Interpolation Functors and Interpolation Spaces: 001

INTERPOLATION FUNCTORS AND INTERPOLATION SPACES Volume I

North-Holland Mathematical Library Board of Advisory Editors.

M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, L. Nachbin, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 47

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

Interpolation Functors and Interpolation Spaces VOLUME I

Yu.A. BRUDNYI N. Ya. KRUGLJAK Yuroslavl State UniversiQ Yarosluvl,USSR

1991 NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands

Distributors for the United States and Canada. ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

Translated from the Russian by Natalie Wadhwa Library of Congress Cataloging-in-Publication

Data

Brudnyi. f U . A. I n t e r p o l a t i o n f u n c t o r s and i n t e r p o l a t i o n s p a c e s : Yu.A. B r u d n y i , N.Ya. K r u g l J a k . v . 1' > , cm. -- ( N o r t h - H o l l a n d m a t h e m a t i c a l library v . 47) Translation from the Russian. I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s a n d index. I S B N 0-444-88001-1 1 . L i o n e a r t o p o l o g i c a l s p a c e s . 2. F u n c t o r t h e o r y . 3. Interpolation spaces. I. K r u g l J a k . N. Ya. 11. T i t l e . 1 1 1 . Series. O A 3 2 2 . 8 7 8 199 1 515'.73--dC20 90-29854 CIP

.

ISBN: 0 444 88001 1

0ELSEVIER SCIENCE PUBLISHERS B.V., 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. / Academic Publishing Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands.

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V

PREFACE

This book is devoted t o a comparitively new branch of functional analysis, viz. the theory of interpolation spaces. It provides a systematic and comprehensible description of many fundamental results obtained i n the initial stages o f t h e development o f this theory, starting from 1976. We shall confine the description t o areas where the investigations have reached a certain level o f perfection (properties o f interpolation functors, general theory of perfection o f the real method and some of its applications). The number

I in t h e title of the book is connected with these restrictions. The time of appearance o f Vol. II (and the list of i t s authors) will depend on t h e pace of research into t h e unexplored regions o f t h e theory. According t o t h e plan worked out mainly by the first author, the second volume will deal with the general theory o f the complex method and t h e methods t h a t are abstract analogs of the Calder6n-Lozanovski’i construction. The authors’ inability t o answer some “simple” questions in this field has forced them t o put off the work on Vol. II for t h e time being. But even this hypothetical Vol. II does not contain all t h e ideas worked out by the authors. We have in mind even a third volume of this course, devoted t o applications (pseudo-differential operators, approximation theory, geometry of Banach spaces, operator ideals, nonlinear functional analysis, etc.). Such a detailed account o f our intention is due t o the fact that the “power of the Unrealized” has definitely influ-

enced the contents and style of the present volume. It contain, besides the finished and rigorously proved results, which constitute the main text, also certain facts which have been mentioned without proof. It would be natural t o present these proofs in the following volumes. This material is mainly contained in supplementary texts which serve as reviews o f the corresponding subjects. Although such a method of description violates the inherent integrity of ideas, it is apparently unavoidable when one is dealing with a

Preface

vi theory which is in a stage o f intensive development.

The theory o f interpolation spaces owes its origin to three classical interpolation theorems obtained by M. Riesz Marcinkiewicz

(1926), s.

(1939).l The significance o f these

Thorin

(1939) and J.

results became clear much

later, mainly due t o the efforts of A. Zygmund and his colleagues and students

(I.D. Tamarkin, R. Salem, A. Calderbn, E. Stein and G. Weiss). This 1950’s,provided some

stage o f development, which was concluded in the

important generalizations of the classical interpolation theorems and many brilliant applications o f these results in analysis. Significantly, the analytical foundation for a further development of the theory was laid at this stage. The next stage o f development, which began in the early sixties, is reminiscent of a phase transition in view o f its intensity and short duration. The analysis is carried out on a new level o f abstraction, and the entire theory is treated as a branch o f functional analysis. The initiator o f this movement was N. Aronszajn, who raised the problem in a letter t o J.- L. Lions in

1958.’ The first publications in this field were made by J.-L. Lions (1958-1960),E. Gagliardo (1959-1960),A.P. Calder6n (1960) and S.G. Kre’in (1960). The

fundamental role i n the further development of the theory is played by the papers by Lions and Peetre (21 (the real method w i t h power parameters) and by Calder6n [2] (the complex method). This was the time of important developments like the appearance o f the K-functional and an elegant “perestroika” of the real method theory

(J. Peetre), the solution o f the “basic

problem of the theory” for the couple

(I&,&)

(A.P. Calder6n and B.S.

Mityagin), and the first attempts t o theoretically systematize the accumulated material (N. Aronszajn, E. Gagliardo).

Let us consider i n detail the

which appeared considerably ahead o f its paper Aronszajn and Gagliardo [l] time. Motivating the need t o carry out this analysis, the authors state that “in view o f the existence of such a large number of interpolation methods3, ‘Naturally, these results also have a past history and are associated with names like I. Schur, W.H. Young, F. Hausdorff and A.N. Kolmogorov (see Sec. 1.lla). ’See introduction to the paper Aronszajn and Gagliardo [l]. 3This is how things appeared in 1965. Fifteen years later, it was found that the number of interpolation methods at our disposal is not large (the real and the complex methods, and the abstract analog of the pmethod); see in this connection Sec. 4.2

Preface

vii

it seems t o be pertinent to study the general structure of all the methods, t o define them and t o analyze the properties that are common for all of them”. This paper contains important concepts like relative completion (in Gagliardo’s sense) and its connection with duality, the interpolation method (functor) as a constructive element o f the theory (each interpolation space is generated by one of such functors), and the extremal properties o f orbit and coorbit interpolation functors. In the introduction to the paper, the authors promise to continue the subject in a following paper, which was supposed t o include the conjugate and self-conjugate interpolation functors, and to study the prevailing “specific” methods in light of the developments of the theory. Unfortunately, this promise was never kept (ironically, a similar promise has been made above by us!), since the programme of action outlined in the last sentence was fulfilled only in the early eighties. The corresponding results are presented in Chaps. 2 and 3 o f this book.

A considerable advancement was made during the period 1965-1975 in applying the methods developed i n the preceding five years. Significant achievements were made in the computation of interpolation spaces for specific functional Banach couples. A detailed description o f the results obtained in this direction can be found in the books by Bergh and Lofstrom [l], by KreYn, Petunin and Semenov

[l], and by Triebel [l]. Hence we shall con-

fine ourselves merely to the statement that a certain decrease i n the interest towards the theoretical side o f the problem was observed during this period. Since 1976,theoretical investigations have been evoking an undiminishing interest. This interest is mainly due to a need to systematize the huge material compiled by the researchers during the preceding decade. However, the present stage of development corresponds t o the works carried o u t i n the early sixties, presenting a sort o f synthesis of the “concrete” approach (associated with the real and complex methods), and the “abstract” approach adopted by Aronszajn and Gagliardo. This inevitably introduced a new level of abstraction in the scientific practice, as was reflected i n the active use o f and Sec. 2.6 for the concept of the “interpolation method”. An affirmative answer to the question as to whether other interpolation methods exist could not be vital for the development of the theory.

...

Preface

Vlll

concepts like interpolation functor, dual interpolation functor, interpolation method, etc.

The main advances during this period have been reflected t o various extents in this book. It provides a possibility of looking at the results of t h e above mentioned books from a new point o f view (although t h e material contained in this book is completely different from t h a t of the books mentioned above). Since the book takes into consideration the interests of beginners of this field, a good deal of efforts went into making the material comprehensible t o readers of this category (unfortunately, this has resulted in an increase in the size of the book). A normal acquaintance with functional analysis and t h e theory o f functions is sufficient for reading this book.

All

information t h a t is not covered within the framework o f functional analysis and t h e theory o f functions is included in this book. Necessary references and remarks are covered in Part A o f the sections “Comments and Supplements” included a t the end of each chapter. There are no references t o t h e literature in the main text, but the names of the authors of the most important results have been included. The contents o f t h e book reveal the material and the order in which it is presented. Note that a reference o f the type (z,y,z) indicates formula (z) from Sec. (y) in Chap. (z), while a reference o f t h e type “see z.y.z” (without parentheses) means t h e

bearing this number

(by result we mean a definition, theorem, proposition, corollary or remark). In conclusion, we would like t o express our gratitude t o the mathematicians who encouraged this venture. In the first place, our thanks are due t o Prof. J. Peetre, who came up with the idea o f publishing our deposited work Brudnyi’ and Krugljak [3] of 1980, based on the results o f investigations carried out by the authors in the second half of 1978 and in 1979. The results presented in that report in a revised and updated form constitute the main part of Chap. 3 and the first part of Chap. 4. Naturally, it would have been more appropriate t o thank Prof. Peetre for his enormous contribution t o the development o f the theory, and also for inventing the K-functional. Unfortunately, it is not customary t o express such kind o f gratitude. Secondly, we are thankful t o those mathematicians who informed us about the results of their investigations in the field under study through

Preface

ix

preprints, letters, and also through personal contacts. We would like t o specifically place on record the contributions from M.Kh. Aizenste'in, M. Cwikel,

S. Janson, P. Nilsson, V.I. Ovchinnikov, 0.1. Reinov, E.M. Semenov, P.A. Shvartsman and M.N. Zobin. Last but not least, we are indebted t o Prof.

S.G.Kre'in,

whose inspiring

lectures (Novgorod, 1976) attended by one of the authors played a significant role in furthering our activity in the field o f interpolation spaces.

Authors

This Page Intentionally Left Blank

xi

PREFACE TO THE ENGLISH TRANSLATION

The theory o f interpolation spaces has its origin in the classical work o f

M. Riesz and J. Marcinkiewicz but had its first flowering in the years around 1 9 6 0 4 am referring t o the pioneering work of N. Aronszajn, A.P. Calderbn, E. Gagliardo, S.G. Kre'in, J.-L. Lions, and a few others. It is o f some interest t o note that what at the beginning triggered off this avalanche were concrete problems in the theory o f elliptic boundary value problems related t o the scale o f Sobolev spaces. Later on applications were found in many other areas of mathematics: harmonic analysis, approximation theory, theoretical numerical analysis, geometry o f Banach spaces, nonlinear functional analysis, etc. Besides this the theory has a considerable internal beauty and must by now be regarded as an independent branch o f analysis, w i t h its own problems and methods.

A new era in the theory of interpolation spaces begins in the mid 70'st h e authors of this book mention the year 1976 as being crucial for themselves; as told in their own preface their interest in interpolation was awoken by a series o f lectures delivered by Kre'in at a summer school i n Novgorod.

It meant a greater focusing on the theoretical questions and a return and a reworking o f the foundations. Among the leaders o f this development we encounter, besides the names Brudny'i and Krugljak and those of their numerous coworkers i n Yaroslavl', also names such as M. Cwikel,

s. Janson,

P.

Nilsson, V.I. Ovchinnikov, who all have in various ways furthered this area of mathematics. The most important single achievement here was however the solution by Brudny'i and Krugljak in 1981 o f one o f the outstanding questions in t h e theory o f the real method, the so-called K-divisibility problem. In a way what this book does harvest what has come o u t of this solution. In addition the book draws heavily on a classical paper by Aronszajn and Gagliardo, which appeared already in 1965 but whose real importance was

xii

Preface t o the English translation

not realized until a decade later. This includes in particular a systematic use of t h e language, if not the theory, of categories. In this way the book also opens up many new vistas which still have t o be explored. In short, I am convinced that the Brudny'i and Krugljak treatise will be the beginning o f yet another era in t h e theory of interpolation spaces and that it will set the mark for all serious work in this area o f mathematics for

the coming decade, if not longer. B y publishing this book in the West, the publisher North Holland undoubtedly is doing a great service to the entire mathematical community. Writing these lines I remember how m y own involvement in this project began, in the summer o f 1982 during a brief visit t o Amsterdam, where I came t o meet Einar Fredriksson. Actually, this volume, mainly devoted t o the real method, is just the first of several planned volumes. Thus Part T w o will be devoted t o t h e complex method and Part Three, not less important, is meant t o deal w i t h the applications. L e t us hope that the authors will have all the time and energy and good health t o accomplish their project.

Jaak Peetre

...

xlll

CONTENTS

PREFACE

V

PREFACE TO THE ENGLISH TRANSLATION

xi

CHAPTER 1. CLASSICAL INTERPOLATION THEOREMS

1 1 3 8 13 23 31 34 39 48

1.1. Introduction 1.2. The Space of Measurable Functions 1.3. The Spaces L, 1.4. M. Riesz’s “Convexity Theorem” 1.5. Some Generalizations 1.6. The Three Circles Theorem 1.7. The Riesz-Thorin Theorem 1.8. Generalizations 1.9. The Spaces L,, 1.10. The Marcinkiewicz Theorem 1.11. Comments and Supplements

66

84 84 87 87

A. References B. Supplements

1.11.1.

The Riesz Constant

1.11.2. The Riesz Theorem as a Corollary of Theorem

88

1.7.1

1.11.3. The Meaning o f the Theorems of Riesz and Thorin for pi

<1

1.11.4. Interpolation of Quasilinear Operators 1.11.5. Interpolation o f Spaces H p

89 90 90

Contents

xiv

CHAPTER 2. INTERPOLATION SPACES AND INTERPOLATION FUNCTORS

91

2.2.

Intermediate and Interpolation Spaces

91 113

2.3. 2.4.

Interpolation Functors

140

Duality

174

2.1.

2.5. 2.6. 2.7.

Banach Couples

Minimal and Computable Functors

211

Interpolation Methods

245

Comments and Additional Remarks

254

A. References B. Additional Remarks 2.7.1. Category Language 2.7.2. Further Extension of t h e Concept of Couple 2.7.3. Density of the Set of Dual Operators for

254

Finite-Dimensional Couples CHAPTER 3. T H E REAL INTERPOLATION METHOD

3.1. 3.2.

I<-divisi bility

3.3. 3.4.

The K-method The 3-method

The K - and J-functionals

3.5. Equivalence Theorems 3.6. Theorems on Density and Relative Completeness 3.7. Duality Theorem 3.8. Computations 3.9. Comments and Supplements

A. References B. Supplements (Computation of I<-functionals and Real Method Spaces)

3.9.1. General Approach 3.9.2. Couples of Banach Lattices 3.9.3. B M O and H p 3.9.4. Differentiable and Smooth Functions

258 258 270 279 289 289 315 338 360 387 409 422 438 459 459 464 464 467 473 482

xv

Contents

3.9.5. Interpolation of Operator Spaces

487

3.9.6. Some Unsolved Problems

490

CHAPTER 4. SELECTED QUESTIONS OF THE THEORY OF THE REAL INTERPOLATION METHOD

493

4.1.

Nonlinear Interpolation

493

4.2.

Real Interpolation Functors

505

4.3.

Stability of Real Method Functors

553

4.4.

Calder6n Couples

578

4.5.

Inverse Problems of Real Interpolation

610

4.6. 4.7.

Banach Geometry of Real-Method Spaces

634

Comments and Supplements

667

A. References B. Supplements

667 675

4.7.1. Real Method for Finite Sets of Banach Spaces 4.7.2. Calder6n Couples

675 682

4.7.3. Some Unsolved Problems

684

REFERENCES

687

SUBJECT INDEX

715

This Page Intentionally Left Blank

1

CHAPTER I CLASSICAL INTERPOLATION THEOREMS 1.1. Introduction The theory

of interpolation spaces’ deals wi t h the methods o f construc-

tion and properties of Banach spaces which are interpolation spaces w i t h respect t o a certain fixed family of spaces. Roughly speaking, a space X is

{ X , } , if T ( X ) c X for ‘‘all’’ linear operators T such that T ( X , ) c X , for any a. Of course, it is assumed here that X,, as well as X , are embedded in some “universe”, viz. a linear space on which the operators T are defined. It is also assumed t h a t the topologies in the spaces X , are matched i n a certain sense. In the most thoroughly investigated part of the theory, the family {X,} consists o f two an interpolation space for t h e family

spaces ( B a n a c h couple). Most o f the results described in this book correspond t o this particular case. The development o f the theory for arbitrary families

{X,, a E A}

is an important problem, although not much is known

in this field (even for the case card A

< co).

However, even the part of the theory that has already been developed has many important and interesting applications. Chronologically, the first step in this direction was the “interpolation” proof o f the

F. Riesz-Hausdorff-

Young theorem (1923) by M. Riesz i n 1926. This proof was based on the well-known “convexity theorem” for bilinear forms proved earlier by M. Riesz. This result and its profound generalizations put forth by Thorin (1939) and Marcinkiewicz

(1939) served as the core around which the theory presented

in this book was developed. This chapter is devoted t o a description o f these three classical theorems and some results associated w i t h them. W e believe that a study of this material is even today the best way of acquainting the beginners with the subject. It will be shown tha t the theorems presented in this chapter are interesting not only from a historical point of view. Even

‘This theory is also known as the theory of interpolation of operators.

2

Classical interpolation theorems

now these results are widely used in modern analysis and as before stimulate the development of t h e theory. Moreover, the reader will find that even these classical theorems have not been studied completely so far.

3

The space of measurable functions 1.2. The Space o f Measurable Functions The classical interpolation theorems operate with a scale o f spaces

15 p 5

00,

and with some o f its extensions ( L , with p

<

L,,

1, “weak”

spaces L,, etc.). It is convenient t o take the space o f measurable (classes of) functions M ( f i , C , d p ) as the universe which contains all such spaces and from which they inherit some of their properties. We shall now consider the definition and formulation o f some properties o f this space.* Here and below, we shall denote by (52, C, p ) a measure space, i.e. a set R with a given a-algebra of subsets C and with a measure p : C -+ [0, +m]. The subsets C will be called p-measurable. It is assumed that p is a-finite and complete. Thus, M

(1.2.1)

R=

u

52,

where

anE C and p(Rn) < m .

n=l

Besides, (1.2.2)

AcBEC

,

p(B)= 0 + A E C ,

p(A)= 0 .

We introduce the following equivalence relation on the set C :

(1.2.3) If A

- B,

A

-

B

* p(A\B) + p(B\A) = 0 .

we write A = B (modp).

If some property is satisfied for all

points S except the set A =

0 (modp),

everywhere. A function f : R

is called p-measurable if for any a E

+

R

it is said to be satisfied p-almost

R,the

Lebesgue set

is p-measurable. We introduce the following equivalence relation on the set of p-measurable functions: 2For missing proofs of results described in this chapter, see remarks in “Comments and Supplements” at the end of the chapter.

4

Classical interpolation theorems

f

(1.2.4)

-

f(x) = g ( x )

g

p-almost everywhere

.

T h e set of equivalence classes has a natural structure of vector space over

R.T h e space

define a metric i n

obtained i n this way is denoted by

M(C2,C,dp).3 W e

M , setting4

v is an arbitrary p-measurable positive function for which J v dp = 1. W e can take for v,for example, t h e function Cr=p=I xn, . [2"p(R,)]-', where R, are t h e sets in (1.2.1) and X A is a characteristic function of A . Here

Theorem 1.2.1. (a) T h e metric p, defines i n

M

a topology of a separable linear topological

space, which is independent of t h e choice of

v.

(b) T h e space M is complete in this metric. (c) T h e convergence of t h e sequence

{f,};" c M

to t h e element

f

in the

m e t r i c p,, is equivalent to convergence in measure.

0

It should be recalled t h a t a sequence (9,);"of p-measurable function

converges in measure p to a function for any

E

> 0 we

'p

if for any finite-measure set A and

have

lim p { x E A ; l P n ( x ) - c p ( ~ ) l I E } = 0 .

n-co

It follows f r o m t h e definition t h a t cp is also p-measurable. T h e convergence to a sequence

{f,}? c M

i n measure is then determi-

ned with t h e help of a sequence {q,,}? of functions, where q n belongs t o t h e class

fn.5

3Everywhere below, we simplify this notation to M [sometimes to M ( d p ) ] . A similar simplification also applies to the spaces L,(Q, C, dp) and other spaces of p-measurable functions. 4The notation ":=" means "by definition". 51t can be easily seen that such a definition is consistent (i.e. it does not depend on the choice of the functions belonging to pn).

The space of measurable functions

5

Henceforth, we will not distinguish between the notion o f a class and a function belonging to this class. Thus, if f E

M,

then f ( z ) denotes the

value o f a function which is a representative of the class f a t the point In view of this remark, we define an order i n (1.2.6)

f 5g

f ( z ) 5 g(z)

M

2.

assuming that

p a l m o s t everywhere

Let us now define the exact upper and lower bounds of the subsets of

M . Recall that the element f is called the exact upper bound o f the subset 3 o f a certain ordered set if f 5 fo for all f E 3 and if from the condition that f 5 g for all f E 3 and g 5 fo follows that fo = g. The exact lower bound is defined similarly. We shall use the notation V 3 and A 3 for exact upper and lower bounds respectively.

If F = {fn}yis a countable subset of M bounded from above, i t s upper bound can be easily determined as follows. We take the function

2 -+

-

(P,

E

fn

and consider

cp,(z). Since a countable union of sets of measure

zero is also a set o f measure zero, the functions obtained i n this way are independent o f the choices o f representatives o f class in

fn

and belong t o the same

M . It can be easily be verified that this class coincides with V{fn};".

Naturally, such a construction is not possible for uncountable sets. Nevertheless, the following theorem is valid. Theorem 1.2.2.

If 3 c M is bounded from above, then a v 3 exists. For some countable subset {fn}yc 3, we have

Corollary 1.2.3.

If there exist functions

vf E f

for which the function z

is finite y-almost everywhere, V F exists. 0

-+

supfEF{pf(z)}

Classical interpolation theorems

6

Obviously, t h e same is t r u e for t h e exact lower bound of t h e set L e t us define t h e absolute value of t h e class

(1.2.7)

If1

:= f V

L'O'

:=

f E M by t h e formula

(-f)

and single o u t t h e subset LloCc

(1.2.8)

F.

{f E M ;

M of locally p-

J If1

integrable functions, setting

< m for p ( A ) < m} .

dp

A

T h e f a m i l y of semi-norms

{f

+

J

If1

dp ; p ( A ) < m} determines o n LloC

A

a structure of a complete separable locally convex topological vector space.

It is clear t h a t

LloC is

a (linear) subspace of

M

a n d t h a t t h e embedding

operator is continuous. W e denote such an embedding by t h e symbol -+. Thus,

(1.2.9)

M

LloC-+

Another i m p o r t a n t subspace M consists of functions with a support bounded i n measure. T h i s subspace will be denoted by

(1.2.10)

Lo := {f E M ; p(suppf)

Here, s u p p f denotes t h e support of

Lo. Thus,

< m} .

f , i.e.

t h e set { z E

fl; f(x) # 0)

defined with an accuracy up to p-measure zero. It can b e easily verified t h a t

p(suppf) defines o n Lo a metric which is invariant for translation. W e will introduce the notation t h e function f

(1.2.11)

---t

llfllo

:= 4SUPPf)

'

Proposition 1.2.4.

Lo is a complete metric space.

