . Th e chang e o f this ideology was facilitated to a significant exten t by the imbedding theorems of S . L . Sobolev , i n whic h a number o f fundamenta l theorem s an d inequali -ties o f analysi s wer e interprete d a s assertion s concernin g th e imbeddin g o f one Banach space into another. Imbedding theorems arose in connection with problems o f the theory of partia l differential equations , in which for the study of smoothnes s o f solution s a serie s o f space s i s introduced ; fo r th e stud y o f the behavio r nea r th e boundary o f th e domai n o r nea r som e singula r point s other type s o f space s ar e introduced , th e stud y o f value s o f solution s o n manifolds o f smalle r dimensio n i s performe d i n stil l othe r spaces , etc . Th e abundance o f variou s space s require d a detaile d stud y o f th e interrelation s between these spaces. Thus a new level of abstractio n appeared, on which the Banach space s themselve s ar e considere d a s element s o f som e category . Th e interpolation theor y fo r linea r operator s expounde d i n th e book i s t o a grea t extent connected with such an approach.

The firs t interpolatio n theore m i n operato r theor y wa s obtaine d b y M . Riesz i n 192 6 i n th e for m o f a n inequalit y fo r bilinea r forms . A sharpenin g and operato r formulatio n o f i t wer e give n b y G . O . Thorin . A n essentia l further ste p was th e interpolation theore m o f J . Marcinkiewic z (1939) , whos e proof wa s publishe d b y A . Zygmun d i n 1956 . I n th e fiftie s importan t generalizations o f th e Riesz-Thori n an d Marcinkiewic z theorem s wer e ob -tained by E . M. Stei n and G. Weiss . However , al l thes e and othe r communi -cations wer e concerne d wit h Lp space s o r space s simila r t o them . Th e

ix

X FOREWORD

development of general interpolation theorems for families of abstract Hilbert and Banach spaces began in 1958 independently in several countries. The first publications are due to J. L. Lions (1958-1960), E. Gagliardo (1959-1960), A. P. Calderon (1960), and S. G. Kreln (1960). The work of J. A. Peetre played an essentia l rol e i n th e sequel . Severa l method s hav e bee n create d fo r obtaining interpolation theorems , which have deep interrelations. Moreover, it became clear fairly soo n tha t th e interpolation propertie s o f space s inter-mediate between two Banach spaces are consequences of the functoriality of the methods of construction . Therefore, th e main emphasis has been shifted to th e stud y o f propertie s o f intermediat e interpolatio n space s obtaine d by various methods, and to their realization. Along with this, in the work of W. Orlicz, A . P . Calderon , G . G . Lorentz , E . M . Semenov , an d other s dee p results have been obtained concerning the interpolation of linear operators in spaces of measurable functions.

It i s impossibl e t o expoun d al l result s o f interpolatio n theor y fo r linea r operators in one book. W e have trie d to illuminate only som e o f th e main directions in its development: th e real and complex methods of constructing interpolation spaces , th e method of scale s o f Banac h spaces , and interpola-tion i n space s o f measurabl e functions . Supplementar y informatio n i s con-tained in remarks and references.

In the development of interpolation theory for operators many new general notions of functional analysi s have emerged. These notions and their interre-lations ar e studied i n th e firs t chapte r of th e book. Th e exposition i s based essentially o n th e wor k o f N . Aronszaj n an d E . Gagliardo . T o rea d thi s chapter one needs to know only the basic principles of functional analysis.

The secon d chapter , devote d t o interpolatio n i n space s o f measurabl e functions, make s u p a significan t portio n o f th e book . I t ca n b e rea d independently o f th e firs t chapter , from which only th e simplest definition s are needed . Th e chapte r contain s a theore m describin g al l interpolatio n spaces between Lx and L^, and a theorem which is a further extension of the Marcinkiewicz theorem. The exposition is pursued as far as concrete applica-tions, for example, the theory of orthogonal series: convergence properties of Fourier serie s an d th e basi s propert y o f a functio n syste m ar e studied . Moreover, th e chapter contains muc h auxiliary materia l fro m the theory of functions which is discussed little in the literature. Decreasing rearrangements of measurabl e function s ar e studied in detail , functio n space s symmetric in the sense o f E . M. Semenov , an d in particular , Lorent z and Marcinkiewicz spaces, ar e discusse d (i n th e foreig n literatur e simila r space s ar e calle d invariant with respect to permutations). Sharpenings of classica l inequalitie s of analysis (the Hardy-Littlewood, Hilbert, and other inequaities) are given.