Proof. Let {f,,}? c LO b e a

fundamental sequence. Without any loss of

generality, it can be assumed t h a t

Then

f, = fj, j 2 n, outside t h e set A,

view of t h e preceding inequality, we have

:=

Uj2, s u p ~ ( f j +-~fj),

and in

7

The space of measurable functions p ( A ) L:

c

Ilfj+l

- fjllo

-+

0

as

72

+ 00

.

jln

This means that the sequence function

f

coincides with

f,

{f,} converges i n measure, and the limiting

outside the set A,. Consequently,

Finally, we single out another (linear) subspace S

c M,

consisting o f

functions that assume a finite number of values. Thus, (1.2.12)

S :=

{f E M ; f = C

where {cq.} is a finite sequence from p-measurable sets [i.e. A,

n A,

,

( Y ~ x A ~ }

R3 and { A k } is

= B(m0dp) for k

a family of disjoint

# s].

Proposition 1.2.5.

f , there exists a sequence {f,}? c S such f(x) and If,(x)l 5 If(x)l for all x. I f f 2 0, we can assume

For any p-measurable function that f,(x) -+ that 0

{f,}

is an increasing non-negative function.

Classical interpolation theorems

8 1.3. The spaces L, For 0

< p < 00,

we put for a given function

f

For p = 03, we assume that

Jlfllm

(1.3.2)

:= esssuplfl

.

It should be recalled that, by definition, the right-hand side of (1.3.2) is equal t o inf {supz lp(z)I ; cp Th e function

f + llfll,

E

f}.

satisfies the following important inequality:

where p* := min(1,p).

5

For p

1, inequality (1.3.3) is a direct consequence of the elementary

+

inequality la

bJP

5

+ J b J p ,0 < p I 1. For 15 p <

JaJP

coincides with Minkowski’s inequality.

03,

the inequality

Again, the latter inequality is a

corollary o f the following fundamental fact. Theorem 1.3.1 (Hdder’s inequality).

If 1 5 p

I 03,

(1.3.4)

and p’ is the conjugate index, i.e.

1

1

-+ -= 1,

P p ’

and the functions

f ,g are such that llfllp <

00

and Ilgllpi

<

03.

then the

following inequality is valid:

(1.3.5) For 1 < p

IJ

ghdpl

< 00

I IlsllPllhllP~.

and nonzero g and h, the equality is attained only when the

following two conditions are satisfied: (a) for a certain constant X

(b) 0

191 = X IhlP’-’ sgng = sgn h

> 0, p-almost everywhere. p-almost everywhere.

9

The spaces L, As usual, we define the space

(1.3.6)

L, :=

{f E M

;

IlfllP

<

It is well known that Lp is a Banach space for 1 5 p

5 co and a quasiBanach space for 0 < p < 1. The latter statement means that t h e function f -+ llfll, is a quasinorm for p < 1,6 and the topology given by it defines a complete separable linear topological space.

Let us outline some properties of the space L,, which we will be used later. Theorem 1.3.2. (a)

L,

L,

M.

(b) The set So := S n Lo of simpfe functions is a dense set in L, for p<

00.

(c) The norm in L, has the Fatou property, i.e. from convergence in measure o f the sequence {f,,)bounded in

f

belongs t o

L, it follows that

the limiting function

L, and

Remark 1.3.3. The Fatou property is equivalent t o the closure o f the unit ball

B(L,)

:=

{f E Lp ; l l f l l p I 1) in M . Remark 1.3.4. For p

< 00, the (quasi)

norm in Lp is also absolutely continuous. This means

that

Ilf X Q , llP = 0 61n other words, instead of the triangle inequality, a weaker inequality Ilf f Ilgll} with a constant 7 > 1 is satisfied. In the present case, 7 = 2'/P-'.

7 {Ilfll

+ 911 5

10

Classical interpolation theorems

for any decreasing sequence

(0,) c C with

nr=t"=l R, = 0 (modp).

Finally, we shall point t o some statements which follow directly from Holder's inequality and which will prove t o be useful later. Proposition 1.3.5 (Linearization of the Lp-norm). For 1 5 p

< 00,

we have

For p = 00, the same is also true if max is replaced by sup.

Remark 1.3.6. We replace sup (inf) by max (min) if the corresponding upper (lower) bound is attained.

Proposition 1.3.7 (Convezity inequality). For any p0,pl E (0,+00] and 19 E (0, l), we have (1.3.8) Here (and below), p(19) is defined by the equality

[ 1.3.9)

1

Po

1-0 8 ..- -+ - , Po

PI

0

Proposition 1.3.8 (Factorization of the L p norm). For p := p ( 6 ) , we have :1.3w

IlfllP

= inf { l l f o l l y

llfl,;I 1

7

Nhere t h e lower bound is taken over all representations

fl = IfoI'-8 3

lfi1'* fi

E Lp,.

If1

of the form

The spaces L,

11

In order t o formulate a certain corollary of Proposition 1.3.8, we introduce in the linear space L,

+ L,

:=

{f = fo + fl ; f; E L,,}

a (quasi) norm in

accordance with t h e formula

(1.3.11)

IIfllLpo+Lp,

:= inf{IlfollPo

+

IlfillPI

Similarly, we introduce in t h e linear space L,,

; f = fo

n L,

+ fl) .

a (quasi) norm defined

by

(1.3.12)

:= max{llfllPo,

IIfllLw"L,,

IlfllPll

*

From t h e general results of Chap. 2, it will follow t h a t t h e spaces introduced in this way are Banach spaces for p;

2 1, i = 0,1, and quasi-Banach

spaces

otherwise.

--

Corollary 1.3.9. Lp,

nLp,

0

L,

L ,

+ L,,'

Po

< P < Pl.

If the measure of R is finite, this corollary can be amplified. To be more precise, the following proposition is true. ProDosition 1.3.10.

If ~ ( 5 2 < ) 00, we have for pl (1.3.13)

L,(R)

L)

Lpl(R)

< PO

.

Besides,

lim

q-p-0

Ilfll, = IlfllP .

0

Remark 1.3.11. In the case of a discrete measure, t h e converse embedding t o (1.3.13) holds. Namely, consider an L,-space

it by I,. Thus,

corresponding t o such a measure and denote

12

Classical interpolation theorems

Besides the spaces

L,,

we introduce the weight spaces L p ( w ) with a

p-measurable positive function w (weight). By definition,

The spaces

L p ( w )have properties similar t o those formulated above for L,.7

For example, we can mention the analog of the convexity inequality: (1.3.17) Here,

IlfllLde)(we)

I llflli;&o)

Ilf1119p,(wl)

’

p ( 9 ) is the same as in Proposition 1.3.7, and

w8 :=

wA-’ww;’.

’Since they are L,-spaces constructed with the measure dji := w l / P d p .

M. Riesz’s ‘(ConvexityTheorem”

13

1.4. M. Riesz’s “Convexity Theorem” Let

I,”

coincide with

I I ~ I I:=~

R”provided with the (quasi)

{k i=l

lfilp}*’p

norm

.

For the formulation o f t h e theorem, we need Definition 1.4.1.

The function f : L -+ (0,+m), defined on the linear space L (over B ) ,is called logarithmically convez if the following inequality is satisfied for any x,y E L and for any 6 E (0,l): (1.4.1) Thus,

f((1- 8)z

f

+ fly) L f(z>’-’f(y)’

.

is logarithmically convex iff logf is convex. It should be obser-

ved t h a t it is sufficient t o verify (1.4.1) for just one value 6 = 6(z, y). This follows from a well-known fact contained in Proposition 1.4.2.

I c R,is continuous and for each pair z,y E I there exists at least one 6 E ( 0 , l ) such that If the function

‘p

:

I

-+

R,defined on

an interval

then the function ‘p is convex. 0

Let us now suppose that T :

I,”

-+

1; is a linear operator (matrix) and

Mpqi s its norm. Theorem 1.4.3

(M.Riesz).

The function ( l / p , l / q ) triangle

4

Mpq is logarithmically convex in the “lower”

Classical interpolation theorems

14

Proof. It

is sufficient t o prove that the function Mpq is convex on each

segment lying inside A. In this case, the statement can be extended t o the entire triangle A by continuity. Thus, taking into account Proposition 1.4.2, we must establish that for given pi and qi (1.4.2)

1 < p i 5 q,

< 00 ,

Z

= 0,l

,

there exists a number 6 = I9(pi, q i ) E ( 0 , l ) such that (1.4.3)

M 5 M;-'Mf

.

Here, we have put

Mi

:=

Mp,q,

M :=

;

Mp(9),,(9)7

while p(I9) and q(I9) are defined similarly t o (1.3.9).

Let us first define 6. In view o f (1.4.3), p: 2 q:, and hence there exists a certain number r satisfying the inequalities (1.4.4)

polpi

57- 5 4 0 / 4 : .

We then put (1.4.5)

I9

1 r+l

:= -

and verify that (1.4.3) is valid for such a value of 6. For this purpose, we take a E 1; such that

(1.4.6)

A4 = llTall, ,

llullP = 1 .

Here and below p := p(I9) and q := q(29). Next we choose b E 1:) such that n

(1.4.7)

1( T a ) i b i = llTallq ,

Ilbll,) = 1 .

i= 1

In view o f Theorem 1.3.1, we have for a certain X

Ibl = X 1Ta14-'

,

sgn b = sgn(Ta)

>0

.

W i t h the help of the first equality in (1.4.7), we find that X = M1-, and hence

15

M. Riesz’s (‘Convexity Theorem” (1.4.8)

lTal = M lbIq’-’

. Tt :

Passing t o the conjugate operator

Zrl -+ Z;l

and using the fact that

its norm is also equal t o M , we establish the following equality in analogy with (1.4.8):

(1.4.9)

ITtbl = M lalP-’ .

From this equality we get

we obtain together with the preceding inequality

I Ml

M(llall(P-l)P:>P-l

Ilbll9;

.

Similarly, with the help of (1.4.8)we get ~(llbll(q~-l)qo)q’-l

L Mo IlallPo

*

Raising t h e last two inequalities t o appropriate powers and multiplying them together, we obtain M(ll~ll(P-l)P;) (P-V(

IM,-#M;

Ilql(q‘-1

Ilbl; lla11y

~

qo

)(q’-1)(1--8)

<

*

Let us verify that for the given 9

(1.4.10) ,-” ;1 .1

I ~ l l ~ l l ( p - l ) p ; ~,~ ~ -Ila1l:;~ 8 I (llbll(q~-l)qo)(q’-l)(l-~)

.

Together with the preceding inequality, this gives (1.4.3). For this purpose we put

1 , v := - r P: QO In view of (1.4.4),the numbers p , v E (O,l), while in view of (1.4.5)and

(1.4.11)

L ,L

:=

equalities p = p(19), q = q(9), we have

Classical interpolation theorems

16

q; = (q' - 1)qov

+ q'( 1 - v)

Hence, Holder's inequality gives

whence we obtain t h e first of the inequalities (1.4.10)by taking into account

the second inequality in (1.4.6).The second inequality in (1.4.10)is proved si milarly. 0

Remark

1.4.4.

Let v E

R; and Z;,v be defined

IlfllP,"

{$

:=

by the norm

lfilpv.i}l'p

.

Arguing in the same way as while proving Theorem

1.4.3,we obtain a more

general version of this result. Namely, the following theorem is valid.

1.4.3'. The norm M , := sup {llTzllir,,; llzllpv = 1) as a function of (l/p, l / q ) is logarithmically convex in A. Theorem

0

Remark

1.4.5.

The restriction p ;

5

qi in Theorem

1.4.3cannot be eliminated. Indeed, l e t

us consider t h e operator

T ( 2 , y ) :=

( F T,) X-Y

,

(z,y)ER2.

We denote by llTllPPti t s norm as an operator from 1; in

llTllc01 :=

m a l+lvlv161

2

lit. Since

M. Riesz’s “Convexity Theorem”

17

and, in view of the parallelogram identity,

if Theorem 1.4.3 were valid without the restriction p; I q;, we would obtain the following estimate for 2

I p I 00

and 6 := 2/p:

Thus, the inequality (1.4.12)

( l ~ x~- y + ) U P l’ I(~ l P‘

2 would be valid for 2

5p 5

00.

However, if we take, for example, p := 3,

x := 5 and y := 3, we find t h a t this is not true. It will be shown below (see Corollary 1.7.4) that even if the condition p; I qi is violated, t h e following inequality analogous t o (1.4.3) is still satisfied:

Remark 1.4.6. Since IITII1.. =

$, we can find arguing in the same way as in Remark 1.4.5

but choosing p E [1,2]and

6 := 2/p’ that (1.4.12) is valid for such

a

choice of p . Since

we obtain the following identity for the norm o f

T:

Even this result demonstrates t h e power of the M. Riesz theorem. Indeed, the substitution z := y/x gives

Classical interpolation theorems

18

It is rather difficult, if even possible t o find the latter maximum by emploIndeed, for p > 2, the equality IITllppt= 2-’/P, 1 5 p 5 2 does not hold. Moreover, it turns o u t that the function

ying differential calculus.

l/p .+ IITllm, is logarithmically concave for 2 5 p

5

1.4.7. Putting in (1.4.7)a := l/p, /3 := l/q’ and

t;j

00.

Remark

( x , y ) :=

C,”=l ziyi and

:= ( T e i , e j ) , where

{ e i } ? is the standard basis i n En,we can refor-

mulate the statement o f Theorem

1.4.3as follows.

The function M : .R: + R+, defined by the formula

is logarithmically convex on the set

It was observed by Thorin that M is logarithmically convex i n the entire range o f the parameters a and p. We shall limit ourselves to the following case, which will be used below:

(1.4.15) a + / 3 > 1 , a 2 0 , 0 5 P 5 1 , and consider first only the part of the set Jensen’s inequality

(1.4.15)where a 2 1. In view o f

(1.3.15),we have

If the maximum on the right-hand side equals Mi,(/3), we obtain by putting

x :=

e;,

M. Riesz’s “Convexity Theorem”

19

Hence it follows t h a t

(1.4.16)

M ( a , p ) = max Mi(@).

lsisn But t h e function Mi@) is equal t o

> 0 follows from

(C ltijll/(l-@)l-fl

and its logarithmic

1.3.7. Then the function M is (1.4.16). Thus, M is logarithmically convex on each of two sets, namely, Sldefined by (1.4.14) and

convexity for ,f3

Proposition

also logarithmically convex in view o f

sz :=

{(a,P) : a + p 2 1,

a 2 1, 0

5 p 5 1)

It remains for us t o show that M has the same property on Sl U Sz as well. Otherwise, there would exist a segment 1 intersecting the common boundary of 5’1 and Sz a t a certain point

(1, P o ) , such t h a t the convexity of the function MI1 is violated a t this point. Let ( a ( ~ ) , p ( ~be) )a linear parametrization of I such t h a t ( Y ( T ~= ) 1, ,B(T~)= 1. Let 5,fi E Elnbe maximizing vectors for M(1, P o ) . Then

Besides, l o g N is a concave function and

However, this equality contradicts inequality (1.4.17) when the convexity of logM(1,P) a t point Po is violated (see Fig. 1). Since for 0 < p 5 00, 1 5 q 5 00 we obviously have

t h e statement o f Theorem 1.4.3 can thus be extended t o a wider range o f values p , q:

Classical interpolation theorems

20

Figure 1.

The same is also true for the version

1.4.3’ of this theorem (see Remark

1.4.4). In most applications of Riesz’s theorem it is sufficient t o use a weaker inequality than

(1.4.18)

(1.4.3):

M 5 k M,’-’M;

with a constant

k = k(fi,pi,q,).

There exists, however, a small number

of problems i n which the knowledge o f the exact value of the constant is essential. As an example, let us consider the proof of the inequality from which the uniform conwezity o f the space Recall that the Banach space

convesity bX(&) is

> 0 for

E

> 0.

L, follows

for 1 < p

< 03.

X is uniformly convex if i t s modulus of Here, the modulus of convexity is defined

by the formula

(1.4.19)

~ x ( E ):=

inf(1-

~

2

+

; z,y E S ( X ) , JIz- yII = E }

,

M. Riesz’s “Convexity Theorem” where S ( X ) := {z E Thus,

X , llXll

~ X ( E estimates )

chord [z,y] of length

E,

21

= 1) is t h e unit sphere in

X.

from below the distance from the middle of the

whose endpoints lie on

S(X).

Theorem 1.4.8 (Cladson).

The spaces L, are uniformly convex for 1 < p < 00. &f.

We shall make use of the inequality

Here 1 < p

< 00

and P := max(p,p‘). If f , g E

it follows from (1.4.20) and (1.4.19) that for (1.4.21)

SL,(€)

2 1 - [l - ( E / 2 ) y r > 0

E

S(L,) and

Ilf

- gllp = E,

E (0,2] we have

,

and the uniform convexity is established. In order t o prove (1.4.20), we write i t s left-hand side in the form

Since P is chosen in such a way that the number s

5 1, it follows

from the

inverse Minkowski inequality’ and inequality (1.4.12) that the left-hand side of (1.4.19) does not exceed

5

{ I ( If I” + ,’”.)’”

.

191“

It should be noted that the application of (1.4.12) is justified since by hypothesis P = max(p,p’)

> 2.

Therefore, for

t

:= p / r ’

(t

> 1 in accordance

with the choice of r ) , the left-hand side of (1.4.20) does not exceed

22

Classical interpolation theorems

It should also be noted that if we would use instead of inequality (1.4.3)

the weaker inequality (1.4.18) with k

> 1, this constant would appear in the

inequalities (1.4.12) and (1.4.16). Thus, the estimate (1.4.21) could not be obtained.

23

Some generalizations

1.5. Some Generalizations

A. From Theorem 1.4.3(and

Remark 1.4.7)we can easily obtain a more

general version o f it t o be considered here. Let us suppose that, as before, 0

< 19 < 1 and that

Further, let T be a linear operator acting from the space So := S ( d p ) fl Lo(+) t o the space Llw(dv). (The functions in So := S fl LO are called simple functions.) Theorem

1.5.1.

If under the assumptions made above the inequalities

f E So hold, then T extends by continuity t o an operator acting from L,(,q(dp) into L,(,q(dv),and i t s norm does not exceed M,'-'M,9.'

for

Proof. Let S$(dp) be t h e subspace o f t h e space So(dp), which consists of functions of the form Cy=l a i x ~ , where , A := {A;} is a fixed family of disjoint p-measurable sets. We define on S$(dp) the operator

RA

:

St

--+

R" by the formula

For vi := p(Ai), we have in the notation o f Remark 1.4.4

Next, l e t 23 := {Bi}T=l be a family of v-measurable sets, analogous t o the family

A. Suppose that RB : S t ( d v ) + R" is defined i n the same way

as

RA. Finally, we use the formula 'The theorem is valid for q(9) = 00 (i.e. for qo = q1 = m) only if L , is replaced by the closure of the set So in this space. Henceforth,we shall always mean this substitution for the space L , when speaking about the extension of the operator T by continuity.

24

Classical interpolation theorems

t o define the averaging operator. It then follows from Holder's inequality that

and, hence, for w, := v(Bi),we obtain (1.5.5)

IIRaPaflll~,,= IIPBfllP

I IlfllP .

With the help of T,we can define the operator T := PL"

+ PL" by the

formula

T = R5P5 TRAl . In view of the assumptions of the theorem, as well as the relations (1.5.5) and (1.5.3), we have

llT411~',w I ll(TRA1>41P,I Mi llRA141P, = Mi Il41;,," . Hence, we can apply to the operator

T the version of M. Riesz's theorem

described in Remark 1.4.4. This gives

ll~~lll;8),wI

11411~,),"

7

2

E

R" .

In view of the equalities (1.5.3) and (1.5.5), it therefore follows that

(1.55)

IlP5Tfll,(t9) I M,-$M,9

llfllP(19,

for an arbitrary simple function. Let us now suppose that in

S,(dv), which

put

B,

xn

:=

Ey=l (Y;,,xB,,,

15 n

< 00,

is a sequence

Tf in L,($)(dv)[see Theorem 1.3.2bl. 5 n}. Then PB,(xn) = xn, and so, taking

converges t o

:= {B,,,, 1 I i

account (1.5.5), we get

IITf - p~n(Tf)IIqI IITf - XnIIq

+ IIPB,(T~- Xn)IIq

I211Tf-xnllq+0

.5

asn-+m.

We into

25

Some generalizations Hence, applying (1.5.6) t o

PB,T passing on t o the limit, we get

IITfllq(e) 5 M,'-'M:

(1.5.7)

Ilfllpcs)

7

f E So(dP) .

It should be noted that we can assume that p(19) < 00. This means that So is dense in L,($), and therefore (1.5.7) leads t o the statement of t h e theorem. 0

B. Using the method o f M. Riesz, some similar results can also be proved. Thus, introducing obvious changes in the proof of Theorem 1.4.3, we obtain the following result (real-vaZued analog of the Stein- Weiss theorem). Theorem 1.5.2. Suppose that under the conditions of Theorem 1.5.1 the following inequalities are satisfied instead of (1.5.2)":

1-8 8 og := 00 v1

,

W$

:= w1-8 0 w1

.

T extends by continuity t o an operator acting from L,(~)(W@ ; dp) in Lq(d)(w8; d v ) , and i t s norm does not exceed MJ'-'M:

Then operator 0

We leave it for the reader t o prove this statement as an exercise. Remark 1.5.3.

The estimate of t h e norm in Theorem 1.5.2 (and, hence, the corresponding estimate in Theorem 1.5.1) can be obtained from the following less stringent inequality :

(1.5.9)

IITIILp(u,dp)~L,(tu,dv)5

ma(M0, Ml).

Indeed, putting o; := Mivi, we can write (1.5.8) in the form "The space L p ( w )is defined by formula (1.3.16).

Classical interpolation theorems

26

I llfllL,i(e,;d”)

JJTfIJL*;(w,;dv)

= 071

7

.

We then obtain from (1.5.9)

It Tf IIL d a )(weid”) 5 IIf I1Ld*)(% ; d r ) = M,’ -$M: II f lIL++Ja;dr) Since Mo-9M:

5 max(Mo,MI) as well,

.

Equation (1.5.8) is indeed equiva-

lent t o the estimate i n Theorem 1.5.2.

C . The possibility o f using interpolation theorems for nonlinear operators plays a significant role in applications. W e shall specify the classes of operators for which such theorems can be obtained. Definition

1.5.4.

The operator T mapping the linear space led subadditive if for any z,y E

L

into the space M ( d p ) is cal-

L,the following

inequalities are satisfied

p - a Imost everywhere:

If instead o f this inequality the following inequality is satisfied:

with 7 > 1, operator T is called quasiadditive. Definition 1.5.5.

A subadditive operator T is called s u b h e a r if it is positive homogeneous, i.e. if

(1.5.12)

IT(kz)J= ICI IT(z)I

for any scalar

k.

A quasiadditive operator T is called quasilinear if for a certain y and all

k, the following inequality is satisfied:

>

1

27

Some generalizations Let us consider some important examples of such operators. Example 1.5.6 (Hardy-Littlewood mazamal operator). Let

M

:

(1.5.14)

Lp((ER,dx)3 M ( R , d z ) be an operator of the kind

7h

-!2h

( M f ) ( z )= sup

Ifldx .

2-h

h>O

The measurability of the function M f follows from the semicontinuity from below of the upper bound o f a family of continuous functions, while i t s sublinearity is verified directly. Example 1.5.7.

Let T, : L ( d p ) + M ( d p ) , n E N ,be a sequence of linear operators. We define t h e maximal operator of this sequence by putting (1.5.15)

P(f):=

SUP

ITnfI

n

Obviously, this operator is sublinear. Let us consider a result that clarifies the role of maximal operators in the investigation of t h e convergence o f sequences {T,f} almost everywhere. Proposition 1.5.8. Suppose that T' is bounded in

(1.5.16)

I Y llfll,

IIT'fllP

>

> 0.

L,. This f

E

L,

means that 7

Further, suppose that Tnf +

f p- almost F which is dense in L,. Then T,f converges p-almost everywhere for any function f in L,.

for a certain constant 7 everywhere for all

f belonging t o

Proof. Suppose that cp

a certain subset

E F is such that

Ilf

- cpll,

<E

for a given

E

Then

(1

KG

n,rn+co

1Tn.f- Trnfl(( L

1

KG n,m+oo

ITncp- Trnvl

1(, +

> 0.

Classical interpolation theorems

28

Since

Tncp-+

cp p-almost everywhere for cp E F , the first term on t h e right-

hand side is zero. In view of (1.5.15) and (1.5.16), the second term does not exceed

Since E is arbitrary, it follows hence that

This means that liq,,,,

lTnf- Tmfl= 0 p-almost

everywhere.

0

Remark 1.5.9.

The above statement also follows from the following inequality which is weaker than (1.5.16):

(1.5.17)

p ( { z ; ( ~ * f ) ( c>) t ) ) I 7~

Here y is independent o f f and

J

VIP+

(t > 0) .

t.

The fact that (1.5.17) follows from (1.5.16) is a consequence of the

C h e b y s h e v inequality

Example 1.5.10. Finally, l e t us consider an example of a quasilinear operator, which is important in the theory o f nonlinear differential equations. Suppose that K is a compact set, p is a Bore1 measure on K and 0 :

R+x K

-t

R+ is

a

function continuous in the first and p-measurable in the second argument.

Let

29

Some generalizations

for all

t , s, k E R+and a fixed y > 1. In view o f these inequalities, L

:=

{f E M ( + )

;

t h e set

J W(.>I,.)+ <}.

is a linear space. It also follows from these inequalities that the operator

is quasilinear.

Let us show that Theorem 1.5.1 (and its generalization 1.5.2) is valid for sublinear operators p;, qi

2 1.

Theorem 1.5.11. Let

T

:

(L,

+ L p l ) ( d p )+ LIOC(dv)be a sublinear operator for which the

conditions (1.5.20)

IITfllqi

I Mi Ilfl ,,

are valid. Here 1 I p;

IITfllq(d,

(1.5.21)

Proof. Let

= 071

7

5 qi < 00.

7

Then

I M,'-Qe Ilfllp(s, .

us start with the special case of finite dimensional Lpi. Then

T f E R"and (1.5.20) has the form

llTfIli~i,w I Mi llfllpi We recall that JIz(Ipy:=

7

i = O,1 .

{xy=lIz;lqwi}

1l q

.

Let us consider the functional

+

ui : f + I(Tf);l, 1 <: i 5 n, defined in the linear space Lpo Lpl. In view of the sublinearity of T , the functional u; is positive homogeneous and subadditive. Hence, by the Hahn-Banach theorem, for a given function fo in

L,

+ L,,

there exists a linear functional

Kfo = I(Tf0)il for an arbitrary

f

in L,

and

T.: for which

IKfl I I(Tf>il

+ LpI. Let us define the linear operator V

L,, -+ R" by the formula

:

Lpo+

30

Classical interpolation theorems

Vf Then

:= (Vlf,

...,Vnf) .

lVfl 5 lTfl for all f and V f o= ITfol. Consequently,

llVflll~,;w 5 IlT-fll~~,.~ 5 Mi IIfIIp,

7

i =O,1

7

and by Theorem 1.5.1, we have

I I ~ f o l l l ~ , ) , w5 M,-9M:

llfOllP(19)

*

Since fo is arbitrary, t h e theorem is proved for this case. A transition t o the general case can be made by using the approximation used in the proof of Theorem 1.5.1. 0

Remark 1.5.12. An application of the complex analog o f the M. Riesz theorem given below

(the M. Riesz-Thorin theorem) makes it possible t o obtain an analog of Example 1.5.10 for complex-valued spaces L p . It is interesting t o note that the restriction p; 5 q; is absent i n this case. Further, application of this theorem allows us t o remove this restriction in t h e real case as well. This leads t o the appearance of the constant

5 fi

on the right-hand side of (1.5.20) (see

Corollary 1.7.4).

It will follow from the Marcinkiewicz theorem proved below that Theorem 1.5.11 is valid for quasiadditive operators as well under the restriction 0

<

p ( 6 ) _< q(6) < 00. It is unknown whether this statement is true without this condition being satisfied.

31

The three circles theorem

1.6.The Three Circles Theorem Let us now consider a generalization of the M. Riesz theorem for complex

L, proposed by Thorin. The analytical foundation of t h e Thorin method is the classical Hadamard three circles theorem. Let

f

be a function analytic in the open ring A := { z E

C,r1 < IzI <

r 2 } and continuous in i t s closure A. For p E (lnrl,lnr2) we put

~ ( f p ;) := max If(e"+")l . ip

Theorem 1.6.1. For any po,pl E (lnrl,lnr2) and 9 E (0,l)

M ( f ; P(9)) 5 M ( f ; P o ) l - f f M ( f ; P d f f . Here p ( 9 ) := (1 - 8)po

Proof.

+ 6pl."

Let us consider the function g ( z )

:= z X f ( z ) , where X E

R is

specified below. If X @ 27,then g is analytic on a Riemann surface lying above A (it is multiple-valued). However, 191 is single-valued and continuous in A. By t h e maximum modulus principle for analytic function, we have

for em

5 1.1 5 eP1.

(1.6.1)

Consequently, for 1.1

= eP('), we obtain

M(p(29)) 5 rnax { e X ( W - - p ( S ) ) M ( p Oe)x, ( P 1 - P ( S ) ) M ( p.l ) }

We minimize the right-hand side with respect t o X (the expressions within the braces will be equal). Then X is defined by the following equation: e-X(P1-Po)

= M ( P1)lMbo) .

Substituting this equation into (1.6.1),we get

M(po)5 e-NP(')-W)M(po) =

(e-NP1-Po)

18W P o )

l'In other words, the function p -+ M ( f ; p ) is logarithmically convex

=

Classical interpolation theorems

32 = M(po)'-SM(pl)S . U

We shall require the following Corollary 1.6.2. The function m :

R + R+, defined

m ( z ) :=

SUP Y € B

where z = z

I$

by the formula

akebkz/ 7

+ zy E C, ak E C and bk E R,is logarithmically

Proof.Suppose first that all bk's are rational and q E N

convex.

is the least common

multiple of their denominators. Then

where s = s ( z )

:= e x p ( z / q ) and mk are some integers. Since s maps

the straight line Rez = zo onto the circle Is1 = e x p ( z o / q ) , we have

m ( z ) = sup, I C ak e x p [ m k ( z / q + icp)] and the logarithmic convexity of m follows from Theorem 1.6.1. Passing t o the limit, we arrive a t the general case from here. 0

Corollary 1.6.2 can be obtained straight away by using the following version of Theorem 1.6.1. Theorem 1.6.3 (three lines theorem).

I f f is a function analytic in the open strip S := { z E C ; 0 < Rez

< l}

.(fI

bounded and continuous in its closure, the function z + s u p Y E ~

and

+ iy)l

is logarithmically convex.

Proof.It is sufficient t o apply the argument of Theorem 1.6.1 t o the function f E ( z ) := e x p ( E z 2 ) f ( z ) E,

> 0, and the rectangle SR

:= { z E S ; IyI

Since fc(z) + 0 for IIrnzl + 00, for sufficiently large gives

R

and

L R}.

19 E ( 0 , l ) this

33

The three circles theorem

It remains to make E tend to zero and 0

R

to infinity.

Classical interpolation theorems

34 1.7. The Riesz-Thorin Theorem

In 1938 Thorin, who was a student of M. Riesz, found a remarkable proof of t h e analog of the M. Riesz theorem for complex-valued spaces L,.

The

fundamental idea of this proof, viz. the analytic continuation t o the complex domain with respect t o the variable l / p , had a significant influence on the development of the general theory. In order t o formulate the main result, we denote by L p ( d p ; C) a space similar t o the real-valued space Lp but now composed of complex-valued function. Suppose t h a t

(1.7.1)

O
Let T : So(&;

l L q i < ~ ,Z=O,l.

C)--+ L"'(dv; C)be a

linear operator. Under the above

assumptions, the following theorem is valid. Theorem 1.7.1.

If in the domain of

T the following inequalities are valid:

(1.7.2)

<M k

llT-fl/qk

k = 0,

~ ~ . >f ~ ~ p ~

7

then T extends by continuity t o an operator acting from L p ( @ ) ( d pC) ; into

L q ( 9 ) ( d p ;C). The norm of this operator does not exceed Mt'-'M;.

Proof. Just

as in the proof of Theorem 1.5.1, everything is reduced t o a

finite-dimensional case. Consequently, we must show that if T : CF

--t

CF

is a linear operator, the function

(1/P,

l/d

+

MP,

:=

is logarithmically convex for 0

{llT4ll;,,(c)

<

p

<

;

Il4ll;,"(c)5 I >

mand 1

<

Q

<

00.

It should be

recalled that

If (z,y) = x k j j k is a scalar product and ( t s k ) is the matrix of the operator T in the standard basis C", then just as in the proof of Theorem 1.4.3, we have

35

The Riesz-Thorin Theorem I

n

Mp* := supl(Tx,y))= sup Here the upper bound is taken for the set of those

x and y from 6'"for

which

c where w' :=

We put

1 X , I P ~ s 51

c

,

51 ,

IYkIQ'WL

u2-q'.

:= l / p , ,8 := l / q ' and make the following change of

CY

variables:

xs := p;e"P' ,

R+and

yk := v 0 k eid'k

,

E R. Let us redesignate Mpq by M ( a , p ) . In the new notation, we have

where pa, q k E

(pd,$k

where the upper bound is taken over the set

vy5

R",which

E

R of vectors p , E~ R: and

are independent of a and

p

and satisfy the following

inequalities: Psvs

51

r]kWL

7

51

+

W e shall show that the restriction of the function

M t o the intersection of

with an arbitrary straight line 1 is logarithmically convex. Suppose that the parametric equation for 1 has the form

u ( x ) = sup n where

tks

:=

2

ssk(qsPk) e

tks(Vk)PoP~.

Replacement o f x by z := z the argument (ps and of the set

i(98-d'k)

s,k

p h p k

+ i y E C leads only t o a shift o f X Inq s in

i n the argument

$k

( s , k = 1, ..., 71). Since

R is invariant relative to parallel translations of the vectors

substitution o f z for z does not change

M ( z ) . Consequently,

'p and

+,

36

Classical interpolation theorems

But the inner supremum is a logarithmically convex function, i n view of Corollary 1.6.2. Consequently,

is also logarithmically convex as the upper

bound of such functions. 0

Let us discuss the relationship between t h e real and complex forms of the interpolation theorems. Namely, we shall show that Theorem 1.5.1 is valid for p ;

2

q, as well, but t h e right-hand side of (1.5.2) then contains

the constant K e ( p ; , q,)

5 4 (the

exact value of this constant is unknown).

For this purpose, we introduce the “mixed” space consisting of p-measurable vector-valued functions

L , ( d p ; I;),

f

:=

15 p

(a, C,p )

5 co,

-+

Rn

for which the norm

is finite. It can be easily seen that this space is a Banach space. Further,

let

L ( X ,Y ) denote the Y.

Banach space of continuous linear operators acting

from X into

We consider the operator r n

:

~ ( ~ p ( d Lq(dv)) ~ ) y

+

~ ( ~ p ( ; dc), p Lq(dv; 1;))

>

defined by t h e formula

Let yn(p,q) be the norm o f this operator. Thus, for any

{f;}:=l C L,(dp)

we have

The following result summarizes our knowledge about the constants yn(p,q).

37

The Riesz-Thorin Theorem Theorem 1.7.2. (a) (Grothendieck’s inequality) There exists a constant

KG such that

(b) (Krivine) For all n E N ,

(c) (Krivine) 72(00,1) =

a.

Remark 1.7.3.

The exact value of the Grothendieck constant Kr: - is unknown. The best 7T = 1,782 ... was obtained by Krivine. estimate KG I 2 In( 1

+ Jz)

Let now

T

be the operator from Theorem 1.5.1, but p ,

carry out the “complexification”

> q; 2 1. We

T, of this operator by putting for f E So(Q:

Thus, we have proved that

Consequently, the application of Theorem 1.7.1 to the operator Tc leads to the inequality

(1.7.5)

I I T c f I I L d e ) ( ~ )L

~ ( 8 ,; q ~j)M j ;-'MB

II.fll~dq(~) 7

38

Classical interpolation theorems

where we have put

Considering in inequality (1.7.5) only the functions from L,(s)(pL) and using items (b) and (c) of Theorem 1.7.2, we obtain

Corollary 1.7.4. For pi 2 q; 2 1, Theorem 1.5.1 holds with the constant in (1.7.6) which does not exceed

a.

0

Remark 1.7.5. In certain cases, a better estimate of (1.7.6) can be obtained. For example, in view of the obvious equality y2(2,2)= 1, we obtain the estimate 2'12 for po = 40 = 2. Remark 1.7.6.

A statement similar to Corollary 1.7.4 is also valid for Theorem 1.5.2 (see Theorem 1.8.1 below).

39

Generalizations 1.8. Generalizations

A. Let us start with generalizations which can be obtained by a direct application of the Thorin method. A slight reconstruction of the proof leads t o the corresponding result for complex-valued weighted spaces. Thus, the following theorem is valid. Theorem 1.8.1 (Stein- Weiss). Theorem 1.5.2 is valid for the corresponding complex spaces for 0 and

< pi 5 co

1 5 q; 5 00.

0

Another generalization is associated with multilinear operators. This is

the term applied t o the mappings normed spaces

T from

the product

ny=,Bj of (quasi)

Bj into a normed space B , which are linear in each argument

and satisfy t h e inequality

The lower bound y is called the norm of a multilinear operator

We shall denote the normed spaces of such operators by Mult

T.

(n: Bi ; B ) .

B$ denotes the space So(Rj,Cj,dpj; C )equipped L k-norm. Similarly, l e t Bk := S O ( f i Z , % , dC) v ; be a space with PJ an L,r-norm. Here, j = 1, ...,n and k = 0 , l . Suppose now that

with an

Theorem 1.8.2.

Bjk; Bk)and its norm does not exceed Mk,k = 0,1, 1 5 q k 5 00, 1 5 j 5 n, Ic = 0,1,then T can be extended by continuity t o an operator from Mult(ny=, L,,p) ; L q ( q ) and its norm does not exceed M,'-'Mf. If

T

E Mult(&

and if 0

5

pfi

5

00,

The proof of this theorem for the finite-dimensional case necessitates t h e establishment o f logarithmic convexity of the corresponding multilinear

form. This proof involves an exact repetition o f the argument o f Theorem

40

Classical interpolation theorems

1.7.1 for t h e bilinear form. The rest o f the proof is based on a passage t o limit similar t o the one carried out in Theorem 1.5.1. 0

Remark 1.8.3. Naturally, a generalization t o t h e weighted case is also possible here. Finally, the arguments of Theorem 1.5.11 can be extended t o the complex case without any change. Hence, the Riesz-Thorin theorem is also valid for sublinear operators.

B. The modern version of the generalization of Theorem 1.7.1 refers t o a continuous family of operators T, : L,(,)(dp ; C) + L,(,)(dp ; C), where z runs through the points of the closure of t h e unit circle XI := { z E

C ;IzI < I} or

a more generally simply connected domain of C. Here, p , q

and z + T,are, o f course, analytic functions o f z . This allows us t o use the powerful apparatus of the theory of analytic functions. We give here some information about this theory which we will require later. Definition 1.8.4. The analytic function

f

:

XI

+ C belongs t o the Nevanlinna class N ( B )

if sup

J

r
aD

log+ If(reiq)ldm < oo

.

Recall that dm := (1/2w)d'p and log' z := max(1og I,0).

I f f E N ( D ) , then for almost all cp E aD the limit limt,l

(f

exists along any path y : [0,1] -+ XI which is non tangential t o

o y)(t)

and

which ends at the point exp(i'p). This limit is independent of 7 and hence we can define a boundary function

41

Generalizations Moreover, log

If1

E

L(dm).

1.8.5. A function f in N ( D ) belongs t o the Smirnov class N + ( D ) if Definition

lim r-1

I

log+ If(re’qP>ldm=

am

I

log+ ~j(((p>ldm .

C 3 D

0

The following theorem establishes a basic property o f the class N + t o be used later.

1.8.6. I f f E N + ( I l ) , then

for any point z E 4)

(1.8.11

I log ~ f ( ( c p ) ~ ~ ~ ( .c p ) d m

Theorem

log

I~(Z)I

J

aD Here

P, is the Poisson kernel, i.e. the function

Remark 1.8.7.

It follows from Definition 1.8.5 that N + ( D ) is an algebra over C with respect t o pointwise multiplication.

If U c C is a simply connected domain with a simple closed C’-smooth boundary, we can define the Srnirnov class

N + ( U ) by “transplantation” R : D -t U is

of t h e corresponding class defined on 4). Namely, if

a conformal isomorphism (which exists in view o f the Riemann mapping

C belongs t o N + ( U ) if t h e function f o R E N + ( D ) . It turns out that a function f in N + ( U ) has nontangentiai boundary values almost everywhere on dU and thus the boundary function f : dU -+ C is defined. In this case, inequality (1.8.1)

theorem), we say that the analytic function

is replaced by

f : U

-t

42

Classical interpolation theorems

au where dH,(y) is the so-called harmonic measure connected with the Poisson kernel P, by the following formula:

(1.8.3)

J

f(r)dHR(w)(r>=

J

(f

0

~ > ( e " > ~ w ( ( P > d m ( (.P P )

an

au

C . We now have everything for t h e discussion of the corresponding interpolation theorem. T o formulate this theorem, we assume that (a)

(Y

:

[0, m), ,f3 :

-t

Ib

.--)

[0,1] are bounded harmonic functions.

In the sequel we use the notation

(1.8.4) p

:= l / a

,

q :=

1/p

(b) An admissible family of operators {T,, z E

4>}

is given. This means

that T, : S o ( d p , C ) + M ( d v ; &I) and that, for any simple functions

f and g , the function

is continuous in

Ib and belongs t o N + ( D ) .

Under these conditions, the following theorem is valid. Theorem 1.8.8. Let us denote by M ( z ) the norm of T, considered as an operator from into

L+). Then for any point

(1.8.5)

log M ( z ) 5

J

z

E D,

log M(e"+')P,(cp)dm.

am

Proof.For a

and ,f3, we construct the functions a and b analytic in

such that

(1.8.6) F k a = a ,

Fteb=p.

D

and

Generdiza tions

43

It is well known that such functions are determined uniquely. For a given zo E

XI, let f

and g be arbitrary simple functions satisfying

the following conditions: (1.8.7)

J

lflp(-)cip

J

=

Iglq’(zO)dv =

1.

Here, as usual, q’ := q / ( q - 1) is the conjugate exponent.

We put sgn h := h/lhl (assuming that 0 / 0 := 0) and assume that :=

fz

sgn f

(f(4Z)P(.o)

,

gz :=

(gl(’-P(.))e’(zo)

sgng

.

In view of this definition and (1.8.6), (1.8.4) and (1.8.7) we have

Let us consider the function

F ( z ) := / ( T , f z ) , , d v ,

z E XI

,

N + ( D ) . Since f and g are simple functions, F

and show that it belongs t o

is a linear combination of the functions

z

+

where X , p E

J (Tz X A

ewz)+L44

R

and A ,

B

X B dv

7

have finite measure. Therefore, in view of

condition (b) of the theorem and Remark 1.8.7, it is sufficient t o show that exp(Aa+pb) E

N + ( D ) . But according to condition (a) and the mean value

theorem for harmonic functions, we have

J

(log le~n+pbl>(re+)dp =

am

J am

+

( ~ apP)(re+)drn =

= X40)

+ pP(0)

7

+ pb) E N + ( D ) . N + ( D ) , and in view o f (1.8.1),

which proves that exp(Xa Thus, F E (1.8.10)

log

IF(ZO)I

I

J

aD

log ~ ~ ( c p ) ~ ~ ~ .~ ( c p ) d m

44

Classical interpolation theorems

Further, it follows from (1.8.8) that (1.8.11)

F(z0) =

1

(T'f)gdv .

Besides, in view of condition (b), the function z

4

exp(Xa+~b)1(T,~a)~8dv

has the limit as JzI t 1, so that

P(cp) for all cp E

:= lim F(re'V) = T+1

J

(Teiqfe,q)g,iqdY

alD.

Consequently, taking into account (1.8.9),we have for z :=

eiV

lF(cp)lI M(e"P) . Combining this inequality with (1.8.10) and (1.8.11), we get

log

IJ( T z o f ) g d v lL J

~ ( e " ~ ) ~ , , ( c p ).d m

aD It remains for us t o take the upper bound for all

f and g satisfying conditions

(1.8.7). 0

In order t o verify that Theorem 1.8.8 actually contains the Riesz-Thorin theorem as a special case, we consider the following corollary. Let

I? be vectors in BZ:,

Suppose that

5,
such that

fik

= 1.

Further, we put

Finally, suppose that

T

:

So(&; C)t M ( d p ; C)is a linear operator

and that its norm as transform from L P i ( d p ;C) t o L , , ( d v ; C) does not exceed valid.

hfk,

k = 1,2, ..., n. Under these

conditions, the following corollary is

Generalizations

45

Corollary 1.8.9. The operator T is extended by continuity t o a linear operator acting from

L p ( ainto Lq(qand its norm does not exeed

n;=, M.:'

Proof.We break dID into n arcs Ak so that

We define the harmonic functions values on

Q

and /3 by prescribing their boundary

dlD. Namely, we put a(.)

:= 1/pk, P(Z) := l / q k

for Z E Ak

, k = 1,..., 12 .

Then, by the mean value theorem for harmonic functions, we have

Similarly, p(0) =

1 T.

q p

In view of (1.8. ) and the definition of M , it follows that

(1.8.12)

M(0) = ( I T ( I L ~ ~, + L ~ ~ ~

Further, by the choice of the boundary values for a and

M ( z ) = ~ ~ T ~ ~ L P k ~ L<Mk -' ? k

p, we

have

f o r z E A k , k = l , ...,n .

Consequently, the application o f inequality (1.8.5) with z = 0 gives the following estimate for norm (1.8.12):

J

log M(O) 5

5

M ( t ~ ' ~ ) d5m

Ij)k

log

Mk

.

k=l

am This proves the above statement. 0

Similarly, for the domain

U

considered above, using inequality (1.8.2) we

obtain in an analogous situation t h e following estimate:

(1.8.13)

log M(z)

5

/ log M ( r ) d H , ( r ) ,

au

Z'E

U

.

46

Classical interpolation theorems

It should be recalled that the harmonic measure dH, is calculated w i t h the help of equality (1.8.3). Concluding the section, we note that using (1.8.3), we can extend the concept of harmonic measure t o unbounded domains w i t h a sufficiently nice boundary as well. Then inequality (1.8.2) is satisfied under certain restricti-

F in N + ( U ) a t infinity. Thus, if U coincides with the strip S := { z E C, 0 < Fkz < l}, it is sufficient that

ons imposed on the growth o f function for a certain a

(1.8.14)


sup e-alhPI log IF(z)I

< 00 .

zES

Using this fact, we immediately arrive a t the following theorem.