FOREWORD XI

In th e thir d chapte r th e theor y o f scale s o f Banac h spaces , develope d mainly i n th e publication s o f S . G . Krei n an d Ju . I . Petunin , i s expounded . The prerequisite materia l fo r thi s i s containe d i n th e firs t chapter . Importan t properties o f th e scales , i n particula r thei r "almost " interpolatio n propertie s are als o expounde d i n th e fourt h chapter . I n th e las t sectio n o f th e thir d chapter propertie s o f th e classica l scal e o f Holde r space s importan t i n applications are studied in detail .

In th e fourt h chapte r tw o method s o f constructin g interpolatio n space s enjoying th e larges t numbe r o f application s ar e describe d i n detail : th e method o f comple x interpolatio n propose d independentl y b y A . P . Caldero n and J . L . Lion s an d extensively develope d b y Calderon , an d th e metho d o f constants an d averages du e to J . L . Lions an d J. Peetre . The latte r method i s expounded i n th e mor e genera l for m whic h i t acquire d i n th e wor k o f V . I . Dmitriev (wh o too k th e mos t activ e par t i n writin g th e correspondin g section). The fourt h chapte r can b e rea d independentl y fro m th e secon d an d third chapters.

The boo k doe s no t includ e interpolatio n theor y i n space s o f smoot h functions an d its applications.* Thi s theor y develope d unde r th e influence o f the work o n imbedding theorem s by S . L . Sobolev, S . M. Nikol'skii an d their students an d followers . The abstrac t theor y di d no t ris e immediatel y an d easily to the level o f concret e imbedding theorem s obtained by special means. However, no w suc h a theor y ha s bee n created . It s expositio n apparentl y needs anothe r book. On e can ge t acquainted with i t partly in the book [7 ] by P. L . Butze r an d H . Berens . I t i s expounde d mor e completel y i n Han s Triebel's ver y recen t boo k Interpolation theory, function spaces, differential operators (publishe d b y VE B Deutsche r Verla g Wiss. , Berlin , 1977 , an d b y North-Holland, 1978).( 1) One can get acquainted with the applications of thi s theory t o th e stud y o f boundar y valu e problem s fo r partia l differentia l equations i n th e boo k o f J . L . Lion s an d E . Magene s [27 ] an d i n Triebel' s book mentione d above . A t th e en d o f th e boo k ther e i s a bibliograph y covering, in addition, the indicated part of interpolation theory .

As we noted above , some parts of th e book were written by V. I . Dmitriev . I. Ja. Snelberg provided us with invaluable help. He participated in writing § 1 of Chapte r IV and read a significant portio n of th e book. Hi s critical remark s

* Editor's note. For thi s translatio n a ne w Chapte r V wa s adde d b y th e author s t o cove r thi s subject.

(^The author s ar e gratefu l t o Professo r Triebe l fo r makin g th e manuscrip t o f thi s boo k available.

Xll FOREWORD

enabled u s t o remov e a numbe r o f inaccuracie s an d improv e som e proofs . The authors express their gratitude to both of them.

Finally, we thank al l participants of th e Voronezh seminar on interpolatio n theory fo r linea r operators , and , i n particular , M . S . Braverman , A . A . Dmitriev, E . A. Pavlov , P . A. Kucment , an d A. A . Sedaev , fo r thei r constan t help in th e preparation o f th e book .

The authors

NOTES ON THE LITERATURE

Chapter I §1. A s ha s bee n mentione d i n th e Foreword , th e notio n o f imbeddin g o f Banac h space s firs t

appeared in a paper by Sobolev [355] . The important notion of relative completion has been used implicitly b y S . G. Krei n and Petuni n [184] ; i t has been introduce d explicitl y b y Gagliardo [139 ] and studied in detail by Aronszajn and Gagliardo [55].

§2. Lemma s 2.1-2. 3 an d Theore m 2. 3 are containe d i n [55] , Theore m 2. 1 i n [184] , an d Theorem 2.2 in Berens' book [4].

§3. Intermediat e space s fo r a coupl e o f Banac h space s wer e firs t considere d b y Lion s [202] . Formula (3.6 ) an d it s consequence s ar e du e t o Sedae v [41] . Lemma 3. 4 i s take n from Caldcro n [95]. Th e dua l space s o f a su m o r a n intersectio n o f space s wer e studie d b y Aronszaj n an d Gagliardo [55 ] (see als o [232]) ; the y als o introduce d th e su m an d intersectio n o f a famil y o f Banach spaces.

§4. The notion s o f interpolatio n triple s and interpolation space s have been introduced i n on e form o r anothe r i n al l publication s devote d t o th e abstrac t theor y o f interpolatio n o f linea r operators. Th e mos t importan t result s o f § 4 ar e du e t o Aronszaj n an d Gagliard o [55] . I n connection wit h subsection s 4 an d 5 , se e [37] . Th e importan t Theorem s 4. 9 an d 4.1 0 wer e obtained in [55].