T, : So(&; C) -+ M ( d v ; C ) ,z E S be a family of linear operators that for any simple functions f and g there exists a number a =

Let such

a ( f , g ) < 7r for which the function

F belongs t o

: z

--t

/(T,f)gdv

N + ( S ) ,is continuous in S and satisfies condition (1.8.14)

In these assumptions, the following theorem holds. Theorem 1.8.10 (Stein). Suppose that for certain functions Mk :

R

+

00,

k =0,l.

and any simple f's, the

following inequalities are satisfied:

where y E

lR and 1 2 pk 5

00,

12

I&

5

If, in addition, the following inequalities are fulfilled for a certain b sup e-bly' log M k ( y ) < 00

,

k = 0,l ,

Y€m then there exists a function

M : ( 0 , l ) + lR+ such

that


Generalizations for all simple functions

47

f.

0

Remark 1.8.11.

Of course, we can estimate M by Mk with the help of the harmonic measure of the domain S.

Classical interpolation theorems

48 1.9. The Soaces Lnn

A new interpolation theorem was proved by Marcinkiewicz in 1939. The spaces Lqiare replaced in the formulation o f this theorem by wider spaces L:, . Nevertheless, the statement o f this theorem coincides w i t h the statement o f Theorem 1.5.1 to within a constant.

It will be shown in Sect. 1.10 t h a t such

a generalization is meaningful. Here, we will consider the properties of the family of spaces

L,, (which contains the spaces L, as well as the spaces Lf)

that will be required in the sequel. This family o f spaces plays an important role in modern generalizations o f the Marcinkiewicz theorem as well.

We shall start with the definition o f Lf, ( " w e a k " L,). Definition

1.9.1.

The space L z ( d p ) consists o f all p-measurable functions

Ilfll;

(1.9.1)

:=

( sup

t P p { z ; If(z)l

f for

which

> t } y p< 00 .

t>O

Here 0

< p < oz). For p

:= co,we assume that Lf, := L,.

0

In the study of the properties of function

(1.9.2)

d(f) defined d(f ; t )

:= p { z ;

If(.)[

and the decreasing r e a r r a n g e m e n t

(1.9.3)

f'(t)

L;, it is convenient t o use the distribution

by the equality

:= inf { s

>t} ,

t >0

,

f*,puttingI2

> 0 ; d(f; s ) 5 t > .

If d(f) is a bijection o f the set (0, +m), then f' is obviously the inverse function t o

d(f).

In the general case, this is not true due t o the presence

of constancy intervals and discontinuities. The following properties o f the functionals introduced above follows almost directly f r o m their definition.

''By convention, info = +m. Thus, f' and d(f) may assume the value of +co

The spaces L,

49

Proposition 1.9.2. Let @ be one of the functionals f + d(f) or f + f’. Then the following statements are valid.

0 maps M ( d p ) into the cone of nonincreasing and con).. into [0, + .]. tinuous t o the right functions that map (0, +

(a) The functional

(b) The mapping @ is monotonic (i.e. 0(f)5 @(g) for

If1 5 Igl).

(c) If the functions

f and g are eqaimeasurabZe13,then f’ = g*.

(d) If f*(t), d ( f , t )

< co for t > 0, then

(1.9.4)

( 4 f ) 0 f*)(t)2 t

(f’

7

0

d(f))(t) I t .

Henceforth, we shall also need the “approximation” definition of the functionals under consideration.

For this purpose, we shall use the following

genera I definition. let

Let X o and X1 be linear metric spaces,14 embedded into M ( d p ) , and llzll, be the distance from 0 t o the element z in Xi. Suppose that

&(Xi) :=

{ z 6 Xi ; ((zl(i5

t } is a

closed ball of this space.

Definition 1.9.3.

+ X I by

The E-functional o f the couple ( X 0 , X l ) is defined on the set X O

the formula

Here t

> 0, and the usual convention

inf 8 = +oo is used.

0

Proposition 1.9.4.

The following equalities hold: 13The functions g E M ( d p ) and h E M ( d v ) are equimeasurable if p ( z ; Ig(z)l > t ) = lh(z)I > t ) for all t > 0. 14This means that the metric d, which is invariant to translations is defined on X i .

”(2;

Classical interpolation theorems

50

Proof. Suppose t h a t Ilgtl103 5

t

f x ~ where ~ , At

gt :=

and in view of t h e definition of

:=

(2;

lf(z)I <_ t } . Then

Lo, see (1.2.10),

we have

If for some g we would have

then

f

= g on a certain set A such that

P(Q\A) = Ilf - 9110 < P(fi\At) =

4 f ;t ) .

Then, contradicting the second inequality in (1.9.7), we would have 19) > t on the set A\A, with a positive measure. Similarly, suppose that h, :=

f x ~where ~ , At

:= {z ; lf(z)1 > f*(t)}.

Taking into account the inequalities (1.9.7) we then get

Ilf

- htllO3

I f'(t)

and

llhtllo = d(f ; f * ( t ) )I t

.

If for some h we would have (1.9.8)

[If-

hllm :=

then supp h 3 { z ; lf(z)I

t 2 llhllo 2 P(.;

T

< f * ( t ) and

> T},

llhllo 5 t

and hence

If(.)l >

=

4f; .

In view of (1.9.3), it follows, however, from this that diction t o (1.9.8). 0 Corollary 1.9.5.

.

In the notation of Proposition 1.9.2,

@(f

+ g ; + t ) I @(f ; .) + @(g; .)

T

L f'(t),

in contra-

The spaces L,

51

Proof. Since Bt(Li)

+ Bs(Li) c Bt+s(Li)

i = 0700

9

in view of (1.9.5) and (1.9.6) we have

@(f

+ 9 ;s + t ) =

Ilf

inf

+g -

hIll/i

I

hEBt+,(h)

Let us show that the (quasi) norms terms of the rearrangement

Ilfllt;

and

0


Ilfll,

can be expressed in

f*.

Proposition 1.9.6. The following inequalities hold:

(1.9.9)

Ilfll,

,

=

Proof. In view of (1.9.3)

I 0 0

.

we have

sup f*(t) = f*(O) = inf {s

> 0 ; d(f; s) = 0)

=

llfllco ,

t>O

and (1.9.9) is proved for p := Suppose now t h a t p

s d ( f ; s)l/P

< 00

CQ.

and s

> 0 are given. Let

us verify t h a t

5 sup t’l”f’(t) . t

>o

If s 2 f * ( O ) , then the left-hand side is zero. Otherwise, there exists a t s > 0 for which

Classical interpolation theorems

52 f*(t. - 0) 2 s 2 f*(t.) Then for any 17

> 0 we

sd(f;

S)1/P

Since for a given

E

.

have, in view of (1.9.4),

5 f * ( t s - q ) d ( f ; f'(ts))llP

> 0 and

a sufficiently small 17

- 17) + E 5

does not exceed (t. - q ) ' / P f * ( t .

5 t y f * ( t ,- 7 ) ) .

> 0,

the right-hand side

SUP^,^ t ' l P f * ( t ) + E , t h e required

inequality is proved. Taking the upper bound with respect t o s

Ilfll; I

SUP

> 0, we get

t'l'f*(t) .

t>O

The opposite inequality is proved similarly with the help of the second equality in (1.9.4). In order t o prove t h e second equality in (1.9.9) (for p by [ f ]the ~ truncation of the function f on the level N :

if

If(.)l

< m), we

denote

5N

in the opposite case

.

Then [f]; = [ f * ] ~and , in view of Proposition 1.9.2.(c) we have

d([flN) = W

* I N )

.

Since in the definition o f t h e Lebesgue integral of a bounded nonnegative function only its distribution function is used,15 we can write 00

J

J

V*IW .

I [ ~ I N I=P ~ ~ 0

The spaces L,

53

It only remains for us t o pass t o t h e limit as N

--f

00.

0

Corollary 1.9.7.

L, c LI and, if the measure space (R, C , d p ) does not consist of number of atoms, then

Proof.Since f* is a

a finite

L, # L;. Here 0 < p < 00.

nonincreasing function, we have

whence we obtain the inequality

and the required embedding.

Let us now verify that

LI # L,.

Indeed, in view of the condition on

a,

there exists a countable sequence { A n } n Econsisting ~ of pairwise disjoint sets with positive measure. Without loss of generality, we can assume that

p ( A , ) 2 p(A,+1), n E IV. Let us consider two cases. (a) The series

CnE=

(b) The series

CnEmp ( A , )

p(An) is divergent. is convergent.

In the former case, we denote (1.9.10)

~j

:= p(A1)

+ ... + p ( A n )

We put

f :=

c

UyXa,

jdV

Then it can be easily seen that

where uo := 0 and hence

.

Classical interpolation theorems

54

On the other hand,

Indeed, if

Sn is a

partial sum o f t h e series in the right-hand side, then taking

into account (1.9.10), we have

2

- S,

S,,,,

1 (u,,+~ - u,,)= 1- -+ 1 6 ,

bn+p

for p

in view of (a). Thus, we have proved (1.9.11). So, in the case (a) In the case

(b), it is sufficient

f

c

:=

--+

00

Un+p

L,

# Lz.

t o put

PjllPXA,

7

j=N

where

Then

and

f * ( t )= 0, t >_ p1, so t h a t

Ilf 1;

= 1*

A t the same time, in view of (b) and the choice of pj we have

which is proved in the same way as (1.9.11).

Definition 1.9.8.

The set L,,(dp), 0

f for which

< p , q 5 00 for p < 00

consists of p-measurable functions

55

The spaces L,

For q = 00, the right-hand side is replaced by SUP^,^ t””f*(t). Finally, when p = 00, we assume that L,, := L , for all q. 0

Thus, in view of Proposition 1.9.6,

L, = L, and L,, = L;.

Theorem 1.9.9.

L,, is a quasi-Banach space16 continuously embedded in the space of all

(a)

p-measurable functions.

(b) The quasinorm (c)

11 . llpq

L,, possesses the

(d) For q

< 00,

is monotonic and rearrangement-invariant.”

Fatou property.

the quasinorm

11 . IJw

is absolutely continuous, and the set

So is dense in L,,. (e) For q1

5 qz, the following continuous

(1.9.12)

L,,,

r-t

embedding is valid:

LPqa.

Proof. (a) Among the properties of (quasi) norm we only need t o prove the inequa-

lity

Since it follows from Corollary 1.9.5 that

161t becomes a Banach space for 1 5 q 5 p (see Remark 1.9.15). 171n other words, equimeasurable functions have equal quasinorm.

Classical interpolation theorems

56

multiplying this inequality by t'fP and then applying t o both sides the L,-(quasi) norm (in the measure d t l t ) , we obtain the required statement.

The completeness of L, and the embedding L,

L)

M are special cases

of the general fact concerning approximation spaces considerd in Chapter

4 (see Sec. 4.2). Of course, these statements can also be easily proved directly.

(b) This statement immediately follows from Proposition 1.9.2. (c) Suppose that

{ f n } n E ~ is

contained in the unit ball

ges in measure t o the function

any set f i k with p(fik)

< 00

f. We

we have for

t >0

In view of (1.9.4) it follows from this that for any

(1.9.13)

((f - fn)xn,)*(t)

+0

According t o Corollary 1.9.5, for any

as n + 00 E

B(L,,) and converf E B(L,,). For

must verify that

t >0

.

>0

(fxn,)*(t) I ((f - fn)xn,)*(Et)+ ~ ( ( 1 -EP) . We multiply this inequality by t'/P and take the L,-(quasi) norm for the interval ( a , b ) , where a > 0 and b < 00. Then, taking into account

(1.3.3) and the monotonicity of the rearrangement we obtain

where ii := min(1,q). Making n tend t o infinity, using (1.9.13) and the fact that

llfnllpq

I 1, we get

The spaces L,

57

Passing t o the limit as

E +0

and ( a , b ) + R+, we obtain

If {a,} is an increasing sequence of finite-measure sets which converges to

R,

then

(fxn,)’

increases monotonically and converges t o f * point-

wise. Therefore, in accordance with the B. Levy theorem, we obtain for

k

+ 00 from (1.9.14) the inequality

llfll,, 5 1.

(d) The density of the set of simple functions in L,, for q < 00 (and even of the wider set Lon L,) is a consequence of a more general statement concerning approximation spaces (see Sec. 4.2).

Let us verify the absolute continuity of the quasinorm

11 . Ilp,

for q

< 00.

Let {Rk}kEmbe a decreasing sequence of p-measurable sets with an empty intersection. Then the sequence

{(fxn,)’} decreases monoto-

nously and converges p-almost everywhere t o zero. Each element of this sequence is majorized by f* so that passing t o the limit in the integrand, we obtain

(e) We shall need the following

Lemma 1.9.10.

If the function g :

R++ R+is nonincreasing, then for any a E [l,m]

the following inequality holds:

Proof. It is sufFicient t o verify this inequality for step functions g of the C; gkX(ak-l,ak), where gk > 0 is a nonincreasing sequence and 0 = a0 < al c ... < an. In this case, we can write the inequality in the

form

form

58

Classical interpolation theorems

where we put 19 := 1/a

5 1 and bk

:= gr.

We shall prove (1.9.15) by induction by n. For n = 1, t h e left-hand side is equal to bl(al - uo) = blal as well as the right-hand side. Suppose now t h a t (1.9.15) is valid for any bk, above conditions. For a,+l

Then for n

:= b.,

15 k

5 n, that

satisfy the

we put

+ 1 terms, inequality (1.9.15)

Since cp is concave on and for bn+l

> a,,

ak,

can be written as follows:

IR+,it is sufficient t o verify (1.9.16)

for b,+l

:= 0

But in these cases inequality (1.9.16) follows from

(1.9.15). 0

Suppose now that q1 5 q2 [f*(zP”Jz)]ql and

(Y

< 00.

Taking in Lemma 1.9.10 g ( z ) :=

:= q z / g l , we obtain

After substitutions in the integrand, we obtain (1.9.17)

P Ilf llpm I(-) 92

1/92

( 91P

Ilf IlPIl .

The spaces L ,

59

Consequently, the embedding (1.9.16)is proved for the limit, we can obtain the case q2 = 00.

Remark

92

< 00.

Passing t o

1.9.11.

The example of the function f

:= X A , p ( A )

< 00,

shows that inequality

(1.9.17)is exact. Let us finally consider the question of normability of the space Lpq,Here we shall limit ourselves t o the measurable space ( a , C , d p ) containing no atoms. Theorem (a) For

1.9.12.

1 < p 5 00 and 1 5 q 5

the topology of t h e space L, is defined

00,

by a norm which is equivalent t o the initial quasinorm. (b) In the remaining case, the space

L,, is not normable.

Proof. (a) Let us consider an operator

f

t

f** defined by the following formula:

Let us verify the validity of

l t (1.9.19) f**(t) = 7 f*(s)ds

.

0

Since the expression is obvious for t 2 p(C!), we assume that t

It follows from the definition of

< p(R).

p,see (1.9.3),that for any n E N

60

Classical interpolation theorems Passing here t o the limit as n -+

00

and using the fact that t h e limit

in measure o f an increasing sequence o f sets is equal t o the measure of their union, while the limit in measure o f a decreasing sequence of sets is equal t o the measure o f their intersection, we obtain

Since

R

does not contain atoms, there exists a set

At o f p- measure t

which contains the smallest, and is contained in the largest of the sets

(1.9.20). If g := ~ x A then ~ , in view o f the choice of A t , g* = f*x(o,*), and, according to (1.9.9), we have

appearing in

t

Hence it follows that

Thus, equality (1.9.19) is true. Let us now define t h e functional

and show that this formula defines the norm for 1 5 p

q

5

00.

<

00

and 1 5

Since the nondegeneracy and positive homogenity are obvious,

it only remains for us t o verify the triangle inequality. However, it follows immediately from (1.9.18) that

The spaces Lp,

61

(f

+ g)** 5 f**+ g** .

So it remains t o use the triangle inequality for the &-norm. Finally, we establish the equivalence of norm (1.9.21)and the initial quasinorrn for 1< p

< 00

and 1 5 q

5 00.

Since f**

2 f * by (1.9.19),

we have

In order t o prove the opposite inequality, we need an auxiliary statement which will be used in other cases.

We denote by D the cone o f nonnegative functions g that satisfy the inequality

(1.9.22)

sup

g(s)

5 yg(ct) ,

0 < t < 00

t/25s5t

for some constants y

2 1 and c E [0,1].Then we

define for X E

R3 the

operators

Lemma 1.9.13.

If g E D,0 < q

5 00

and X

The same is also true for

< 0, then

H i . Here, q*

Proof. Suppose that cp(t)

:=

:= t X SX

and the condition X

< 0, we

t/2

have

min(1,q).

fi. Then in view o f (1.9.22) s

Classical interpolation theorems

62

(1.9.25)

2 4 -1

cp(t) I 7g(ct) , -A

0 < t < 00

.

It follows from t h e identity

c m

(H?g)(t) =

2kX$0(2-")

k=O

and the inequalities (1.3.3) and (1.9.25) that for X

< 0,

The second inequality is proved similarly. 0

Let us now apply (1.9.24) with X = l / p - 1, 1 < p

< 00

and 1 5 q

5 oa

t o the function g ( t ) := t'/'f*(t). In view of the monotonicity of

f*,

condition (1.9.22) is satisfied for y = 1 and c = 1 / 2 . Consequently, (1.9.24) gives in this case

Thus, the equivalence of the norm (1.9.25) and the initial quasinorm is proved.

(b) For proving this part, we shall use the following

63

The spaces Lp, Theorem (Kolmogorow).

In order that the topology in a Hausdorff topological vector space V be defined by a certain norm, it is necessary and sufficient that V has a bounded convex neighbourhood of zero. U

As applied t o the situation we are dealing with, it follows from this state-

Lpqis not normable if t h e convex envelope of the unit ball B(LPq) is unbounded. Here, min(p,q) < 1, or p = 1 and 0 < q 5 00. In order ment that

t o avoid some technical details, we shall limit ourselves only t o a sufficiently typical case of the space Lpm(B,dt),0

< p 5 1, and

suggest that the

reader considers t h e general case as an exercise. For a given n E

N ,we define t h e set {f,}:’-,

by putting

Since

and the rearrangement o f each function on the right-hand side is l / t P , then

f;*(t) 5 l/tP as well. Therefore,

(fi)c B(~PC0)Further, let us consider the function g in the convex envelope convB(L,,),

equal t o

& Cz-,

f;. Since

we have

,

n

,

However, the quasinorm on the right-hand side is equal to (2n)’lp so that

Ilsllpa, 2 7n”p-l

2(i + 1

;=o

1)

+oo

forn+oo(p
64

Classical interpolation theorems

Thus the set convB(L,,)

is unbounded.

0

Remark 1.9.14.

If (Q, C,p ) contains atoms, t h e space L,,(a, C, d p ) decomposes into the direct sum

Lpq(!&) @ &(ad),

union of atoms of

fl and Q,

where the “discrete” component

:=

ad

is the

n\f&. Consequently, for Theorem 1.9.12

to be valid in the general case, it is sufficient tha t it is valid for a d . It can be verified that the statement of Theorem 9.1.12 (a) is true. For this purpose, in defining

f**

we must take in (1.9.18) the upper bound for all

A with

p ( A ) 2 t. As t o the statement of Theorem 1.9.12 (b), it is not always true. It can be easily verified, however, that this statement is valid for the space of sequences 1, defined by the quasinorm

replacing the right-hand side by supN’/Pf,’ for q :=

the rearrangement

{ Ifn]}

Since l o g ( l + l )

-

00.

Here

{f;} denotes

in decreasing order.

;,la

the space,Z

coincides w i t h

dp is the “counting” measure ( p ( { n } ) = 1for n E

L , , ( N , d p ) , where

N). and the quasinorms

of these spaces are equivalent. Remark 1.9.15. Let us verify that for 1 _< q

5p

the quantity

11 . I p q

is a norm so that

the utilization of the norm (1.9.21) is superfluous in this case. This can be established i n the easiest way for the spaces

Zpq. In the general case, similar

arguments are applied to simple functions and then a passage t o the limit is made. Thus, we must verify that the triangle inequality holds for (1.9.26) when

1 5 q 5 p . For this we shall use the simple fact t h a t for ak,bk 2 0, ‘“The notation f

-

g means that for a certain constant c > 0 we have 1/c

5 f/g 5 c.

The spaces L, (1.9.27)

65

akbk

5

c

.

In the case of two addends, the validity of this inequality is obvious. If the number of addends is n

> 2, we can, while

proving (1.9.27), assume without

loss of generality that { b k } is nonincreasing ( b k =

i <j

b z ) . When a; < aj for

we change their places on the left-hand side of (1.9.27). Making use of

the validity of (1.9.27) for two addends, we note that the left-hand side does not decrease in this case. Consequently, by consecutive transpositions of aj,

C sib;.

we can transform the left-hand side o f (1.9.27) t o the larger sum Passing then t o the limit as n

--t 00,

we obtain the required statement.

For f , g E lpq, we have

Ilf + sllw= =

(c (c

+g d * I y = [r(n>-e~ f n+ 'Iq In-Wn

9n11q)

1 1 for a := - - - ( 5 0); here r :

P

RV

+ RV is a certain bijection. Then in

9

view of Minkowski's inequality, we have (1.9.28)

(c

Ilf + 911, L UI.(.[

Applying (1.9.27) with a, :=

Ifnl]q)lIq+ ~ ( n ) - and ~ q b,

(c[+la

:=

lfnlq

on the right-hand side, and considering t h a t a: =

b: = (f:)q,

q

IgJl

)

119

.

t o the first summand

n-aq

(since a

5 0)

and

we get

(C(r(n>-aIfnl)q)llq

2

(C I ~ - ~ . C P=) "I~I ~ I I ~ *

Proceeding in the same way with the second term of (1.9.28),we obtain from this inequality

Classicd interpolation theorems

66

1.10.The Marcinkiewicz Theorem A. The results of this section are valid for the scalar field R as well as for

C.

1.10.1(Marcinkiewicz). Let T : (Lpo+ L p l ) ( d p ) -+ M ( d v ) be a linear operator such t h a t T(L,) c L,,- and such that the norm of T on Lp, does not exceed Mi, i = 0 , l . If in this case 1 I p , , q; I 00, qo # q1 and Theorem

(1.10.1) P(8) I q(6) T(Lp(qc ) L,(q and the norm of T I L ~does ~ ) not 1-SM9 exceed Ka(pi, qi)Mo 1.

for some 29 E (0, I), then

Proof. In order t o formulate Marcinkiewicz's idea in a clear form, we confine ourselves t o t h e case 1 I p , 5 pl 5 00, pi = qi and t o the measure dv coinciding with Lebesgue measure on R,. It then follows from the assumptions of t h e theorem that

(1.10.2) ( T f ) * ( t I ) M;t-'lP1IlfllP, Let us decompose

f

f=fo+f1

, i = 0,l

as a sum o f two components

7

fo := f x A r , fl := f - fo and At := { z ; If(z)l > f * ( t ) } . In view of Corollary 1.9.5and the linearity of T ,we then have where

(1.10.3) ( T f ) * ( tI ) (Tfo)*(t/2)+ (Tfi)*(t/2). Hence, taking into account equation

(1.10.2)and Proposition 1.9.6, we ob-

tain n

(1.10.4)

IITfllP(d)

5

C

1

21'p'M; t - l l p

i=OJ

Further, in view of the definition of

IIfiIIp.Ilp(d) .

f, and Proposition 1.9.6,we have

The Marcinkiewicz Theorem

67

We substitute this expression into the right-hand side of (1.10.4). Then, putting

and defining functions gi by the equality

g;(tj := [t’lP‘”’f’(tj]

,

2

= 0,l

,

we can rewrite inequality (1.10.4)in the following form:

(1.10.5) llTfllp(t9)I m u (21’p’Mi) kO.1

C

IIHi,giIILtiF)

i=O,1

It should be recalled t h a t the operators H i were defined in Lemma 1.9.13. Since po

< pl,

we have

Xi

< 0, i

= 0,l and hence Lemma 1.9.13 can be

applied t o the estimate of the right-hand side of (1.10.5). By using this lemma and the definitions of gi and ri, we obtain from (1.10.5)

I I T ~ II I ~Ks(pi)(ma (~) Mi)

C

IIgiIIL??)

-

i=O,l

= 2xS(pi)( m u M‘>11f11p(O),p(t9). i=O,1

This, together with Theorem 1.9.9,leads t o t h e inequality

(1.10.6)

llTfllp(t9)

I PI(J(Pi)( max

Mi)

IIfIIp(t9)

.

kO.1

Let us apply this inequality t o the operator TA := DAT, where Q, is the dilatation operator:

It can be easily verified that (DAg)*= Dx(g*), and hence

(1.10.7)

IlD)xgllpp =

llgllw

.

Classical interpolation theorems

68

Consequently, we obtain the following estimates from (1.10.2) for

Applying (1.10.6) t o

Tx:

TA and taking into account (1.10.7) and (1.10.8), we

get

IITfllp(s)= I l r r X f l l p ( S )

X-'/pnM. :) Ilf

I Ks(Pi)(

IlP(9)

i=OJ

Multiplying both sides by

A1/P(')

and taking the lower bound in A, we obtain

t h e required estimate

Remark 1.10.2. (a) Let us assume that the operator in Theorem 1.10.1 is only quasiadditive, 1.e.

+

L,, L,, and a certain constant y 2 1. Since inequality (1.10.3) is valid in this case also (with the constant y in the right-hand

for all f , g E

side), t h e proof is also valid for quasiadditive operators.

(b) As 8

+

0 or 1, the constant Ks(p,,qi) -+ 00. With the help of the

general theory which we shall develop below, it is possible t o obtain the

Ks at infinity (as a function o f 8). The exact value K&;, q;), however, is not known.

order o f growth of of

(c) Although the condition po

#

pl plays an important role in t h e above

proof, actually only the condition qo (d) The restriction 1I p i , qi

5 00

# q1 is essential.

can be removed; this also is obvious from

the proof of the special case considered here.

The Maxcinkiewicz Theorem

69

We shall show that Marcinkiewicz's theorem is not valid without condition

(1.10.1). For this purpose, l e t us consider

Examde

1.10.3.

We choose p i , q; E [l;w ) such that

1 1 (1.10.10) - - - = a , Qi Pi where a

> 0.

i=O,l,

Then for any 29

E (0, l), we have

1 1 -a>o, (1.10.11) -- -Q(4 P(29) and thus the condition (1.10.1)is not satisfied in this case. Consider the linear operator T : L'""(E&+)+ M ( R + ) ,defined by the formula a

( T f ) ( t ) := t-"-'

f(s)ds

, t E R+.

0

It follows from the definition o f f * that

J

I(Tf)(t)l I t-"-'

t

IfX(0,t)I

ds

5 t-'*-l

R t

J

f*(s)ds .

0

Hence, in view of the monotonic decrease of the right-hand side, we get t

( T f ) * ( t5 ) t-=-'

J

f'(s)ds = t-*f**(t).

0

From the equivalence of the norm

llfllA,

see

(1.9.21),and the quasinorm

~ ~ f ~equation ~ p q , (1.10.10)and the inclusion (1.9.12),we obtain

llTfllq,mI

SUP

t-a+*'qtf**(t) = llfll~,, 5

t>O

I~ Thus, the operator

Ilfllp,m 5 %pi)

b i )

Ilfllp,p,

= ?(Pi) IlfllP,

.

T satisfies the conditions o f Theorem 1.10.1 with Mi :=

? ( p i ) . Nevertheless, we will prove that T(L,(#))is not contained in Lq(s)for any value of

i3 E (0,l). For this purpose, we note t h a t if f

and nonincreasing, the function

Tf will

is nonnegative

have the same properties. Hence,

70

Classical interpolation theorems

.

f(s)ds 2 t-”f(t)

(Tf)*(t) = t-a-’ 0

Thus, taking into account such

(1.10.12),(1.10.11)and (1.9.9),we obtain for

f

2 Since p ( 6 )

&

1

1/9(’)

Itl’p(’)f(t)p

t dt

> g(I9), then by putting the function f

= llfltP(’),d’)

.

equal tot-’/P(’)llog-P(l/t)

in a small neighbourhood of zero and equal t o zero outside of it, we obtain for l / P ( d ) < P

< l/q(G)

Together with the previous inequality, this means that T(Lp(q)! !$

Corollarv

L,(q.

1.10.4.

If T : (Lp0+ Lpl)(dp)3 M ( d u ) is a quasiadditive operator, such that

for all f E Lp,, i =

0,1, then the following inequality is valid for each function f E Lp(8) under the restrictions on p i , q i , 19 similar t o those in Theorem 1.10.1: Il~(f>IIq(19) I K s ( p i ; gi)Mi-’Mf

IIfIIp(s) .

The proof of this corollary follows from the inequality (see Proposition

11 . llq,m 5 1) . ]Ip,

1.9.6)and Remark 1.10.2(a).

D

B. None o f the previous ways of proving M. Riesz’s theorem leads t o the generalization contained in Corollary 1.10.4. This circumstance, together

The Marcinkiewicz Theorem

71

with the fact that condition (1.10.1) cannot be removed, indicates t h a t in spite o f the similarity i n appearance between Marcinkiewicz’s theorem and

M. Riesz’s theorem, the two are different in principle. A comparison o f the proofs confirms the validity of this assumption. Indeed, the key role in t h e proofs o f the theorems o f M. Riesz and Thorin is played by multiplicative inequalities, while the proof of Marcinkiewicz’s theorem is based on the possibility o f representing the elements o f Lp(8)as a sum of components from

Lpi,

i = 0 , l . This difference between the classical interpolation theorems has led t o two different methods in the general theory for constructing interpolation spaces, viz. the real method (derived from Marcinkiewicz’s theorem) and the complex method (derived from the M. Riesz-Thorin theorem). The first ideas about the complex method are given in Theorem 1.8.8. The modern generalization of Marcinkiewicz’s theorem given below demonstrates some basic aspects of the real method o f interpolation. Theorem 1.10.5.

Let

:

+

(LPoro

L p l r l ) ( d p ) -+

M ( d v ) be a quasiadditive operator, such

that

(1.10.131 IIT(f>lls,s, 5 Mi

Ilfllp,r,

for all f E LP,‘,,i = 0 , l . Further, suppose that

(1.10.14) 0 < p i , ~ i , ~ i ,500 s i 9 PO # P I

7

Then the following inequality is valid for any

(1.10.15)

llT(f)IIq(8)r

# QI .

QO T

E

I YKBM:’-’M,S IIfIIp(8)r

(0, +00] and 19 E (0, l) : ;

Here, y is the constant i n (1.10.9) and K8 is independent o f f and T . The proof, which will be outlined here, develops the main idea of Marcinkiewicz, which involves the construction o f an “intermediate” space Lp(8) from the sum fo

+ fl of functions

fi

belonging to the “boundary” spaces

L;; . For the general case of a couple of Banach spaces, such an approach was first suggested by E. Gagliardo (1959) on the basis of the concept which is

72

Classical interpolation theorems

now called “Gagliardo’s diagram”. An equivalent, but a lot more flexible and convenient method for applications was later proposed by J. Peetre in 1963. This approach is based on the simple but extraordinarily extensive concept of the A’-functional o f a couple of linear metric spaces. We shall consider i t s definition for the particular case considered in the present context.

X; L-) M(d,u) be a linear metric space with an invariant metric and let IIXII; be the distance from 2 t o zero in Xi, i = 0 , l . Let

Definition 1.10.6.

The K-functional of the couple (Xo,X,) is the transformation from the sum

Xo+X1 into the cone of nonnegatie concave functions defined on R+, given by the formula (1.10.16)

K ( t ; 2 ; X o ; X , ) :=

inf

{llzollo+t

llxllll} ,

t >0.

o=zo+x1

The K-functional can be used t o determine the family of linear metric

(X0,X1)Bq,where 0 < 6 < 1 and 0 < XO+ X I , we put spaces

with the usual modification for q := (1.10.18)

00,

q

5

00.

Indeed, for

2

E

and define the linear metric space

(X0,Xl)Sq := {. E xo + Xl ; 1141(Xo,Xl)eq < I.

.

We shall show t h a t an analog of M. Riesz’s theorem is valid for t h e family of spaces introduced. For this purpose, we consider a couple (Yo,Y,) analogous t o ( X o , X , ) , where yi

L)

M ( d u ) , i = 0,1,and assume that the

metric in yi is monotonic. Thus, (1.10.19)

14 I IYI * 1141Y, L IlYllY, .

Next, suppose that such that

T

:

Xo + X1

+

Yo+ Yl is a quasi-additive operator,

73

The Maxcinkiewicz Theorem

ProDosition

1.10.7.

where 7 is the constant in

(1.10.9). In particular, 7 = 1 for a linear or

sublinear operator.

Proof.We consider an arbitrary representation (1.10.21) z = 20 + 2 1 ; of a given element x in Xo

Then in view of

zi E xi,

+ X1 and put

(1.10.19)and (1.10.20),we obtain

Y;, i = 0,l. Also, since yo (1.10.16)

Thus, y; E view of

i = 0,l ,

K ( t ; T ( z ); Y0,K) I

+

llVOllY0

= T(z)by definition, we get in

+ t Ilylllyl

I

Taking the lower bound in this inequality for all representations we obtain

Mi K ( t ; T(z) ; Yo,Yi) 5 7MoK( ~0 t ; z ; Xo, Xi) . Hence, taking into account

0

(1.10.17), we get

(1.10.21),

Classical interpolation theorems

74

The application o f this proposition t o the situtation encountered in the theorem also requires the proof of the isomorphism

which, together with Proposition 1.10.7, directly leads t o the statement o f Theorem 1.10.5. In order t o formulate the general result of the theory which leads t o

(1.10.22) as a particular case, we introduce the family of spaces E&(Xo,Xl), cy

> 0, 0 < q 5 00.

For this purpose, we use t h e concept of the E-functional,

see (1.9.5), and put

k

It"E(t; 2 ; xo,xl)Iq7

(1.10.23)

II5 lIE&(XO,X*) :=

for z E Xo

+ XI,with the usual modification for q

:=

i'^

00.

One of t h e fundamental results o f the real method theory can be formu-

lated as follows. Theorem 1.10.8 (Peetre-Sparr).

The following isomorphism is valid:

where

(~(29)

:= (1 - 29)cr0

+

0

Initially, the proof of this theorem was quite complicated. It also left open

the question concerning such a remarkable stability o f t h e family of E-spaces under t h e action of the contruction

( . ) d q . At present, we are in

a position

t o give a simple explanation for this and many other facts of this kind. It was found that all of these facts are based on a fundamental property of the K-functional

(K-divisibility), which was established some 20 years after the

definition of this functional. This property will be described in Chapter 3. For t h e present, we shall derive (1.10.22) form Theorem 1.10.8. For this purpose, we just have t o note that in view of Proposition 1.9.4

75

The Marcinkiewicz Theorem

LP, = E;,p,,(Lco) Lo) From this and Theorem 1.10.8, we obtain

C . In order t o demonstrate the significance of the generalization of the Riesz-Thorin theorem proposed by Marcinkiewicz, let us consider a few examples. Example 1.10.9 (generalized Bessel inequality).

Let {pn}Fbe an orthonormalized system i n the Hilbert space L z ( d p )and l e t (cn(f)):

be a sequence of Fourier coefficients of the function f E L z ( d p ) .

Then Bessel's classical inequality has the form

II(cn(f))r112 :=

(1.10.24)

Let us consider the question extending this inequality t o the space L p ( d p ) ,

15 p

5 2.

(1.10.25)

Here, we assume t h a t

M := sup

Ilv)nllco < 0 0 .

n

This inequality ensures the existence o f Fourier coefficients of any function

f

in L,(dp) for 1 5 p

5 2.

Indeed, in view o f Holder's inequality and

(1.10.25), we obtain i C n ( f ) l ~

I I ~ I II ~~ ~ I I IM~* !

11fllp

llvnll? = M

p)-2 p

IlfllP

*

Next, l e t us consider the (Fourier) operator 7 ,defined by the formula

Then in view of (1.10.24), we have for simple functions

f

76

Classical interpolation theorems

while in view o f (1.10.25)

we obtain for the same functions

I l ~ f l l mI M llflll .

(1.10.27)

Application o f Theorem 1.5.1 in the real case or Theorem 1.7.1 in the complex case with po := 1, qo :=

03,

pl = q1 := 2 and 19 := 2/p’leads t o

the following result. Theorem 1.10.10 For 1 5 p

(F. Riesz).

5 2, we

have

Another generalization of Bessel’s inequality was obtained by

G.H. Hardy

and J.E. Littlewood in 1926 for a trigonometric function and was exten-

ded in 1931 by R. Paley t o general orthonormalized systems with condition (1.10.25). In other words, t h e following theorem is valid. Theorem 1.10.11 (Paley). For 1< p 5 2, we have

(c

(1.10.28)

1 /P 2 2

lcn(f~lpnp-2)

I 7 ( P ) M

IlfllP

.

ndV

Proof (Zygmund,

1956). Simple examples show t h a t this inequality is not

valid for a trigonometric system for p

:= 1.’’

Hence it is not possible t o

apply the Riesz-Thorin theorem, while the application of the Marcinkiewicz theorem almost immediately yields the desired result. In order t o use this theorem, we consideron the point n E operator

T

:

PV. f

IN

a discrete measure v which is equal t o nW2a t

On the set of simple functions S(dp), we consider the

+ (nc,(f))r.

llTf(l2,dv

:=

{

Since 112

ll(~(-f))~l/2

/c.if)ni’n-’}

9

ndV

lgFor example, we can take for f the function z +

Ey

belonging to L i ( T ) .

The Maxcinkiewicz Theorem

77

Next, we consider the set

for n E Mt, putting

t N := -we obtain M Ilf 111

Here we have used the inequality it is proved that

CnEm5 < 2 for N < 1. Consequently,

Applying Marcinkiewicz’s theorem 1.10.1 with po = qo := 2, pl = q1 := 1 and 29 := 2 / p - 1, we obtain inequality (1.10.28) from (1.10.29) and (1.10.30). 0

Example 1.10.12 (conjugation operator). Suppose that the function f E L ( T ) and that its Fourier series expansion has the form

f

- 2+ c

(a,

cos

n x + bn sin n z > .

n E N

Let us consider two harmonic functions u j , v j : f and defined by the formulas (1.10.31)

U,(z)

:=

V,(z) :=

a0 +

c n E N

c

( a , cos nx

+ b,

ID --t R,connected with sin nx)rn ,

nEN

(-b, cos n z +a, sin nx)rn ,

78

Classical interpolation theorems

where z :=

T

exp(is). According t o Fatou’s classical result (1906), we have

f(z) = lim Uf(reiZ> r+l

for almost all

2.

Privalov’s theorem (1909) states t h a t a similar limit exists

almost everywhere for V f . Denoting this limit by

f , we obtain, for

almost

all x,

In case

fl is integrable, its Fourier expansion has the form f”

-

C

( - b , cos nz + a , sin nz) .

ndV

Hence, in particular, we have 112

(1.10.32)

llIllz = C (niN

0:

+ b:

1

However, t h e conjugation operator

I IIfIIz .

f

fl is

i

not bounded for p

:=

1

(N.N. Luzin, 1913). Hence, we cannot apply the Riesz-Thorin theorem for extending inequality (1.10.32) to the case of the space L p . However, Marcinkiewicz’s theorem can be applied, in view of the following fundamental result. Theorem 1.10.13 (Kolmogorov, 1925). There exists a constant y

mes ( 2 E

> 0, such that for

any

t >0

7; If(.)l > t ) I ( y l t )llflll .

Thus, the conjugation operator is a bounded map from L1 into L1, and an application of (1.10.32) together with Marcinkiewicz’s theorem leads to the inequality

Since the conjugation operator satisfies the identity

The Marcinkiewicz Theorem

-

fl.gdx=

J

-

II

79

f .jdx,

II

Lp and g E Lpr, 1 < p < 00, the preceding inequality can easily be extended t o the interval 2 I p < 00. Thus, the following important

where f E

theorem, which was initially proved by the methods of the theory of analytic functions, is valid. Theorem 1.10.14 IlflllP

( M . Riesz).

I T(P) I l f l l P l 1< P < 00.

0

Example 1.10.15 (Hardy-Littlewood maximal operator). Consider the sublinear operator M : L';"(R) -+

M ( B ) i n Example 1.5.6.

Thus,

(Mf)(x) := sup f(4 where I(s) is an interval from (1.10.33)

llMf llco I Ilf

I103

R2 with x as its centre.

Obviously,

.

If a similar estimate were valid in L1, we could conclude with the help of M. Riesz's theorem t h a t M is bounded in L , ( R ) for 1 5 p I 00. However, we have for the function

1 (MX[O,lI)(X)= g Hence M is not bounded in (1.10.34)

mes {z E

for 1 5 x

< 00.

L l ( n t ) . However, l e t us verify t h a t

a;(Mf)(x) > t } I ( 2 / t ) [If111

so t h a t Marcinkiewicz's theorem, consequently, is applicable. For this pur-

Et the set in (1.10.34). If x E €t then by the definition of M there exists an interval I(x) for which pose we denote by 3

(1.10.35)

mes I(x)

/

w

IfldY > t

Classical interpolation theorems

80

Since the centres of the intervals I(z) cover a countable number (In:=

Et, we can

choose a t t h e most

of these intervals such that their

I, n in+^ = 0 for K 2 2. Then the exceed C mesI,, and mesI, < $ J lfldy in view

union covers Et and, a t the same time, measure of

Et

does not

1,

of (1.10.35). Thus, considering that the multiplicity of the family

{In} is

not more than two, we have

which proves (1.10.34). An application of Marcinkiewicz’s theorem now leads t o Theorem 1.10.16

(Hardy-Littlewood, 1930).

Example 1.10.17

(the Halbert transform).

Finally, consider the operator

( H f ) ( z ) := lim E-0

It can be shown that i f f E L,(B2), this limit exists for almost all 2. In this H is unbounded in &(EL). Nevertheless, we

case, however, the operator

can apply Marcinkiewicz’s theorem, and this leads t o the inequality

Let us confine ourselves t o the case of the discrete Hilbert transform, which

is rather easy t o consider. For a given two-sided sequence we put

(hf)n :=

C

fn-m

7,

mcZ\toI In view of the elementary identity

n E z .

f

:=

(fn)nGz

The Marcinkiewicz Theorem

81

and the inequality

we obtain

which means that h is a bounded operator in 12. At the same time, 00,

where Si = (0,

...,0,1,0 ,...), and so h is unbounded

in

ll.

IIh(S,)lll =

However, we

shall show that h ( l l ) c Il,. For this purpose, we estimate the number of elements in the set {n E Z ; I(hf),,l > t } . Without loss o f generality it can be assumed that

f

has a bounded support. Further, t h e set in which we are

interested is included in t h e union of four sets {n

E Z; f(hf,t), >

:},

f* := max {kf,0). Hence it remains t o move the estimate card{n E Z ; (hf), > t } for the case when all nonzero fn have the same

where

sign (say, plus). Thus, let

> 0 and the prime indicates that the terms with zero denominator

where fmJ

have been omitted. Let us replace n by x E

& > t } . It

112 and let Ei := {x E 112;

can be seen from the graph o f the function

N

x+C fm, xj=1

mj

(see Fig, 2) that &: is a union of intervals

(mj,xj),

1 5 J’

I N , where xj is

:= €:

n Z lying in the

a root of the equation N

(1.10.37) j=1

Since mj E interval

fm -t x-mj

Z the number o f integers of the set €t

(mj, xj)

does not exceed its length. Consequently,

82

Classical interpolation theorems

i(

I

I

I

Figure 2.

c N

(1.10.38)

card&

5

j=1

mj) =

z; j=1

c N

N

(zj -

mj

.

j=1

T h e sum of the roots of equation (1.10.37) can be found from Vieta's formula reducing this equation to a common denominator and finding the coefficients of xN and z N - ' . This gives

Substitution of this sum into (1.10.38) leads to the estimate n

The Maxcinkiewicz Theorem Hence

83

h : l1 + 11, and an application of Marcinkiewicz's theorem together

with (1.10.36) leads t o the inequality

IlhfllP L T(P)I l f l l P Since

7

1
I2 .

( h f , g ) = ( f , h g ) for simple f and g , we can use Proposition 1.3.5 t o

extend the obtained inequality t o the interval 2 following theorem is valid.

Theorem

1.10.18.

The following inequality is valid:

5 p < 00

also. Hence the

Classical interpolation theorems

84

1.11. Comments and Supplements A. References Sec. 1.1. The interpolation o f infinite families of Banach spaces is de-

scribed in papers by Coifman, Cwikel, Rochberg, Sagher and G. Weiss and Krei’n and Nikolova

[l-31

[1,2].

Secs. 1.2, 1.3. In presenting basic concepts and results of the theory o f

[l]. For the properties o f t h e M and L, spaces, see Dunford and Schwartz [l]. Sec. 1.4. The original proof o f Theorem 1.4.3 is due to M. Riesz [l] and measure and integral, we follow the book by Halmos

it is not well known a t present. In the same paper, the unremovability of the condition p

5q

in this theorem is demonstrated. Remark

from the paper by Thorin

1.4.7 is borrowed

[2].

1.4.8 due t o Clarkson [I] is probably new. It is interesting that for p > 2 inequality (1.4.21) cannot be The “interpolation”

improved in order as

E

proof of Theorem

4

0.