Chapter II Introduction. Ideal Banac h lattices are also called Banach functio n spaces ; their properties are

described i n [29 ] and [49 ] (see als o [95] , [188], [25 ] and [47]) . Concernin g Lebesgu e space , se e [305].

§1. Theorem 1. 1 was obtained by Peetre [284] (concerning (1.7) , see [122]). For Lemma 1.1 , see [107]. Fo r mor e detai l o n logarithmicall y conve x functions , se e th e boo k [39] . Lemma 1. 4 wa s obtained in [59].

§2. Theorem 2. 1 was obtained b y Krein and Semenov [191] . The books [8] , [17], and [50] have much informatio n o n rearrangement s o f measurabl e functions . Man y propertie s o f rearrange -ments hav e bee n establishe d i n [87] , [88] , [95] , [172] , [317] , an d [312] . InequaUt y (2.40 ) i s published here for the first time.

§3. Formula s (3.4 ) an d (3.5 ) wer e obtaine d b y Peetr e i n [262 ] an d [261 ] (see als o Oklande r [250]). The important inequality (3.14) was established by Lorentz and Shimogaki [227] , Orbits of the semigroup of contractive operators were studied by Ryff [311] .

349

350 NOTES ON THE LITERATURE

§4. Symmetri c space s wit h th e additiona l assumptio n tha t th e nor m i s semicontinuou s wer e introduced b y Lorent z unde r th e nam e spaces invariant under permutations i n the book [28 ] (see also Luxembur g [231]) . Without thi s assumptio n the y wer e studie d b y Semeno v [331] ; th e mai n results o f subsectio n 1 wer e obtaine d b y him . Th e mai n theore m describin g al l interpolatio n spaces between Lx an d L^ wa s proved by Calderon [96] by another method. This theorem has a long history: for integral operators in Orlicz spaces it was first proved by Orlicz [256], for integral operators i n spaces invarian t unde r permutations b y Lorent z [28] , and for arbitrar y operator s i n separable space s o r symmetri c space s dua l t o separabl e space s i n [241 ] (see Theorem s 4. 9 an d 4.10). The theore m has been generalized t o the case of nonlinea r operators satisfying a Lipschitz condition i n [257] , [227] , an d [349] . W e not e tha t Browde r [85 ] has obtaine d a genera l theore m enabling us to obtain interpolation theorem s fo r Lipschit z operators from interpolation theorem s for linear operators.

The hypothesi s o f Theore m 4.3 ca n b e formulate d i n the followin g way : th e assumptions tha t y G £ , x G L x + L^, an d K(ty x) < K(t yy) impl y that x G E an d H* ^ < | | v||£. I n connectio n with thi s ther e aros e the following conjecture . Le t (A0, A x) an d (B& BX) be tw o Banach couples . For th e tripl e (A 0, A x, E) t o b e a n interpolatio n tripl e relativ e t o th e tripl e {B& Bx> F) i t i s necessary an d sufficien t tha t th e assumption s y G £, x G B 0 + B x, an d K{t, * , B& Bx) < K(t,y, A 0, A x) impl y tha t x G F an d \\x\\ F < C||.y||£ . Thi s conjectur e ha s bee n confirme d fo r the followin g couples : 1 ) (L^ , L J, (L q1 L w ) (Lorent z an d T . Shimogak i [228] ; Sedae v [324]) ; 2) (/j * /f») , (/f° , /f 1) (a 0 an d a x ar e weights ) (Sedae v an d Semeno v [327]) ; 3 ) (If, L?*), (Z/o, Z^» ) (Sadae v [324]) ; 4 ) (A^A,) (L£> , L £) (Peetr e [37 ] an d Sadae v [41]) ; 5 ) (A ^ L J , ( L , , / ^ ) (V . I . Dmitrie v [117]) ; 6 ) (LpLp), (L „ LJ (V . I . Dmitrie v [119]) ; 7 ) (Lg, I£») , (I^°, L£ l) (Spar r [361]). In the general cas e th e conjecture coul d not be confirmed (se e [37] and [117]). V . I . Dmitrie v [119] , [10 ] single d ou t a genera l clas s o f couple s o f space s (differenc e couples) for which the conjecture is true.

The action o f dilatio n operator s in symmetric space s has been studie d in [348] , [350] , and [83]. In term s o f th e nor m o f th e dilatio n operator , th e proble m o f interpolatio n o f th e propert y o f complete continuit y o f linea r operators i n intermediat e space s fo r th e couple (L, , LJ ha s bee n solved in [348] . The upper and lowe r dilation exponent s of symmetri c space s were introduced by Boyd [81 ] under th e name upper and lower indices. Lemma 4. 7 i s due t o Semeno v an d play s a n important rol e i n wha t follows . Th e notio n o f fundamenta l functio n wa s introduce d i n [331] . There ar e example s o f space s fo r whic h inequalit y (4.29 ) i s strict , and , eve n more , th e lef t an d right sides have different asymptotic s at infinity (se e [350]).

§5. Lorent z space s wer e introduce d b y Lorent z i n [225] ; h e establishe d tha t thei r dua l space s are th e Marcinkiewic z spaces . Some propertie s o f Lorent z space s wer e obtaine d i n [191] . Th e spaces M j wer e considere d b y Semeno v [330] . The imbeddin g theorem s wer e obtaine d i n [331] . Theorem 5.9 is contained in [342] under different assumptions .

The first example of a noninterpolation symmetric space was constructed by Russu [307]. §6. Inequalit y (6.1' ) i s actuall y containe d i n [347] . Th e mai n interpolatio n Theorem s 6. 1 an d

6.1' ar e extension s o f Marcinkiewicz * theore m [234] , whos e proo f was publishe d b y Zygmun d [407]. Man y author s hav e deal t with th e generalizatio n o f thi s theorem ; se e [64] , [74], [94], [96], [106], [148], [161], [163], [190], [214], [250], [262], [342] , [371], [406], etc. Here the versions of Krei n and Semenov ar e given from [192] . For the case of the Lp spaces an analogue of Theorem 6.1 ca n be foun d i n [334] , wher e th e condition s o n th e spac e E ar e give n i n term s o f th e fundamenta l function q> £. However , th e proof contain s a n erro r an d i s true only i n th e cas e where th e nor m ||<7T||£ o f th e dilatio n operato r coincide s with M VE (concerning (4.29) , se e above) . A correctio n and strengthenin g o f thi s resul t i s expounde d i n § 6 (se e [192]) . Th e firs t theorem s o n th e optimaliry o f interpolatio n triple s o f concret e symmetri c space s wer e obtaine d b y Dikare v an d Macaev [111 ] and Caldero n [96] . Calderon's idea s he a t the base o f th e proofs o f th e optimalit y theorems of [192] .

NOTES O N THE LITERATURE 351

Hardy-Littlewood an d Hilber t operator s an d majoran t function s fo r symmetri c space s hav e been studied in [332], [347] , [255] , [144] , [78] , and [218]. A generalization of th e Hardy inequalit y (6.41) has been obtained by F A Pavlov .

The space s Lp r ar e specia l case s o f th e Lorent z space s A ^ [225] . Interpolatio n theorem s fo r them are contained i n [96]. Theorems 6.1 2 an d 6.13 are new. Applications o f Theore m 6.1 t o the convolution operato r ar e indicate d i n [193] . Theore m 6.1 7 i s a sharpening o f a resul t o f O'Nei l [253].

§7. I n discussing th e properties o f th e Hilber t singula r operator we have followe d Zygmund' s book [51] . The main formula (7.9 ) i s due t o Stei n and Weis s [371] . Theorem 7. 2 wa s prove d b y Boyd [78].

§8 has an auxiliary character . The operatio n o f takin g th e dual in symmetric space s has bee n studied in [133], [144], [254] and [333].

§9. Fo r th e classica l Pale y theorem , se e [50] . The generalizatio n o f i t i n th e presen t for m i s published her e fo r th e firs t time . Th e article s [313] , [314] , [316 ] an d [333 ] ar e devote d t o th e generalization o f th e Hardy-Littlewoo d theore m o n serie s with monoton e coefficient s b y mean s of interpolatio n theorems. For properties of th e Haar system, see [20]. Theorem 9.5 fo r the space Lp was prove d b y F . Ries z (se e [2 ] and [50]) , fo r symmetri c space s i t was prove d b y Semeno v [333]. Theore m 9. 6 fo r th e Lp spaces was obtaine d b y Marcinkiewic z [235] , fo r Orlicz spaces b y V. F . Gaposkin , an d fo r symmetri c space s b y Semeno v [336].( 1) Concernin g Theore m 9.8 , se e [304].

Chapter III §1. Th e propertie s o f scale s o f Ranac h space s ar e expounded i n [186] . Th e notio n o f norma l

scale wa s introduce d an d studie d b y Krei n [181] . Th e proble m o f relate d Banac h space s wa s studied by Krei n and Petuni n i n [184]. The condensation o f a normal scale by means o f relativ e completion i s considered here for the first time.

§2. Maxima l scale s have bee n studie d i n [181] , an d minimal an d regular scale s in [185] ; thei r properties are discussed in detail in [186].

§3. Th e scal e o f Holde r space s wa s considere d i n detai l i n [186] . V . Friedric h ha s kindl y indicated t o us that there are inaccuracies i n [186], which we correct here. The new exposition of interpolation properties of th e Holder scale is based on the work of Petuni n and Plicko [297]. We note that in a more general case a detailed exposition of th e properties of the Holder scale and its dual wit h application t o th e transportatio n problem i s contained in Friedrichs* book [14].