Sec. 1.5. The transition from the finite-dimensional t o the general M. Riesz theorem, carried out i n the proof o f Theorem

1.5.1 is well known t o

the specialists, although it is difficult t o give an exact reference. Th e real-valued analog o f the Stein-Weiss theorem

[l], given inTheorem

1.5.2,can also be obtained directly from the complex-valued theorem by the same authors (see Sec. 1.11.2). Theorem 1.5.11on the interpolation o f sublinear operators was formulated by Calder6n and Zygmund [2].The simple proof o f this theorem presented in this book was proposed (in a more general case) by Janson [3]. Sec. 1.6. Theorem 1.6.1was obtained by Hadamard [l] and generalized by Hardy [l] to the case of the integral p m e t r i c (0 < p 5 m). Theorem 1.6.3 was established by Deutsch [l]. Sec. 1.7.In the proof o f the “finite-dimensional” part o f Theorem 1.7.1, we followed Thorin (11 (see also [2]). Other proofs were proposed by Tamarkin and Zygmund [l] and Calder6n and Zygmund [l]. Item (a) o f Theorem 1.7.2 is a corollary t o the Grothendieck inequality,

85

Comments and Supplements

[l]. The best estimate of the Grothendieck constant KG [2].The statements of items (b) and (c) of this theorem are due t o Krivine [l]. On the complex Grothendieck constant see Pisier [2]. See. 1.8. Theorem 1.8.1is due t o Stein and G. Weiss [l],where p i , q; 2 1 (on the meaningfulness o f the case pi < 1 5 qi, i = 0 or 1, see Sec. 1.11.3 below). Theorem 1.8.2was obtained mainly by Thorin [2].In presenting the see Grothendieck

has been obtained by Krivine

subsequenct material, we followed the paper by Coifman, Cwikel, Rochberg, Sagher and Weiss [3], in which the general situation is analyzed. The starting point of this line o f arguments was Stein’s paper is proved (see also Hirshman for example, Duren

[l], where Theorem 1.8.10

[l]). On the spaces N ( I ) ) and N + ( I ) ) , see,

[l].

Sec. 1.9. The appearance of the “weak” space L, goes back t o the work of Kolmogorov [l], Hardy and Littlewood

[2]and Marcinkiewicz [l]. The

definition of the more general scale o f spaces

L,, for

15 q

5p

was given

[1,2].The role of spaces L,, in interpolation theory was revealed [l], the case of q := 1, O’Neil [l], and the works refferred t o below in connection with Theorem 1.10.5). A number of basic

by Lorentz

a decade later (see Kre’in

properties of these spaces was established in the work of Lorentz mentioned

[2]and Oklander [2]. The “approximation” approach based on Proposition 1.9.4is indicated in the paper by Peetre and Sparr [l].

above and in the papers Hunt

Sec. 1.10. Theorem 1.10.1 for the “diagonal” case pi = qi was formulated by Marcinkiewicz

[l].

Not long before his premature perishing, he

presented the proof in a letter t o Zygmund (see the foreword by Zygmund t o

[2]). Later, Theorem 1.10.1was proved for the general case by Zygmund [l] and, independently, by M. Cotlar [l]. Example 1.10.3 was pointed out by Hunt [l]. Marcinkiewicz’s book

The generalization o f Marcinkiewicz’s theorem t o the spaces

L, and

1.10.5,is mainly due t o Cal[3](see also Hunt [l], Oklander [2],Lions and Peetre [2],Peetre [7] and KrCe [l]).

quasiadditive operators, contained in Theorem der6n

The notion o f “Gagliardo diagram” and the concept o f interpolation

Classical interpolation theorems

86

I < Q ( X ~ , Xcan ~ ) be found in Gagliardo [1,2]. The definition of t h e ( X o ,XI)$* and Proposition 1.10.7 were proposed by Peetre [7]. A similar approach was developed by Oklander [l].The new proof of Theorem 1.10.5, outlined in this section, is taken over from the paper by Peetre and Sparr [l],as well as Theorem 1.10.8. Theorem 1.10.10 for a trigonometric system was proved by Young [l] (p’ E 2 N ) Hausdorff [l](2 5 p’ 5 CQ) and generalized by F. Riesz [l] t o arbitrary orthonormal systems. I t s “interpolation” proof was given by M.

space

I<-functional, t h e space

Riesz [l]. Theorem

1.10.11 was for the trigonometric case obtained by Hardy and

Littlewood [l)and generalized by Paley [l]t o arbitrary orthonormal systems.

The “interpolation” proof of this theorem was given by Zygmund [l]. Basic facts about the conjugation operator and, in particular, the theorems of Privalov [l]and Luzin [l]mentioned here are discussed in t h e book by Zygmund [2]. The Kolmogorov inequality [l]in Theorem

1.10.13 has

played an important role in interpolation theory as the first example of an operator of “weak” type. Theorem

1.10.14 was initially proved by M. Riesz

[2] by methods of the theory of analytic functions. Its “interpolation” proof based on Kolmogorov’s inequality was given by Marcinkiewicz [l]. The maximal operator was introduced in the work of Hardy and Little-

[2], where Theorem 1.10.16 was proved. An elementary proof of inequality (1.10.34) going back t o F. Riesz [2] is based on Besikovitch’s approach [l](see also Guzman [l]). wood

The Hilbert transform and the conjugation operator are typical examples of a broad class of t h e so-called Calder6n-Zygmund operators. Concerning t h e theory of these operators and their role in modern analysis, see the book

by Stein

(21. Inequality (1.10.36) is due to I. Schur [l]. This work also

contains the first example o f “interpolation” of operators.

On the other

hand, the proof of inequality (1.10.39) was given by J. Boo1 as long back as 1857 (in this connection, see Levinson [l],p. 68). Theorem 1.10.18 is due to M. Riesz [2].

Comments and Supplements

87

B. Supplements 1.11.1. The Riesz Constant Suppose t h a t p(19,p,, q;) := inf y, where the lower bound is taken over

all constant y’s in the inequality

where p

:= p ( 8 ) and q := q(I9), 0

< 19 <

1 and 0

< pi,qi 500. We

denote by pc(.19,p;,q;) t h e similar constant in the complex case. According t o Theorems 1.5.1 and 1.7.1 we have for O

< p; I qi I 0 0 , i = O,I , min q; 2 1 ,

(1.11.2)

p(d,p;,q;) = 1

(1.11.3)

pc(19,p;,qi)= 1 for 0 < p,qi

min q; 2 1 .

I 0 0 , i = 0,1,

Besides, from the definition o f -y,(p,q) [see (1.7.4)], we also have the inequality (1.11.4)

P(fl,Pi, qi)

I TZ(PO, ~0)’-~72(~1,q l ) ’ P c ( d , ~ i qi) ,

7

as well as the inequality

Finally, the example in Remark 1.4.5 shows that supp,,*, p(b,p;,q;)

>

1.

On t h e other hand, it was shown by Gustavsson [2] that this upper bound is finite. The exact calculation of the Riesz constants seems t o be a complicated problem. We formulate here conjectures whose proof or disproof will help understand the situation better.

It is true that (1.11.6)

p(d,pi,qi) = 1

for 0

I pi I pi 5 0 0 , i = 0,1 ?

Is it true that

(1.11.7)

p(d,pi,qi) = 1

for 0 < p

I q I 00

?

88

Classical interpolation theorems

It should be recalled that p := p(29) and q := q(6). Besides

where 0

(1.11.2),the following inequality justifies the former conjecture:

< pi 5 q; < 00,

a is an arbitrary quantity greater than zero, and

l/Kp(cY) :=

Here

{C,},~Z

is an absolutely summable sequence whose sum equals unity

(the constant K q ( a )is defined i n a similar way w i t h

PO,

q1 instead of p0,pl).

(1.11.8)is outlined in Problem 18 in Sec. IX.6 o f the book by [l],where it is erroneously stated that p ( S , p i , q ; ) = 1 for all values

The proof of Bourbaki

of the arguments. Taking co := 1 and ci := 0 for

i # 0,

we note t h a t

ITp(&) 2 1, while

the application o f the Holder and Jensen inequalities gives

for q

5

1. Consequently, for q := q(29)

5

1 and a + 0, we obtain from

(1.11.8)an equality similar t o (1.11.5)in which the condition min q, 2 1 is now replaced by the condition q 5 1. The conjecture (1.11.7) seems to be less plausible. If it were valid, a similar equality would also be valid for pc also. This follows f r o m (1.11.5)if we take into account the equality (see Sec. 1.11.2) (1.11.9) T 2 ( p , q ) = 1

for 0

1.11.2. The Riesz Theorem

< p 5 q < 00 .

as a Corollary of Theorem

1.7.1.

It follows from (1.11.9) and (1.11.4) that equation (1.11.2), which is equivalent t o the statement of the M. Riesz theorem 1.5.1,is a direct consequence o f (1.11.3) i.e. o f the M. Riesz-Thorin theorem 1.7.1. This is

Comments and Supplements

89

probably the simplest proof of Theorem 1.5.1, since equation (1.11.9) can be proved quite easily (see, for example, Zygmund [2], Problem 13, Chap.

5q

IV for p := q; the case p of Jensen’s inequality).

is obtained in a similar manner with the help

For some other results concerning the constants

yz(p,q), see VerbitskiY [l], where it is shown, in particular, that for p >_ 2

1.11.3. The Meaning o f the Theorems o f Riesz and Thorin for pi

<1

The statements of Theorems 1.5.1 and 1.7.1 became trivial for

if the measure p is continuous. This is due t o the simple fact that

JqLP(44, L,W) = (01 for p

< q’

when p is continuous. In order t o verify this (6. Day[l]) we take

f

an arbitrary function

in L,(dp) and, making use of the continuity of p ,

consider a subdivision {A;}? of the measurable spaces

J

IfIPdp = l / n

J

IflPdp

,

i

= 1,...)n

R such that

.

n

A,

Here n is an arbitrary fixed number. In this case, for T : L p ( d p ) + L,(dv), we have:

IlTjll, 5

(C I I T ~ ~ I I ; 5* ) IlTllpq(C ”~*

llfill~)l’g’

= lpllpqnl/+l/P here f; :=

Ilf IIP

=

;

f x ~ , Passing . t o t h e limit as n + 00, we find that T = 0.

It should be noted that when p = q

< 1 and p

is continuous, the space

~ ( ~ ~ ( d ~ , p( d) v ,) )is very small.

Obviously, the two above-mentioned theorems are no longer trivial if the measure p is discrete.

Classical interpolation theorems

90

1.11.4. InterDolation of Quasilinear ODerators It can be asked whether Corollary 1.10.4 is valid without the restriction p(29)

5

q(29). The answer t o this question is a matter of principle, and we

shall discuss it in Chap. 3.

1.11.5.Interpolation in Spaces H p Due t o lack o f space, we shall not describe the classical results obtained by Thorin [2], Salem and Zygrnund [l],and Calder6n and Zygmund [l] concerning the interpolation of t h e Hardy spaces

H p . W e refer the reader t o

Zygmund [2], Chap. XII, Sec. 3, for a detailed description.

91

CHAPTER 2

INTERPOLATION SPACES AND INTERPOLATION FUNCTORS

2.1. Banach Couples

A. The main object of study of the theory presented in this chapter is the Banach couple. The description o f this concept includes Definition 2.1.1. Banach spaces X o , XI form a Banach couple if

(2.1.1)

xi x ,

i =0,l

Lt

for a certain Hausdorff topological vector space X . 0

It should be recalled that the symbol embedding. If

X

X

and

Y

‘I-’’

denotes a linear and continuous

are normed spaces, we shall also use the notation

y Ilzllx is satisfied for all z E X . Henceforth, a Banach couple (Xo,Xl) will be denoted by 2. Let

Y when the inequality

( ( z ~ ( Y<_

us

consider several examples o f couples.’ Example 2.1.2. Suppose that

X o & XI. Then ( X o , X 1 )is obviously

a couple. Such coup-

les will henceforth be called ordered couples.

Example 2.1.3.

x’ be a couple and E;: be a closed subspace of Xi equipped by the induced norm, i = 0 , l . Then f = ( Y 0 , X ) is a couple referred t o as a subcouple of 2. Let

“Couple” means “Banach couple”.

Interpolation spaces and interpolation functors

92 Example 2.1.4.

If r? is a couple, then

zT := ( X l , X o )is the couple obtained from r? by

transposition. Examde 2.1.5. Recall that we call the lp-sum (15 p

5 00)

of the family (X,),,,

spaces the space consisting of all elements ( z , ) , , A

of Banach

of the Cartesian product

for which (2.1.2)

:=

ll(za)aEAll

(c

1IP

~

~

~

m

<~00 ~.

~

a

)

CYEA

It is well known and can be easily verified that norm (2.1.2) defines a Banach

-. If X ,

space. We denote it by $ p ( X , ) a E A .

.+

Let now ( X , ) , , A

be a family of couples.

(XOu,Xla),we put

:=

@p(zu>u€A

:= (@p(XOa)aEAt @p(Xla)aEA).

The fact that this formula defines a couple can easily be verified with the help of Proposition 2.1.7 given below. In our analysis, we shall constantely encounter examples of “concrete” couples as well. At the moment, it is appropriate t o mention the couple

L dG )

:=

(Lm(wo),Lpl(w~)),1 5 p o , p l 5 00 (see Theorem 1.3.2).

Each couple

x’

is canonically associated with two Banach spaces, viz.

the sum and i n t e r s e c t i o n . To define these spaces, we put

for z

E X o n X , and

for z belonging t o the (algebraic) sum

xo+x1

:= { z o + z ~ E r ; z i E X i 2=0,1}. ,

93

Banach couples Proposition 2.1.6. Formula (2.1.3) specifies a Banach structure on the linear space

Xo + X I .

and formula (2.1.4) one on the linear space

Xo n X I ,

Proof. Let us consider the subspace D o f the space Xo $, X I ,see (2.1.2), defined by

D

Xo

:= { ( z o , ~ ~ E )

$ ,

Xi;20 = x i ) .

It is a Banach space in the norm induced from Xo

$ ,

XI. Then the

map

j : z + (z,z) is a bijection from X o n X1 onto D ,and in view of (2.1.3),

lIj(~)lIxo~mxl = llzllxonxl X o n X1 is a Banach space. X1 + X is the continuous map defined by the formula s(zo, zl) := zo + zl. Then S is a surjection on XO XI.

Consequently, (2.1.3) is a norm, and Suppose further that s :

Xo

+

In this case, kers

:= {(zo,z1); s(z0,zl) = 0) is a closed subspace of kers + zo z1 is obviously a : (zo,zl) Xo bijection of (XoB1 X1)Ikus onto Xo XI.Taking into account (2.1.4) and the definition of the factor-norm for 2 := (20,z1) ker s, we obtain

+ +

XI, and the map s^

II 2 II(XO& XI

her

= inf {llzo

+ +

,=

+ YOllX0 + lIz1 + Y1I1x1; yo + y1 = 0)

Consequently, s^ is an isometry and X o

=

+ X1is a Banach space in the norm

(2.1.4). Henceforth, the sum of a couple

x'

will be also denoted by C ( 2 ) and

the intersection by A(x'). Obviously, (2.1.5)

A(2)

A Xi L:

C(x'),

and if for some Banach spaces V and W

v L xi A w ,

i=O,l,

the following embeddings are valid:

i

=0,l

Interpolation spaces and interpolation functors

94

V

(2.1.6)

&

A(X)

A

C(f)

and

W

The space X from Definition 2.1.1 is required only for the definition of the sum and the intersection o f a couple and will not be used in t h e further analysis. In order t o avoid the verification of the continuity of embedding in (2.1.1) use can be made o f Proposition 2.1.7. Suppose that Banach spaces X; are linearly embedded into a vector space

X ,i = 0,1, and the following condition o f consistency is satisfied: (A) If a sequence ( z n ) n E E~ zonx, converges t o an element y; in a space Xi,

i = 0,1, then

yo = y1. Then (Xo,Xl) forms a couple.

Proof. Let us show that formula (2.1.4) defines a X o + X1 c X . Obviously, only the property

norm in the linear space

((z((x0+x1 =0 +5 =0

(2.1.7)

requires a verification. Suppose that 1(zllx0+xl= 0. Then for certain sequences (zi)nEm C Xi,

i = 0,1, we have

11~onllxo+ ll~;llxl

+

0

72

+

7

and 5:

+

5;

=5

for all n. Since z i - 5 ; E X ; and zz Ex,,nx,, Here zk + 0 in

- zy = -(.A

- z:),

we have

i=o,1

X ; , i = 0 , l . This means that

zk - zf -+

and from condition (A) we obtain the equality zy = -z:. 5

= 5:

+

5:

-xi, i

=

0,1,

Consequently,

= 0 and property (2.1.7) is proved.

It is now sufficient t o take for

X

in (2.1.1) the normed space X ,

+XI.

95

Bmach couples

B. Let us consider another basic concept of the theory, which involves Definition 2.1.8.

The linear map T : C(x') ---f C(?) is called a h e a r contanuous OpeTUtOT from the couple d into the couple ? if (2.1.8)

Tlxi E L ( X ; ,y t )

, i =0,l .

0

Here and below,

L ( X , Y ) denotes

a Banach space of linear continuous

operators from the Banach space X into the Banach space Y . The norm in this space is denoted by

IITllx,y.*

The set of operators in Definition 2.1.8 forms a linear space in which the norm is defined by t h e formula (2.1.9)

IITllf,< :=

m a {(ITlliI!li.Yi7i = 0,1)

The normed space obtained is denoted by

.

L(-f,?).

Proposition 2.1.9.

L(x',P) is a

Banach space.

Proof. Let us consider the subspace D of the Banach space L(X1,Yl)defined by the condition L(X0, YO)

then, in view of (2.1.9) and the definition of D, the map j is an isometric map of

L(d,?) onto D. Since D

is obviously closed, C ( x ' , F ) is a Banach

space. 0

2 h t e a d of L ( X ,X ) we shall write L ( X ) and instead of IITllx,x,just llTllx. Similar abbreviations will also be used henceforth.

96

Interpolation spaces and interpolation functors

Proposition 2.1.10.

The class of couples and linear continuous operators in the couples forms category3 denoted by

g . Substituting the set L , ( Z , p) :=

a

{ T E L ( T?) , ;

IITlla,p 5 1) for L(x',?) (closed unit ball), we obtain an (incomplete) subcategory of

6 denoted by 51.

Proof. Suppose that 12 denotes a unit and for any T E L(x', ?) we have

map C ( x ' ) . Then 112 E

.Cl(Z),

T 112 = 11pT = 2'. Further, if T E

L(x',?)

and S E

L ( ~ , ~then ) , ST

E

L(x',?), and in

view of (2.1.9), we obtain

IISTlIY.2 I IITll2,P IISllP,Z

*

U

Remark 2.1.11.

B the category of Banach spaces and linear B1 its incomplete subcategory in which t h e morp-

Henceforth, we shall denote by continuous maps and by

hisms are contractions (linear operators with a norm

5

1).

Definition 2.1.12. Banach couples 2 and ? are called linearly isomorphic (or *-isomorphic) if there exist operators T E L(x', ?) and S E L(f,x')such that ST = 112 and T S = 119. 0

The operator S is called the inverse operator t o T and is denoted by T-1. We shall denote the linear isomorphism of couples x' and ? by x'

- ?.

A simple corollary o f Banach's theorem on the inverse operator4 is ~

~

~~

3A brief review of the category language used here and below is given in 2.7.1. 4For the corresponding concepts of functional analysis, see references in the comments at the end of the chapter.

Banach couples

97

Proposition 2.1.13.

If T E L(x’,g) and T is a bijection o f Xi on yi (z’ = 0, l), then X -#

- ?.

0

Substituting in Definition 2.1.12 the morphisms from hisms from

2 1 ,

6 for

the morp-

we obtain the definition of linear isometry (or g 1 - i s o -

morphism) of couples x’ and

? (denoted

by

x’

31

?). Thus, x’

N

? if

f )for which Tlx, is an isometry of X i on x,i = 0 , l .

there exists T E L(x’, Remark 2.1.14.

The gl-isomorphism of x’ and f does not follows from the existence of T E L(d,?) effecting a linear isometry of C ( x ’ ) on C(?). Indeed, l e t x’ be such that Xo A X I . Then

and since the right-hand side is equal t o llzl)xl, C ( 2 ) = X1 (with equality

?

y > 1, and l e t yX denote a Banach space obtained from X by multiplying its norm by y. Then 1 Yo Yl and C(?) = X1. Therefore, the identity mapping is an isometry of C(x’) onto C ( f ) , but X o is not isometric t o Yo.

of norm).

-

Suppose now that

Remark 2.1.15. Henceforth, we shall denote by X

= (yXo,X1) where

-

Y (X

I IY

) a linear isomorphism

(isometry) of Banach spaces X and Y .

L(x’,?). Recall X --f Y is the term

Let us now define the kernel and t h e image for T E that in the Banach case the kernel o f an operator5

T :

applied t o the closed subspace {r E X ; T r = 0) equipped by the induced norm (and denoted by kerT). Therefore, in the case of couples, it is natural t o put 5Here and below, we mean by “operator” a “linear continuous operator”

98

Interpolation spaces and interpolation functors ker T := (ker To,ker Tl) ,

(2.1.10)

where it is assumed that T; = T ( x , . Obviously, kerT is a subcouple of

x'

(see Example 2.1.3).

Further, we define for T : X + Y the image ImT as the linear space

{ T z E Y ; x E X } with t h e norm

Before defining an analogous concept for couples, we prove ProDosition 2.1.16. (a) I m T is a Banach space;

(b) I m T

Y;

(c) T E L(X,ImT), and

Proof. (a) We can assume that

T

#

0 since otherwise everything is trivial. Let

us first show that (2.1.11) is a norm. Obviously, it is only the triangle

inequality t h a t has t o be verified. Suppose that yi

E I m T and E > 0 are

given. We assume that xi E X is such that

Then y 1 + yz = T(z1+ 22). and hence

Thus, (2.1.11) is a norm. Let us now show that ImT is a Banach space. Suppose t h a t ( y , ) n c ~ 2,

c I m I and

that

C

l l y n l l h ~<

E X satisfying the conditions yn = Tx, and

00.

We choose

Banach couples

99

C IIxnllx < co and

hence the series C x, converges to a certain X . Then C y, = T(C 5,) converges i n Y to an element y E Y , and since y = TI, y E ImT. Finally, in view of (2.1.11), Then

element x E

for

N + 00 so that I m T is complete.

(b) If y E ImT, then y = Tx for some x E X , and hence

Taking inf over all such x ' s , we obtain

and the embedding is proved. (c) Since

Let us now define the image o f T E

L(x',?) by

(2.1.12) I m T := (ImTo,ImTl) , where, as before,

:= Tlx,, i = 0,l.

Corollarv 2.1.17. I m T is a couple, and

Interpolation spaces and interpolation functors

100

llTll2,hT I llTll2,P

.

0

Remark

2.1.18.

Henceforth, t h e notation

T ( X ) will be used along with I m T

Using the definitions o f kernel and image of an operator acting in coup-

les, we can define injective and surjective mappings assuming that T

e

C(x',?) is injective if kerT = (0) and surjective if ImT coincides with

?

(without equality o f norms). Some other important classes of mappings are considered in the examples given below. Example 2.1.19 (Embeddings).

K, i = 0 , l . Then, in view of (2.1.6), C ( 2 ) -+ C(?), J E L(x',?) acting as J(z) := z is defined for 5 E C ( x ' ) . We call J the operator of embedding of r? into ? and use the notation x' ?. If in this case IIJlla,? 5 7 , we also write I? & P. Suppose t h a t

Xi

L)

-

and thus t h e operator

Thus, from Proposition 2.1.16 it follows that

ImT

(2.1.13)

-

A ?.

In the further analysis, we shall use the notation and

?

2

when

2

L)

?

x'.In such a case, the Banach spaces X ; and Y , coincide as sets

and have equivalent norms for

i = 0, 1.6

A similar notation will also be used in the category of Banach spaces. For Z Y if X and Y coincide as sets and have equivalent norms.

example, X

Example 2.1.20 (Projections).