Chapter IV Introduction. Theorem 1 wa s obtained by V. I . Dmitriev [116]. §1. The complex metho d of interpolatio n i n th e form expounded here was suggested indepen -

dently by Calderon [95] and Lions [204]. The main results of subsections 3 and 4 are contained in Calderon [95] ; th e usefu l Remar k 2 wa s mad e b y Stafne y [363]. ^ Theorem 1. 4 i s als o du e t o Calderon [95] ; its proo f ha s somewha t bee n simplifie d b y I . Ja . Sneiberg . I n [95] , besides th e spaces [A 0, A x]a, th e interpolatio n famil y o f space s [A 0> Atf* i s als o introduced , an d th e space ^(A 0yAx) ° f function s i n th e stri p n havin g th e followin g propertie s i s considered : 1) ll/WIU+yi , < C(\ + 1*1) ; 2 ) /(* ) i s continuou s i n th e nor m o f A 0 + A x i n II ; 3 ) f(z) i s analytic i n II ; an d 4 ) th e differenc e / ( l + it 2) - / ( l + it x) belong s t o A x an d th e differenc e

(*) Concerning [333 ] and [336] , th e remarks t o §6 must be take n into account .

352 NOTES O N THE LITERATURE

./(if2) — f(itx) belong s to A0; moreover ,

/ 11/(̂ 2 ) -/Oh)II 11/( 1 + / /2 ) - /q + itl )\\ \ max su p , sup = ||/|| § < 00 .

I | | l i~h \ AQ I I h~h \ Ax) The space [A0, A ,]" consists of al l x G A0 + A x for which x = /'(«)> / £ iK^o > ^ 1)' anc *

IWU^ f = ™ ll/Hs-/ ( « ) - • *

It turns out [95] that (M0> ^ I D' * s isometrically isomorphi c t o the space [A' 0, A\Y (i f ^ 0 n ^ 1 is dens e i n ,4 0 an d > 4 j). I n subsectio n 6 th e dua l o f [A 0, A x]a i s describe d i n term s o f relativ e completion (Theore m 1.6 , I . Ja . Sneiberg) . Fro m wha t ha s bee n sai d w e obtai n th e equalit y Mo> ^ ir ~Mo*^ilo - Change s hav e als o bee n mad e i n th e proo f o f Calderon' s reiteratio n Theorem 1.7 . W e not e tha t i n [320 ] i t i s indicate d tha t th e conditio n tha t A 0 n A , i s dens e i n Aa n Ap i s superfluous; however , the proof o f thi s fact contains an error.

The notio n o f a n analyti c scal e o f space s was introduce d b y Krei n [180] ; h e establishe d th e connection o f thi s notion with the complex method of interpolation [186] .

The theor y o f Hilber t scale s o f space s was constructe d independentl y (i n differen t terms ) b y Lions [200] and Krei n [ 180]. A n importan t rol e i s played by familie s o f topologica l linea r spaces obtained fro m a Hilber t scal e b y mean s o f projectiv e o r inductiv e limits . B y mean s o f the m a number of delicate properties of spaces of analytic functions ha s been studied.

Theorems 1.1 1 an d 1.1 3 wer e obtaine d fro m othe r consideration s b y E . Heinz , an d wer e sharpened b y T . Kat o (se e [23 ] an d [24]) . Hein z ha s prove d a n inequalit y mor e genera l tha n (1.49), i n whic h th e fractiona l powe r o f th e operator s j an d j \ i s replace d b y mor e genera l functions. Le t u s conside r th e clas s o f function s positiv e o n th e semiaxi s [0 , 00) , admittin g analytic continuation t o the complex plane with the negative semiaxis removed, which results in a function mappin g th e uppe r half-plan e int o itself . Le t th e functio n <p(t) b e suc h tha t <p 2(f,/2) belongs to the indicated class. Under the conditions of Theorem 1.1 3 we have the inequality

(Tx,y) < \\<pU)x\\Ho\KUi)y\\H 0> wher e 9# ( / ) - t/<p(t).

All interpolatio n space s with interpolation constan t 1 between a couple o f Hilber t space s have been describe d b y mean s o f function s o f th e indicate d clas s (Foia § an d Lion s [134] , an d Donoghue [120] ; see also [160]).

The famil y o f space s XQ~°X X wa s introduce d an d studie d b y Caldero n [95] . Here w e d o no t discuss it s connectio n wit h hyperscale s (se e [188]) . Lozanovski i [230 ] constructed a n exampl e i n which th e triple s (X 0, X Xy X^~ aXx

a) a n d (*o > y i> y o ~ a * 7 ) o f i d e a l lattice s ar e not interpolatio n triples. Sestakov [340] , [341 ] showed tha t in th e genera l cas e [X 0, X x]a i s th e closur e o f X 0 n X x

in AQ1 ~aXxa

y and consequently is a closed subspace of it . For further study of th e family XQ ~aXxa,

see [401] and [229]. We not e tha t Schechte r [320] , [319 ] constructe d a generalizatio n o f th e comple x metho d o f

interpolation base d on th e idea tha t the intermediate space i s constructed no t from th e values of functions analyti c i n a stri p or thei r derivatives bu t rathe r fro m th e value s o f som e generalize d function wit h compact support defined o n these functions .

Interesting result s concernin g th e unambiguou s solvabilit y o f linea r equation s an d th e spec -trum of linea r operators i n the famil y o f space s [A 0, A x]a hav e been obtained by Sneiber g [353] , [354] and Stafney [364] .

§2. Th e method s o f constant s an d average s originat e i n a pape r b y Lion s an d Peetr e [214] , where the y ar e constructe d fo r th e cas e tha t E 0 an d E x ar e Lp spaces with powe r weights . Th e generalization of thes e methods to the case of arbitrar y ideal lattices discussed here was proposed by Peetr e in [35] and [264] , and developed b y Dmitrie v in [114] , [116] and [118] . The appearanc e of space s o f th e Caldero n scal e i n th e reiteratio n Theore m 2. 8 was unexpecte d (V . I . Dmitrie v [59]).

NOTES O N THE LITERATURE 353

The %- an d % -methods were proposed by Peetre [262] in the case where E i s an Lp space with power weigh t (se e subsectio n 8 ) an d hav e bee n th e mos t widel y disseminate d o f al l rea l interpolation methods . These methods were generalized to the case of mor e general ideal lattice s by Peetre [35] , Bennett [67] , and other authors. For a special cas e of Theorem 2.10, see [262] and [73].

Theorems 2.11 and 2.12 were proved by Lions and Peetre [214]. Theorem 2.1 3 o n th e numbe r o f parameter s wa s obtaine d b y Peetr e [261] . I t ha s bee n

generalized to the case of arbitrary ideal lattices E0 and Ex by V. I. Dmitriev [118]. Extreme spaces (subsection 9) were studied by Hayakawa [156]. The duality of th e methods of constants and averages (in the simplest case when they coincide)

was studie d b y Lion s an d Peetr e [214] . Her e w e hav e expounde d th e result s o f V . I . Dmitrie v [116].

Applications o f th e method s o f constant s an d average s t o quasinorme d space s hav e bee n studied in [168] and [157].

The connectio n betwee n th e methods o f constant s an d average s an d th e theory o f scale s an d the almost interpolation properties of scales were studied by Petunin [186].

Lemma 2.2 0 an d Theore m 2.2 3 wer e obtaine d b y Lion s an d Peetr e [214] . Th e proble m o f conditions on the commutativity of the functors corresponding to the method of averages and the complex metho d ha s been studied by Grisvard [153] .

Chapter V §1. Th e approac h t o th e construction , b y mean s o f a n abstrac t approximatio n process , o f

intermediate spaces between a Banach space and the domain of an unbounded operator acting in it i s expounde d her e fo r th e firs t time . Concret e realization s o f i t hav e bee n studie d b y man y authors. Closel y relate d bu t differen t construction s ar e foun d i n Berens ' boo k [4] , I t was apparently ther e tha t th e connectio n betwee n th e behavio r o f a n approximatio n proces s an d relative completio n (Lemm a 1.1 ) wa s notice d fo r th e firs t time . See [24] for mor e details on th e subordinate operator s o f subsectio n 4 . The approximatio n proces s constructe d fro m a power o f the resolven t was firs t studie d i n Grisvard' s fundamenta l pape r [153] , th e result s o f whic h ar e discussed her e onl y partially . I n particular , i n i t th e relatio n o f th e intermediat e space s con -structed there to the complex interpolation method is studied. Lemma 1.1 0 is due to Ljubic [220]. For operator s satisfyin g conditio n (1.15 ) fractiona l power s ar e define d (se e th e book s [23 ] and [24]), an d therefor e th e theor y expounde d i n subsection s 5 an d 7 ca n b e carrie d ove r t o space s constructed fro m fractional power s of th e operator. The connection betwee n interpolation space s and domains of fractiona l power s of operators was studied by Lions [209] for accretive operators in a Hilbert space , and i n th e general cas e in a long serie s of article s by Komats u [177] , and by Sobolevskil in [356]-[359] (see also [72], [246], [392], and [398]).