Let

? be a subcouple o f r? and let the operator P E L(r?, ?) be such that

PIX, projects X ; onto Y,, i = 0 , l . Then P is called a projection and

is

60bviously, when d A Y' and f A d , we have d = Y'. It should be emphasized that the equality sign is used only in this case.

Banach couples

101

a complemented subcouple o f

2

It should be noted that, unlike the Banach case, the subcouple

finite-dimensional (dimY,

< 00, i = 0 , l ) and

? can be

uncomplemented in

x'. In

f := (Yo,&)of the couple ( X , X ) , which consists o f finite-dimentional subspaces Y, o f the space X that are nonisomorphic t o each other. I f ? is complemented and P E ,C(x') is the corresponding projection, then P ( X ) = Y,, i = 0 , l 50 that y0 E Yl

order t o see this, let us consider a subcouple

in spite o f the assumption. Let us show that if the subcouple (2.1.14)

Yo

? is finite dimensional,

&

S

+

is not only a necessary, but also a sufficient condition for

mented in

the condition

Y

t o be comple-

2.

Indeed, let (2.1.14) be satisfied and let {zl,...,Z N } be the basis in Yo(or

K).

{fi, ...,f~}c

Suppose that

Since

Yo is finite

induced in

is the dual basis (so that f;(zj) = & j ) .

dimensional, the functionals

f; are

bounded in t h e norm

Yofrom C ( 2 ) . Let 6 be an extension o f off; on C ( 2 ) obtained

with the help o f the Hahn-Banach theorem. We put

c N

P :=

jj

@

" j .

j=1

P maps X i continuously into Y,, i = 0 , l . Besides, P ( q ) = C zjfj(z;) = z;,1 5 i 5 N , so that P is a projection and ? is In view of (2.1.14)

complemented in

2.

Remark 2.1.21. In the Banach case, the norm o f a projection does not exceed

d m , but

for Banach couples the norm o f projection can be as large as we desire even for d i m x = 1, i = 0 , l . It can be easily shown that the following sharp estimate is valid here:

Interpolation spaces and interpolation functors

102 where

2

E Yo\{O}.

Example 2.1.22

(Linear continuous functionals). +

Let us consider for a given couple X a space of linear continuous f u n c -

tionals L ( r ? , R ) , where

k

=

(R,R). In spite

of the fact t h a t such a

definition o f a functional on a couple is quite natural, the space

L(2,k)

X * . We shall show, for

differs considerably from its Banach counterpart

example, t h a t in the case under consideration, there is no even weak version of the Hahn-Banach theorem. For this purpose, l e t us consider the couple

( X , X ) and subcouple f consisting o f the subspaces Yo,

of the space X

,

t h a t have t h e following properties: (a)

Yon K = ( 0 ) ;

+ K is dense in X

(b) Yo (c)

Y:

$ ,

Iff E (f0,fl)

but does not coincide with it;

Y; is not isomorphic t o X * .

L(?,&), f

is, in view of (a), canonically identified with the couple

Y:,

E Y< $ ,

L(?,liz)

N

where f i

:= fly,. Consequently,

Y< $ ,

Y; ,

and moreover

L((X,X),&) If

f

21

X' .

E L ( ( X , X ) , & ) is such that

fly,

(:= f,),then in view o f property

f is uniquely defined on Yo+Y, by the formula f(yo+y1) while in view o f (b) we obtain t h a t

f

(a)

= fo(yo)+fi(yl),

is uniquely determined by t h e couple

(fo,fl). Thus, if a weakened version o f the Hahn-Banch theorem were valid in t h e case under consideration (the extension without preserving the norm), the formula ( f 0 , f I ) + f defines the bijection inverse operator

Yo'

$

Y; on X * .

f + (flfi, flyl) is obviously continuous, X'

-

Since the

Y:

$ ,

Y;

in view of Banach's theorem on t h e inverse operator, which is in contradiction t o (c). On the other hand, it should be noted that if

9 is a complemented sub-

couple of X , an extension operator exists. Namely, if

f E L(?,&), it

is

Banach couples sufficient t o put

103

f"

= f o P , where P E L ( 2 , P ) is the corresponding

projection. Remark 2.1.23.

The space L , ( T ) and its two subspaces Yo= {f ; cn(f) = 0 for n < 0) and := { f ; cn(f) = 0 for n 2 0) serve as a "concrete" realization of the above example. Here c,,(f) =

J f(t)e'"'dm(t) -

is the Fourier coefficient.

II

The fulfilment o f condition (b) is connected with D. Newman's theorem on the noncomplementarity o f ReH(T) in L l ( T ) , while the verification of (c) is associated with Fefferman's theorem describing the dual of R e H ( T ) . Simpler examples o f this kind are likely t o exist.

C. Some problems in the theory necessitate the extension of the concept of a couple. For example, it would be desirable to consider {XG,X;} for a given couple X as a "generalized" couple (the dual spaces do not form a couple since XG n X ; = 0). Several other examples will be considered below. The first step in this direction was the concept of generalized Banach couple, which makes it possible in some cases t o operate with the above object with two dual spaces as well. A more general approach is described I

in 2.7.2. Definition 2.1.24.

A generalized Banach couple is a triple (Xo,Xl,T) consisting of Banach spaces X ; and a closed linear operator T whose domain is a certain linear space V, C X Oand whose range is a certain linear space R, c X1. 0

Henceforth, we call T the identification operator. The generalized Banach couple is denoted by

2,.

Example 2.1.25.

A couple 2 can be treated as generalized if we put T := la(?). Thus, '0, = R, = A ( 2 ) . Let us verify the property of T being closed. Suppose

104

Interpolation spaces and interpolation functors

x, -+ x in Xo and ~ ( 2 , )+ y in X i . Since ~ ( 5 , )= x,, it follows from (2.1.1)that z and y coincide and belong t o A(X') so that ~ ( x = ) y. that

2.1.26. Let A(x') be dense in every X i , i = 0,l (such couples will henceforth be called regular) and let ? be an arbitrary couple. Let us verify that the Banach spaces L(X0,Yo) and L ( X 1 , X ) form a generalized couple if we define the Example

identification operator

T

as follows. Put

L ( X i , K ) ; TolA(3) =

L := {(To,Ti) E L(Xo, Yo) -4

Since A ( X ) is dense in each

Tt/qa,}.

X i , i = 0,1,the first coordinate of the element

(To,Tl)E L is in a one-to-one correspondence with the second coordinate. Consequently, putting V, := Pro(L), R, := Prl(L) and 7(T0) = Ti, where TOE V, and Ti is determined from t h e condition (%,Ti) E L,we obtain a linear bijection

:

T

follows from the fact that Henceforth, the couple

L

V, + R,. The

is closed in

property of

being closed

ern L ( X 1 , V ) .

L(X0,Yo)

( L ( X o ,X,),L(Xl,

T

K), T)

will be denoted by

,!?(c",?) (do not confuse with L(x', ?)). Example

2.1.27.

If in t h e previous situation ? := k,then L ( X ; , R ) = X:, and hence for regular r? t h e generalized couple formed by conjugate spaces is also defined. We denote if by 2*.

2.1.28. X i , i = 0,1, be normed spaces subject t o condition (2.1.1). Then = ( X o , X l ) will be called a normed coupIe. Let us consider the possibi-

Example Let

x'

lity of abstract completion o f completion of

2. It would

be natural t o assume that the

17 is the family { X i , X , " } ,where X"

is the completion of

X.'

Unfortunately, this family generally does not form even a generalized Banach couple. To be able t o consider the operation of completion, we shall 7Thus,X is isometrically embedded in X" as a dense subset.

Banach couples

105 4

confine ourselves t o only those normed couples X for which the following

condition (A') is fulfilled. (A') If the sequence ( z , , ) ~ € NC XOn XI is fundamental both in Xo and in X1, the conditions

are equivalent.

We will prove that if this condition is satisfied, then (X,",X,") forms a generalized (Banach) couple. For this purpose, let us define

T

as follows.

Let x E X: and y E X t b e such that for a certain sequence (xn)nEm c Xon X I , which is fundamental i n Xi, i = 0,1, we have

Then we put (2.1.16)

T(Z)

=y

.

Let us show that the definition o f

T

is consistent. Suppose that for a given

x E X: there exist y1,y2 E Xt such that for some sequences ( Z ; ) ~ € N , in Xon XI, which are fundamental in X;, i = 0,1, we have

1. - .Illx; 0 119' - .Illx; 0 (n I.. for i = 1,2. Then 1 1 ~ : - z;llx, + 0 and it follows from condition (A') +

7

--f

--$

that

the same is valid for X1. But then IlY1 - YzIlx;

5

c

llYi - dllx;

i=l,?

(n + 00) and hence yt = y2, and

T

+ 114- Z:llx,

+

0

7

is well defined.

It can be similarly verified that

T

is a bijection. Finally, according t o

(2.1.15), the graph of T is the closure o f the set ( X On Xl) in the space X: @ Xi', and hence T is a closed linear operator. Henceforth, (X,",X t , T ) , defined as a generalized couple, will be denoted and called the completion of the normed couple 2. by

106

Interpolation spaces and interpolation functors

Remark 2.1.29.

It can be easily seen that the regularity condition in Examples 2.1.26 and 27 and condition (A') in Example 2.1.28 are not only sufficient but also necessary for the objects under consideration t o form generalized couples. Let us now define the space

L(Z7,gv) of

linear continuous operators

acting in generalized couples. Definition 2.1.30.

The space L(J?,, i;b) $ ,

L(X1, K ) for which the diagram

is commutative (thus, 02'0

= TIT on DT).

We equip L(Z?,,?v) with induced norm. In the situation described in Example 2.1.25, this space coincides with L ( 2 , f ) so that Definition 2.1.30 is a natural generalization o f Definition 2.1.9.

Put now Ila, := ( Uxo, Ilx,)and (%TI> 0 (S0,Sl) := (Toso,TI, S,) for (To,TI)E L ( f v , (SO,SI) E C ( 3 , , F v ) . Using these

z,),

definitions, we can easily verify t h e validity of the following analog of Proposition 2.1.10. ProDosition 2.1.31.

-

The class of generalized couples and linear continuous operators in them form a category (denoted by U

BY).

Banach couples

107

Substituting for C(z,,?,)

the set

L1(zT,?,)that forms a closed unit 4

ball in this space, we obtain the (incomplete subcategory BY1. ---).

Along with the “61-theory” of interpolation spaces, the “BY1-theory” could also be developed. Fortunately, this is not necessary since there exists

-

a “canonical” way of extending all the concepts in the gl-theory t o t h e

BY1-theory. This method is based on the existence of a special covariant +

functor R : BY1-r

gl (the so-called reflector).8

For the formulation of the +

corresponding result, we shall need the concept o f B Y - i s o m o r p h i s m o f the -+

+

4

generalized couples X , and Y, (denoted by X ,

if there exist the operators

ST=

T S = I!qe 7 Obviously, if

d,

T E C1(zT,F,) 112,

-+

Yn).Namely, X ,

and S E

Cl(?,,

-+

N

Y,

zT)for which

.

+

Yo,then X i

N

-+

N

I I

x,i = 0,1,so that the notation

introduced is correlated with the one used earlier. It should also be noted

that in view of Example

-

2.1.25,we can (and shall) assume that

g1is a

(complete) subcategory of BY1. Proposition

-

2.1.32.

There exists a covariant functor

R

: BY1+

61 such that for

2

-

any object

E ~ Ythe , following condition is satisfied: the couple R ( z ) is BYlisomorphic t o the couple 3. In this case, R also preserves the norms of

operators and

R(d)i

I I Xi

2, be

i = 071.

rT is

Proof.

Let

Xo

XI, consisting of the elements ( z , - ~ z ) , z E D,. We define the

Banach space

(2.1.17) Let

Jo

7rT

:

specified and

g(2,)

C(2,)

:=

a closed subspace of the space

by the formulag

(xoel x,)/r,.

be the canonical surjection o f X O @1 XI onto g ( 2 T )and let (0) be the canonical injection. We put

XO + X ,

“The concepts used here are described in detail in 2.7.2. ’The meaning of this notation will be clarified later on in this book.

108

Interpolation spaces and interpolation functors x o := (7rTJ0)(XO).

Similarly, the spacezl can be defined with the help of the injection 51 :

X1

-+

(0) X 1 . Since r, n ( X o B1 (0)) = (0) and the same is true for (0) B1 X 1 , we have

xi x

N-

xi,

i =0,1.

We now put

(2,):= (ZO,Z1). Obviously, in this way a couple is defined, and

E(Xo,Xl) =

fi(2,)[see

(2.1.17)].

Let us now suppose that T := a continuous linear operator from

1

(2.1.1a)

(To, Tl)E L(r?,, gU).We define R(T)as

fi(2,)into E(&)

for which the diagram

-1

*r

is commutative. We leave t o the reader the simple verification of the consistency of this definition and

R( 12,) =

IR(W,,

of the following relations:

WW = R(T)R(S).

7

(To,So,TI,S,). + + R is a covariant functor from BY1 into B1, we must also show that R ( T ) E L1 (R(r?,), R(fu)) for T E L1(ZT,Fu).But it Here T S :=

In order t o verify that

follows from the commutativity of the diagram (2.1.18) that

R ( T ) ( X i )L)

z,

2

= 0,l

,

:= R(g0)i,i = 0 , l . Consequently, R(T)E L ( R ( i T )R(fo)). , Since it also follows from

where we put norm in

L ( f , , q u ) that

the definition of the

109

Banach couples

the required properties o f R are established.

-

It only remains for us t o note that if 2 is a couple, then considering it as an object i n B Y with T := ra(y), we have Z kers, where

rT

( X o , X 1 ) -+ Xo,X1. It was shown in Proposition 2.1.6 that in this + case t h e space in (2.1.17)is linearly isometric t o C ( X ) . It can be easily verified that the operator effecting this isometry generates a BY1-isomorphism s :=

of

-

x’ and R ( d ) .

0

If now F : g1-+ B1 is an arbitrary covariant functor, then with the help of R it can be “transferred” t o BY1 via the formula d

F := F O R . Thus, formula

(2.1.17)defines the sum of a generalized couple. Similarly,

putting

A@)

:=

A ( R ( ~ , ) ),

we define the intersection. It should be mentioned that for the couple sum

d , the

E(d)is only isometric (but does not coincide with C ( 2 ) . The same is

true of

A.

The properties of the sum and intersection that are expressed by state-

(2.1.5)and (2.1.6)for couples have analogs for generalized couples as well. This is considered in greater detail in 2.7.2. ments

Example 2.1.33. A(,L?(J?~,~ N) )L ( z 7 , f q ) . (For the definition o f

2, see

Proposition

2.i.6.)In pariicular, A(d*)N L(Z,&). D. Although this book is devoted t o the theory based on the concept of Banach couple, we shall sometimes encounter more general concepts, viz. Banach families, as well. We shall give several definitions and results which are essential for the further analysis.

Interpolation spaces and interpolation functors

110

Let A be an arbitrary set o f indices. Definition 2.1.34. Banach spaces X, space

a E A, form a Banach family if there exists a Banach

W such that X,

(2.1.19)

&

W

If X := (X), intersectaon

,

a€A .

is a Banach family, the concepts of its sum C ( X ) and

A(X)will be introduced

as follows.

Definition 2.1.35.

The sum of a family

X

is the term applied t o a Banach space X such that

(b) If for a certain Banach space

then

x

1 L--)

Y

we have

Y.

Changing the direction of embeddings, we obtain from here the definition of the intersection of the family

X.

The sum (embedding) is obviously defined unambiguously. Proposition 2.1.36. The sum and the intersection of a Banach family exist.

Proof. (a)

Let us consider the set C ( X ) of those elements x in W [see

(2.1.19)] for which there exists a representation

x=

C

x,

,€A

having the property

,

x, E X ,

,

(convergence i n W )

111

Banach couples

It can be easily seen that C ( X ) is a linear space. We put

and show that a norm on

C ( X ) , which makes it t o a Banach space, is thus

defi ned . For this purpose, we consider the operator S :

@l(~,),E~

4

W defined

on the elements (X,),€A with a finite number of nonzero coordinates by the formula S(3a)aEA

=

Za

*

Since in view of (2.1.19) we have

S extends by continuity t o the entire Il-sum. We now establish, as was done J of the space C ( X ) onto

in Proposition 2.1.6, that there exist a bijection

( & ( X , ) ) / k e r S for which the norm J, in the factor space is equal in magnitude t o (2.1.20). We have thus established that C ( X ) is

a factor-space

a Banach space. Here the fulfilment of properties from Definition 2.1.35

directly follows from t h e definition o f C ( X ) .

Let us further consider a linear space A(X) consisting of those z E

x, for which

We leave t o the reader the simple verification o f the fact that (2.1.21) defines a Banach norm and that A ( X ) has the properties indicated in Definition

2.1.35. Having defined the sum and the intersection, we can now introduce the concept of a linear continuous operator acing from the family X := (X,),,, t o the family is such that

y

:=

T(X,)

(Ya)=E~. Namely, T E L ( X , y )if T

c Y,, cy E A, and

E L(E(X),C(y))

112

Interpolation spaces and interpolation funct o m

It can be easily verified that (see Proposition 2.1.9) expression (2.1.22) de-

C ( X ,y ) . T h e class of families (X,),,, and linear continuous maps of these families forms a category which we denote by B A (thus, g = Bto7'}).Its (incomplete) subcategory B t is defined in an obvious way.

fines a Banach norm on

113

Intermediate and interpolation spaces 2.2. Intermediate and Interpolation Spaces

A. Definition 2.2.1.

A Banach space X is called an intermediate space for a couple

A ( 2 ) ~f X

(2.2.1)

L)

C(2)

x’

if

.

0

We shall denote the set of intermediate spaces for the couple x‘ by I(?). Proposition 2.2.2.

If X , Y E I ( 2 ) and X c Y , then X

Proof. Let

~f

Y.

X c Y is closed. Indeed, if { x , , } , , ~c~ X converges t o x in X and t o y in Y , then in view us verify that the embedding operator

J

:

of (2.2.1) x = y. Consequently, by the theorem about a closed graph, the operator

J

is continuous.

0

The following definition contains one of the fundamental concepts of the theory. Definition 2.2.3.

The intermediate space X E I ( f ) is called an interpolation space relative 2 if for any T E L ( 2 ) we have

t o the couple (2.2.2)

T ( X )c X

.

0

L ( 2 ) + L ( X ) given by the formula 7rx(T) = Tlx is well defined for the interpolation space X . Since In view of (2.2.2), the linear operator x x :

TX

obviously is closed, it is continuous. The norm of this operator is called

the interpolation constant o f the space X and is denoted by i ( X ) . Thus, it follows from (2.2.2) that the following “interpolation” inequality is valid:

114

Interpolation spaces and interpolation functors

Definition 2.2.4. The space X is called an esact interpolation space if

i ( X ) = 1.

0

Henceforth, we shall use t h e notation Intm(d) for the set of all interpolation spaces and I t ( 2 ) for the set of exact interpolation spaces for a couple

d.Obviously,

(2.2.4)

Int,(d)

c I(-?) .

In certain cases, the left- and right-hand sides of this expression coincide. For example, it follows immediately from t h e definitions that these sets are identical for

X I . It can be shown t h a t in all other case Intm(d) # I ( 2 ) .

Xo

Examdes 2.2.5. (a)

A(-?> E Int(d). In a more general form, let x := (x,),,, c Int(r?>. Then A ( X ) E Int(-?) as well. Indeed, in view of (2.2.3) and Remark := SUP, / l T ~ l l xI a SUP I I z I I x ~ = I I Z I ~ A ( X ) . 2.1.21, we have IITzI~A(x)

(b) C ( d ) E Int(2). In a more general form, let

X

be the Banach family

in (a). Then C ( X ) E Int(d). Indeed, in view of (2.2.3) and Example 2.1.20, we have

X E I ( 2 ) . We shall henceforth always use the notation X for the closure o f X in the space C(-?). Obviously, X E I(-?) and if, in addition, X E Int(z), then X E Int(2) as well.

(c) Let

(d) By Riesz's theorem 1.5.1,it can be stated that L,p) E Int(L,-), while in view of Theorem 1.10.5, Lp(+ E Intm(LF). We recall that here p[9) := 1-9 0 < 19 < 1, r being arbitrary and L,-:= ( L m , L p l ) , Pa

Lg,t :=

+ $,

(Lpo.ro 7

,rI

1.

Intermediate and interpolation spaces

115

The relation between intermediate and interpolation spaces is established in Proposition 2.2.6. For X E

I ( 2 ) given, there exists

a maximal (minimal) exact interpolation

(Xmin) contained in it (containing it).

space, X ,

Proof. Let us consider t h e Banach family X

consisting of those X E Int(2)

t h a t are continuously embedded in X. Then, according t o Example 2.2.5(b),

C(2) E

Int(2)and in view

o f Definition 2.1.35, C ( 2 )

-t

X . Moreover,

according t o the same Definition 2.1.35, for any X E Int(2) embedded into

X we have X existence of, X ,

&

C ( X ) . Consequently, C ( X ) coincides with.,,X,

The

is proved in a similar way. 0

It will be shown later [see 2.3.181 t h a t if X E

Intm(2),we have Xmin =

XmX. Therefore, the following corollary is valid. Corollarv 2.2.7. +

If X E Int,(X),

there exists an equivalent norm of X, converting it into an

exact interpolation space.

Proof. In this case, X

2

X,,

(= X-).

0

This corollary makes it possible to confine any theoretical analysis only t o exact interpolation spaces. We shall proceed precisely in this way. The interpolation inequality in Example 2.2.5(d) is stronger than (2.2.3).

It is expedient t o fix similar situations for the further analysis. Namely, l e t 'p :

R+-+ R+be a

concave nondecreasing function and p*(tO,tl) :=

to'p(t1 /to). Definition 2.2.8. The space X E

Intoo(z) is o f interpolation t y p e

'p

if for a certain constant

7 > 0 and for all T E ,C(x') the following inequality is satisfied:

Interpolation spaces and interpolation functors

116

0 In particular, if

cp(t) = t9, 0 < 29 < 1, then X is said

to be of 19 power

type. We shall denote the lower bound y i n (2.2.5) by irp(X).When z,(X)

= 1,

the space is of an ezact interpolation type 9. Thus, L,(s) is of exact, and L,(,+ o f inexact interpolation power type 19 for the corresponding couples.

I ( 2 ) and Int(2) the subsets consisting o f regular spaces. For this purpose, we denote by Xothe closed subspace of X obtained by closing the set A ( 2 ) in X , where X E I ( 2 ) . We distinguish i n the sets

Definition 2.2.9. A n intermediate space X is called regular if A ( 2 ) is dense in X (i.e.

x = XO) 0

W e denote the set of regular intermediate (exact interpolation) spaces for the couple

d

by

p(2)[respectively,

by Into(X)].

T h e following notation will also be used:

(2.2.6)

2'

:=

(X,",X:) ,

where Xi0 is equal to the closure o f

A ( 2 ) i n X:, i = 0,1, i n accordance

with what was said above. Definition 2.2.10.

A couple x' is called regular if 2 ' =d 0 Similarly, the intermediate space X E

I ( 2 ) is regular if Xo= X.

The following obvious statement describes the basic properties of regu-

larization, i.e. the operation of transition

to an object marked by a circle.