The constructio n o f interpolatio n space s b y mean s o f bounde d semigroup s o f operator s wa s first don e b y Lion s and Peetr e [213] , and since the n i t has been studied in many publications — see [3], [69]-[71], [91], [92], [151], [153], [222] and [392] . We note that a fairly complete expositio n of propertie s o f th e spaces constructe d fro m resolvent s an d semigroup s o f operator s i n th e cas e G = Lp ^ i s include d i n th e boo k [7 ] b y Butze r an d Berens . Theore m 1. 9 wa s obtaine d b y Grisvard [153] ; thi s theore m an d Theorem s 1.1 0 an d 1.1 1 wer e generalize d t o th e cas e o f fractional power s of operator s by Muramatu [246] . The proof o f Theorem 1.1 2 given here is also due t o Muramatu . Interestin g bu t apparently no t complet e researc h has been carried out i n th e case where the operators do not commute and are infinitesimal operator s of som e representatio n of a Lie group (see [281]).

354 NOTES O N THE LITERATURE

Grisvard i n [153 ] began t o conside r unbounde d operator s i n a coupl e o f Banac h spaces . Th e imbedding theore m give n here , namel y Theore m 1.13 , i s du e t o Yoshikaw a [394] . I t ha s bee n developed and applied in [245], [396] and [397].

§2. Spaces of trace s were introduced and studied i n a series of publications by Lions (see [201], [202], [205 ] an d [207]) . Th e stud y ha s bee n continue d b y Grisvar d i n [152 ] an d [154] ; th e exposition given here is based on his articles.

§3. A s w e mentione d i n th e Foreword , Sobolev-Nikol'skii-Slobodeckii-Beso v space s an d imbedding theorem s fo r the m serve d a s a guidelin e fo r th e constructio n o f th e correspondin g abstract theor y expounde d i n Chapte r V i n a n incomplet e form . I n almos t al l publication s mentioned i n §§ 1 and 2 ther e are applications t o the theory o f th e indicated spaces . In addition, we mention th e serie s [379]-[386] o f article s b y Triebel . W e have discussed onl y th e fact s whic h can b e obtaine d fro m result s o f §§ 1 an d 2 ; beside s th e abstrac t theory , w e hav e her e use d Minim's [30] and P. I. Lizorkin's [217] theorems on multipliers for the Fourier transformation, th e Hestenes-Whitney extensio n o f smoot h functions , an d othe r technique s whic h hav e becom e standard in the theory of partia l differential equations . Here we have used the monograph [27] in an essential way .

We entirely omitted other classes of smoot h functions , in particular, Lebesgue spaces or spaces of Besse l potentials , which are connected with the complex method of interpolation . Their theory is expounded i n Nikol'skii's boo k [32 ] without thi s connection, and in connection with interpola-tion theor y i n TriebeF s ne w boo k Interpolation theory, function spaces, differential operators mentioned i n the Foreword (see also [93], [212] and [218]).

On othe r classe s o f functions , se e [15] , [86], [87], [90], [95], [99]-[101], [129], [171]-[174] , [194], [198], [211] , [212] , [216] , [218] , [249] , [279] , [321] , [365] , [366] , [373 ] and [374] .

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SUBJECT INDEX

absolutely continuous norm, 4 5 almost imbedde d famil y o f Banac h spaces ,

279 analytic scale of Banach spaces , 23 4 associated space , 4 5 averaging operator, 8 0

B. Levi's theorem, 4 0 Banach couple of spaces , 9 Banach function space , 4 0 basis, 17 8

unconditional, 17 9 Besov space, 33 0 Bochner integrable function , 20 9 bounded operator in couples of Banach spaces,

18 bounded semigroup , 29 6

Co-condition, 29 6 compact scal e of Banach spaces , 18 7 complemented subcoupl e o f a Banac h couple ,

29 complex method o f interpolation, 21 4 concave function , 4 6 convergence in measure, 3 9 conjugation operator , 16 9 convex function , 4 6 contraction semigroup, 8 4 convolution operator, 14 7 covariant functor , 3 5

dilation exponent s o f a function , uppe r an d lower, 5 4

dilation exponents of a space, upper and lower , 99

dilation functions , 5 3 dilation operator, 9 6 dimension of th e norm, 33 4 discretization o f an ideal lattice, 4 4 distribution function , 5 8 dual family o f Banac h spaces , 19 5 dual scale of Banach spaces, 19 6

Egorov's theorem, 3 9 elementary function , 9 2 (Lemma 4.2) equimeasurable functions , 5 8 equivalent functions , 4 8

Fatou property , 4 4 Fatou's lemma, 4 0 Fourier transformation , 32 5 fundamental function , 10 1

generalized function , 30 9 derivative o f a , 30 9 regular, 30 9 trace of a, 31 1

generalized Hardy-Littlewood theorem , 22 4 generalized Paley theorem, 22 4

Hardy inequality , 16 7 Hardy-Littlewood operator , 13 8 Hilbert operator , 14 0

singular, 15 0 Hilbert scale , 23 7 Holder scale , 20 1

ideal Banach lattice, 4 0 ideal lattice, 4 0 imbedded famil y o f Banach spaces, 27 9

373

37 4 SUBJECT INDE X

imbedding constant, 1 imbedding o f Banac h spaces , 1

compact, 2 dense, 1 normalized, 2

incomplete scal e of Banach spaces , 18 8 incomplete scale with base, 18 8 infinitesimal generato r of a semigroup, 29 7 integral, 21 0 intermediate space , 1 5

of typ e 0, 25 7 interpolation constant , 2 0 (Lemma 4.3 ) interpolation functor , 3 5

M o ^ i L * 221 (A0,AX)QP, 27 1

of typ e a, 3 6 interpolation property , 19 5

almost, 28 0 normalized, 19 5 strong, 19 5

interpolation space , 2 0 of typ e a, 2 2

interpolation theorem , 2 2 interpolation triple s o f spaces , 2 0

good, 2 1 interpolation triple s o f typ e a , 2 2

normalized, 2 2 intersection o f th e spaces of a Banac h couple ,

9 isomorphic Banac h couples, 1 3 isomorphic measure spaces, 4 6

Lebesgue space , 4 6 Lebesgue's theorem, 4 0 logarithmically convex function , 5 1 Lorentz space, 10 7

majorizing normalize d scale , 19 7 Marcinkiewicz space , 11 2 maximal scal e o f means , 27 8 maximal symmetri c space, 10 4 method o f constants (3Gmethod), 24 6 method o f means (J-method), 24 8 minimal scale of Banac h spaces , 19 7

Nikol'skii space , 33 0 normalized interpolatio n spac e of typ e a, 2 2 normal scal e o f Banac h spaces , 18 8

continuous, 18 9 maximal, 19 3 regular, 19 6

normative linea r manifold , 7 operation**, 12 4 operator o f fractiona l integration , 149 , 150

operator o f stron g (weakened , weak ) type , 130,131

optimal interpolation triples , 2 7 orbit o f a point, 8 9 problem of multipliers , 32 5 quasiconcave function , 4 9 Rademacher system, 18 3 rearrangement o f a function, 5 9 reflexive couple of spaces , 9 regular domain, 33 5 regular ideal lattice, 4 5 reiteration theorem , 231 , 261 related space , 18 9 relative completion, 3 restriction of an ideal lattice, 4 3 Riesz-Thorin theorem , 2 2 right invertible mapping, 3 3 r-regular domain, 33 5

scale of Banac h spaces , 18 7 scale of means, 27 8 Schwartz space, 32 5 separable measure , 4 5 simple function, 3 9 smallest concave majorant , 4 7 smoothing approximation process , 28 3 Sobolev-Slobodeckfi space , 33 1 Sobolev space, 32 4 space complet e wit h respec t t o anothe r space ,

6

space S(^o» ^i) » 21 6 space of slowl y increasin g Schwart z distribu -

tions, 32 5 space of traces , 31 3 spaces invariant unde r permutations, 35 0 strictly simple function, 3 9 strongly continuou s semigrou p o f operators ,

296 strongly measurable function , 20 9 subadditive function , 5 1 submultiplicative function , 5 2 subordinate operator, 29 0 support of an ideal lattice, 4 5 symmetric linear subset, 9 4 symmetric space , 9 0 total linea r manifold, 8 truncation, 10 2

right, 10 2 two-fold, 10 3

upper and lowe r indices, 35 0

weighted idea l lattice, 4 3

Young inequality , 16 7

NOTATION INDEX

C , imbedding, 2 A^, Lorentz space , 9 8 M^, Marcinkiewiczspace , 9 8 X e{ t), characteristi c function o f a set e, 3 9 xN, truncation , 10 2 xN, righ t truncation , 10 2 *M* two-fold truncation , 10 3

1 C , imbedding with imbedding constant 1 , 1 6

7r,(/lfl,C£), uni t ball in L(/ l£ , CD), 2 3

L{AB,CD), linea r spac e o f bounde d opera -tors fro m a Banac h coupl e A, B int o a Banach couple C, D, 1 9

**(/), rearrangemen t o f x(/), 5 9 /?X(T), distribution functio n o f x , 5 8 M^s), dilatio n function o f >/>, 5 3 <pE, fundamental functio n o f th e space £ , 10 1 ^(R") , spac e o f infinitel y differentiabl e func -

tions with compact support , 32 3 ^ , Fourie r transformation , 32 5 E, closur e o f E (used occasionally )

375

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