117

Intermediate and interpolation spaces Here X, Y E Int(2). Proposition 2.2.11. (a)

Xo is a

closed subspace o f

X , which is an exact interpolation space if

X possesses this property." (b) (X0)O= Xo, (c)

+ (2')' = Xo.

x A Y* x o

z YO.

It would be useful for the further analysis t o have formulas for calculating the intersection and the sum of a couple 3 '. For the intersection, we obviously have

(2.2.7)

A(..$')

= A(x')

.

However, the corresponding result for the sum is less trivial. Namely, t h e following proposition holds. Proposition 2.2.12.

C(2J) = q q o .

Proof.We need the following auxiliary statements. Lemma 2.2.13.

X i n xl-i= x,O_~ (i = 0, i).ll -

Proof.Since the closure of A(&

in Xl-; is contained in A(r?), we have

x,o_~ A xl-in ~ ( 2A)xl-i n x i , and we have t o prove the inverse embedding. Suppose that z E XI-i Then for any

E

> 0 there

exists

2,

E

n Xi.

X i , such that

"More generally, if X , Y E Int(x') and X C Y , then the closure Closy [XI of the set X in the space Y belongs to Int(X'). "Recall that denotes the cloaure of X in C ( x ' ) . Here X E Z(x').

x

118

Interpolation spaces and interpolation funct o m

Consequently, we have the following representation: 5

- 5, = z;

+ zf ,

5;

E xi

xi 3 z; + 2, = 5 - zf-1 E x1-i so that these elements belong t o 1 1 5

- (2,

This means t h a t

5

+ 5f)llXl-,

A(x’)

and

= 115fllx,

E

belongs t o t h e closure o f

. A(2)

in

Xi.

0

Lemma 2.2.14.

If z = z o

+

4

5 1 , 2;

any element y E

E X i , i = 0,1, and z1 @ Xf, then for any couple Y and there exists an operator T E L ( 2 , P ) such that

Proof. In view of Lemma 2.2.13,

t h e fact that zl@ X f implies that XI does

not belong t o the closed subspace X o of the space C ( 2 ) . Then, according t o the Hahn-Banach theorem, there exists a linear functional f E

C(I?)* for

which

It remains t o put T := f 18y. Then T(X0) = (0) c XO and T(Xl) = R y c X1, so that T E L(x’,?) and T(zl) = f(z)y = y . 0

Let us now prove the proposition. Obviously,

E(x’0)

A

C(x’)O,

Intermediate and interpolation spaces

119

and we have t o prove only the inverse embedding. For this purpose, it is

+

if x E E ( d ) O and x = z1 5 2 , x; E X ; , then x; E X:, i = 0,l. We suppose that the opposite is true and that, for example, 31 E X1\Xy. Then, according t o Lemma 2.2.14, 2 1 = Tx for some operator T E ~ ( 2 Since ) . in view of Proposition 2.2.11(a) c(Z)Ois an interpolation space and x E E(d)O, we have x1 = Tx E E(d)O. But

sufficient t o verify that

then, by Lemma 2.2.13,

and we arrive at a contradiction. 0

Along with regularization, the operation of relative completion will play an important role. Let us consider this operation. Definition 2.2.15 (Gagliardo). The relative completion of a normed space X in a normed space Y (continuously) containing X is the term applied t o a normed space whose unit

ball is equal t o the closure o f the unit ball B(X) o f the space X in t h e space

Y. 0

Denoting the space obtained by XC*', we remark that by definition it coincides as a set with

UnE=

n c l o s y [ B ( X ) ] . It can easily be verified that

the Minkowski functional o f the set c l o s y B ( X ) on an element z E equal t o inf{,,) SUP,,=N converging t o

x in Y . Thus,

Nhas t h e same meaning as above. where ( z ~ ) ~ ,c=X Let us agree t o use for X

(2.2.9)

X" := X"*'(')

Y

is

)lxnllx,where { x n } , , , =c ~ X denotes a sequence

,

E I ( 2 ) the following notation:

120

Interpolation spaces and interpolation functors

as well as the notation

2" :=

(2.2.10)

(X,",XE) .

We shall call t h e operations (2.2.9) and (2.2.10) the relative completion of the corresponding objects. Definition 2.2.16. A couple x' is called relatively complete if x" = X E I(d)is called relatively complete if X " = X .

d.

Similarly, a space

0

The set of all relatively complete intermediate (exact interpolation) spaces of the couple

2 will be denoted by I " ( 2 )[accordingly, Int"(x')].

T h e basic properties o f t h e relative completion operation are contained in the following Proposition 2.2.17. Let

X,Y E

A

(a) X

I(d).Then X " ; here X" E I(x') and, if X E Int(x'), then X' E Int(x') as

well;

(b) ( X ' ) " = X ' ; (c)

c

L Y + X " L Y";

(d) X c E C ( I ? ) + X

E C(2).

Proof. (a) In view of (2.2.8), we have

where 7 is the norm o f the embedding operator of X into C ( 2 ) . It follows from the same formula (2.2.8) that for z E X ,

121

Intermediate and interpolation spaces

Thus, we have established that prove the inclusion

A(2)

~t

X

~t

C(x'). In order t o

X " E I ( f ) , it remains t o verify that X " is complete.

be a fundamental sequence in X".In view o f (2.2.11), it will be also fundamental in C ( 2 ) . Hence, (2,) converges t o some x in Let

(Z,),~N

this space. For any E > 0 and sufficiently large k and I, the inequality ( I x k - 2111~~ 5 E is satisfied. This means that x k - x1 belongs t o the closure in c(2)of the ball B,(X). Since xk - 21 + x k - 2 in c(X) -#

as 1 -+

00,

llxk - Z/IXC

the element xk

I E , and since E

-2

also belongs t o this closure. Thus,

is arbitrary,

If, in addition, X E Int(2) and T E x in C ( x ' ) we have

(2,)

converges t o

.C(x'),then for

(2,)

2

in X " .

cX

conver-

ging t o

Taking sup for

Consequently,

TI

E

N

and inf for

(x,), we obtain

x cE Int(21.12

(b) Suppose that z E

(X")". Then for some sequence (z,,),,,=N c X " ,

~~x~~= ( x11zn(lxC c)c and

lim

2,k

=2

in C ( 2 )

.

k-m

In view of these and previous relations, one can choose a sequence (z$k))kCm

for all

c

so that

k.

Then z belongs t o

X " , and according t o (2.2.8).

I2The same line of reasoning can also be used t o prove the following more general fact. If X , Y E Int(x') and X LI Y ,then Xcpy E Int(x').

122

Interpolation spaces and interpolation functors

1

( X " ) " L+ X " , and since the inverse inclusion is also true by virtue o f (a), ( X " ) "= X " . Thus,

(c) This property immediately follows from (2.2.8).

(d) Suppose t h a t X

9

C(@. We shall show t h a t in this case X " differs

from C ( 2 ) . For this we require the following Lemma 2.2.18. If a Banach space X is embedded in a Banach space Y and does not coincide with it, the closure o f t h e ball B ( X ) in Y is not dense anywhere in

Y.

Proof.Suppose that the opposite is true and that closy[B(X)]contains the ball Bz,.(zo) of the space Y with the centre z o and radius 2r. Then the

c l o s y [ B ( X ) ]contains the ball B,(O) as well, in view o f the convexity and central symmetry. Consequently, the closure in Y o f the ball B , ( X ) (:= a B ( X ) ) contains the ball B,,(O) o f the space Y . Further, let y be an arbitrary element in Y.We shall show that y E X so that Y = X contrary t o the assumption. Without loss o f generality, we assume that y E B,(O). Then, in view of inclusion B,(O)c c l o s y [ B ( X ) ]there , exists an element z1 E B ( X ) for which IIy - zlIIy 5 r / 2 . Then, for t h e same reason, there exists z2 E B I p ( X ) for which IIy - z1 - zzlly 5 r/4, and so on. Thus, we have constructed the sequence ( Z ~ ) , , ~ Nc X such that

set

llznI(x 5 2-",

nE

N

and y =

C

2,

inY

.

nEN

Since the series y E

C

llznllx converges,

C

z, also converges in X , and hence

x.

0

Let us apply t h e lemma t o the spaces X and C ( x ' ) . Then the closure of

B ( X ) in C ( 2 ) is not dense anywhere i n C(x'). But as was mentioned earUnEmclosE(m,[B(X)]. Therefore, for X " Z C ( X ) we would obtain a representation of C(x') in the form of the union of a lier, X " coincides with

Intermediate and interpolation spaces

123

countable family o f sets that are not dense anywhere. Since C ( 2 )

IS '

com-

plete, it is in contradiction t o the Baire category theorem. 0

Remark 2.2.19. Some other properties of relative completion associated with duality are described in Sec. 2.4. For the further analysis, we shall need formulas for calculating the intersection and the sum of the couple (2.2.12)

C(2") = C ( 2 )

2'. Obviously,

.

The following proposition is less trivial. Proposition 2.2.20.

A(??)' = A(Zc).

Proof.We require Lemma 2.2.21.

and

Ilzllx: = sup t - ' K ( t , z ;

2).

t>O

Here the K-functional of the element z E C ( 2 ) is defined by the formula (cf. Definition 1.10.6) (2.2.13)

K ( t , s ; 2) :=

inf

+ IIZIJIX~} .

(Ileollxo t

x=xo+aq

k f . It should be noted that in view of (2.2.13), we have (2.2.14)

t-'K(t,z;

x') = K ( t - ' , z ; x'T)

124

Interpolation spaces and interpolation functors

so th at only the first statement has t o be proved. Suppose t h a t z E (Z,,),~:~V c

llznllxo= Ilzllx;, Then for an arbitrary 6 E.

nE

PV

and

in C ( 2 )

z = lim z,

X ; , i = 0,1, such

(2.1.4)], there exist zb

- 2, =

Then z = (5,

+ 2:)

+ and + I:and hence 2 ,

~ ( t ;,2)5

5 Since E is arbitrary,

(2.2.15)

.

> 0 and sufficiently large n,we have 112 - z n l l z ~<~ ~

This means that according t o the definition of the norm i n

2

Xk and

X o is such that

1

that

IIz:/l~o

+ 11zk11x1 <

IIzn + ~ O ~ I I X +~ t ~lzfll5 11~11x;

C ( X ) [see

+ (1+ t ) E

IIICnIIxo

E

.

+ (1+ t

) I ~

*

it follows hence that

K ( t , z ; 2)5 1Iz11x; .

sup t>O

Conversely, suppose that X and a given

E

> 0, there 0

z = 2,

+ z,

1

-+

:= suph’(t,z;

exist zb

X)<

00.

Then for

t

:= n

E X ; , i = 0,1, such that

Ilz”,Ix = n llz!,llxl I X + E .

and

Consequently, we have (2.2.16)

1 1 2

- e:llc(a,

5

11411x1 I X +ne + O -+

so th at ( Z : ) ~ ~ N converges to

I in

C(X).

Since according to (2.2.16) and (2.2.8)

II”llx0. ISUP I1411xoI+ E , n

in view o f the arbitrariness of

E

we obtain the inequality inverse t o (2.2.15).

0

W e can now prove the proposition. Since the embedding

125

Intermediate and interpolation spaces

(2.2.17) A(2)'

A

A(..?')

is obvious, only the converse has t o be proved. Suppose that 2

E A(x").

By the lemma,

114la(a.,

= m=(Ao,

XI> > +

:= s u p t - ' K ( t , z ; X),i = 0,l.

where A;

Hv there exist elements xi,yh E X i , i = 0,1,

By (2.2.13),for a given n E such that 0 2 = 2 : + z , =1Y , + Y ,

Here E

1 7

> 0 is an arbitrary fixed

number.

Then we have

and, moreover, 115

-

(4- Y:)llc(R)

Thus, the sequence (2:

(2'2'18) 11511A(aF

- y,?,)

I IlY:llxo + 114llXl I E A(..?) converges to

: . 11

I

- Y:llA(y)

L

A0

+ +2E + o . A1

72

i n C ( i ) , and hence

*

n>N

But since

: . 11

+ IIY:llxo

- Y:llxo

5 Il~:llxo

- Y:llxl

= 112: - YillX 5 A 1

A1 + E

I A0 + E + n

and, similarly,

: . 11

the right-hand side of

+E , +E + n A0

(2.2.18)does not exceed max(Ao, A,) (2.2.18)leads t o

Since E and N are arbitrary,

+E + O(l/n).

126

Interpolation spaces and interpolation functors

and thus the embedding inverse to (2.2.17) is proved. 0

In view of what was proved above, the operations of regularization and relative completion have the following property:

However, these operations can be applied alternately and lead t o a seemingly unbounded sequence o f new couples. As a matter of fact, only one new couple can be obtained i n this way. This follows from the statement t h a t will be proved only in Chapter 4 (see 4.5.15).

Proposition 2.2.22. (20)C

E

( 2 C ) O .

Corollary 2.2.23. (a) if

2 is regular, 2"is regular as well. +

2 is relatively complete, X o is also relatively complete. Proof. We have (2")' S (do)'= d cso that I?" i s regular.

(b) If

The second

statement is proved similarly. 0

The successive application of the above operations t o intermediate spaces may lead t o a certain confusion.

For example, in contrast t o +

(r?")',

where the operation is applied t o the new couple X " , for X E I ( X ) these +

operations are applied t o the same couple X . Hence it i s surprising that Proposition 2.2.22 does not hold for intermediate spaces. This is confirmed by

Intermediate and interpolation spaces

127

Example 2.2.24. Let p be a prime number and let ep :=

epn :=

I

1

,

n=p;

p-'

,

n = p k , ICE

0

9

n#pk.

be such a sequence that

nV\{l);

We shall consider the normed space m of sequences of the form z = C a p e p , where

Since the Banach space q, (consisting of sequences with the norm ~ ~ z := ~ ~ c o rnaxnEmlxnl, which converge t o zero) is also continuously embedded into

+

l ~ ( ( n - ' } )we > , can consider the sum m co. Let us now consider the couple := ( X o , X l ) ,where X o := m +Q,X I := Zl({n-z}), Let us show

x'

t h a t this couple is regular. Indeed, if

xn is t h e characteristic function of the

..., n} and z E Il({n-2}), then zxn E Q and

set {1,

X1, and since X o A(x') (= X O )is dense both in X Oand X1.

for n +

00.

Thus, X o is dense in

L)

X1, we see that

Further, we show that

Xg = I , .

(2.2.19)

Since it is obvious that X o

A ,I

-

X1, we have Xg (l,)', where the right-hand side contains the relative completion of I , in X1 := I I ( { ~ - ~ } ) . However, it can be easily verified that I , is relatively complete in this space. 1 Hence Xg L) I, and it remains t o prove the inverse embedding. Suppose

that

I 1. Then IlzXnIl~., 5 1, and IIz-zXnllxl 5 Ck
]IZ/II,

+

as n -+ (2.2.19) is proved.

(X;)" does not coincide with (X:)". Since X O , we only need t o establish that A(x') = X o is not dense in

Let us now verify that

X:

=

Interpolation spaces and int erpofation funct o m

128

X: (=

Zoo).

For this purpose, we take an element 2 in,,Z

to the characteristic function o f the set

E

:= {6';

which is equal

k E N}.Since the

elements in m have zero coordinates numbers not equal to the powers o f a prime number, for any z E

Xo := m

+

Q

we have limk,,

26k

= 0.

Therefore,

distrm(2,Xo)= 1 and

,

X o is not dense in XA. Thus, we have proved that

(X3'

# (XE)O

and the required example has been constructed. 0

Remark 2.2.25.

It will be useful for the further analysis t o point out one more property of the regular ordered couple

d just constructed.

(X;)' and Xo do not coincide. Moreover, the nomrs 11 . Ilx0 and 11 1 1 ~ ; are not equivalent on XO, so that X o is not even closed in X:. Indeed, l e t the above norms be equivalent on X O . Since m n co = {0}, T h e spaces

we obtain for any prime numer p

llep - ePXPllXo = IlePllrn

+ IlePxPllu,

=2

(here we took into account that ep E m while e p X p E co). Consequently, due to the equivalence of the norms mentioned above, we have

But Ilep - epXpJ(I, = l / p -+ 0, and we have arrived a t a contradiction.

It should be noted that the relative completion operation is also applicable for a normed space N e

A(d) continuously

embedded i n C(I?). In

this case, N' is a Banach space belonging to I ( @ , N position

1

N' (cf. Pro2.1.18),the relative completion also exists for embedded normed L)

129

Intermediate and interpolation spaces

spaces. Obviously, it depends on the enveloping Banach space. An intermediate position between these two types of completion is occupied by the

Cuuchy completion which we shall sometimes use. Definition 2.2.26.

If a normed space N is continuously embedded in a Banach space X , i t s Cauchy completion is the term applied t o the set of z E exists the fundamental (in

X for which there

N ) sequence ( z n ) n Ec~ N , converging t o

z in

X. 0

We denote this set by N"' and put

The norm

N"" can obviously be also defined as follows:

(2.2.20')

llzllpc :=

lim ] l z n ] l.~

inf

n+m

(Zn)nqN

Hence it follows immediately that if N is a Banach space, then N"" = N . In particular,

(N"')"' = Nac.13It also follows from the definition that N"" does

not depend on the enveloping space

X. In other

words, if we provisionally

denote the space in Definition 2.2.26 by Nac7x,for any Banach space Y e N

N"",', which leaves N intact. The relation between completions N", N"' and Ncvx leads t o

there exist an isometry o f Naclxonto

Proposition 2.2.27. (a)

N"C

A

NCJ;

(b) N" coincides with NaC,i.e. these spaces, which leaves

there exists an isometric isomorphism of

N intact if and only if the following condition

is satisfied. 13We leave to the reader the simple verification of the facts that NaCis a Banach space and that N L Nacc,X .

130

Interpolation spaces and interpolation functors

(A") Every sequence that is fundamental i n N converges to zero in X , converges t o zero i n

N as well.

Proof. (a) This property is obvious.

(b) L e t cp : N" (Z,),~N

--t

N"' be the isometry specified in the statement and N and converging to zero in

be a sequence fundamental in

X . W e denote by

N " , which is defined by (z,). Let [z] E N" be the image of the element z E N under the canonical embedding of N into N". Since (z,) converges to zero i n X , i n view o f embedding N"" ~f X , the same is true for N"" as well. Consequently, y the element o f

p ( y ) = n-oo lim (~([z,]) = lim z, = O (convergence in n-oo lim

N""). Consequently, y = 0, and hence

IIYI~No

11znIlN =

=0

*

Condition (A") is thus satisfied Let us now suppose that condition (A") is satisfied and that z E N"". Then there exists a sequence (z,)

cN

converges to z i n X . If (y,) (2,

- y),

cN

which is fundamental i n N and

is another sequence o f this type, then

is fundamental i n N and converges to zero i n X . Therefore,

it follows from ( A N ) that

112,

- ynllN -+ 0 as n

-+ 00.

Thus, all such

sequences define a single element i n N " . We denote this element by cp(z). Since a stationary sequence corresponds to the element

2

in N ,

y ( z ) = [z]. Furthermore, a fundamental sequence in (2,) c N , which

defines a given element y in N " , converges in X t o a certain element z E N""so that cp(z) = y and cp is a surjection. Finally, taking (z,) so that

= liq-,oo

llznllN, we have

which means that cp is an isometry. U

cN

Intermediate and interpolation spaces

131

Concluding the discussion o f the Cauchy completion, it should be noted that this operation was implicitly encountered while defining the sum of the

C (X,),,, coincides with Nac,where N is the linear space formed by finite sums C zai Banach family (Xa)aeA. Indeed, it can be easily verified that

and supplied with the norm

IlYllN

inf

:=

u=C Zq

(cll%llxJ

*

B. Let us consider the arrangement of interpolation spaces in a couple. T heorem 2.2.28 (Aronszajn, Gagliardo). If X E Int(2), one of the following statements is valid:

x

s

(b) X

~t

(a)

(c)

Xi

4

~(2); C(2)';

X

c--t

X;,

i = 0 or 1.

b f . We require Lemma 2.2.29.

x; = + x;-;(i = 0,l). xi

-

Proof. The embedding

xi + x,o_i 1 x; follows from the fact that each term on the left-hand side is contained in the right-hand side. Conversely, suppose that z E

xi-1

+

Then for any

z l - i , x j E X j , j = 0,1,we have E X; X ; = X;.Therefore, according t o Lemma 2.2.13, E XI-; n X;= Xy-;. Thus, x is contained i n X ; + x Y - ~ .

representation of z in the form z; 51-i

Xi.

=

2

- x;

+

0

Let us now suppose that the interpolation space z E Int(2) differs from

C ( 2 ) and is not contained in C(2)'. consider its representation

We take an element z E X and

132 (2.2.21)

Interpolation spaces and interpolation functors

+ 21 ,

z = I0

z; E

x;.

Suppose that among these representations, there exists such that Then, by Lemma 2.2.14, for any y E for which

Tx = y.

$1

@ X:.

X1there exists an operator T E L ( 2 )

Since X is an interpolation space, y = Tx E X as well.

Consequently, as y is arbitrary, we have X1LS X.A similar line of reasoning can be applied t o t h e case when among t h e representations (2.2.21), there

xo $ X,O(in this case, Xo c+ X).If, however, for any z E X and any of its representations (2.2.21) we had z; E Xi”, i = 0,1,

exists one such t h a t

then in view of Proposition 2.2.12 we would obtain z =

20

This means that

+ 21 EO, +x,o= C ( 2 ) O .

X

~t

E ( d ) O in spite of t h e assumption about X

Thus, there always exists z E

and for a certain j we have zj E x j L)

X

for which

Xj\Xj” so that

x

If, however, this embedding is satisfied for any x E

X

and any of its repre-

sentations of the type (2.2.21), zl-; can no longer belong t o Xl-;\X:-i. Indeed, if such an element z existed, we would also have the embedding

X I - ; ~t X . which leads t o X;+XI-; = C ( x ’ ) L) X in contradiction to the assumption about noncoincidence of X and C ( 2 ) . Thus, for any z E X in the representation (2.2.21), we have xi E X; and zl-; E X:-,. Hence x E X, X:-;= X;according t o Lemma 2.2.29.

+

Combining embedding

Xi

L)

xavxLtxi.

X

with Proposition 2.2.2, we obtain

133

Intermediate and interpolation spaces Corollary 2.2.30.

If z E Int(x') and is closed in C ( x ' ) , then X coincides with one of the spaces ~ ( x ' )c(Z>O , or Xi, i = 0,1. 0

Corollary 2.2.31.

If A(x') is closed in Xi, i = 0,1, then

Int(2) =

{x0, xl, ~ ( x ' ~) ,( x ' ). )

Proof.According t o Theorem 2.2.28,

it is sufficient t o verify that A(x') and

Xi, i = 0, 1, are closed i n C(2). Since A(x') is closed i n Xi, and in view of Lemma 2.2.29, we have

Further, usingthe property of closure, we obtain -

~(2 = xo ) n X, = xon x,

(=

~ ( 2 .) )

Let us now consider an important problem concerning the extent t o which a couple is determined by its interpolation spaces. It is natural t o expect

that the following conjecture is true. Conjecture 2.2.32.

If Int(x') = Int(?), then

x' 2 ? or x"

?.

0

Here, we shall confine ourselves t o Theorem 2.2.33 (Aron~zajn,Gagliardo). Suppose that

?

:=

(A(l?),E(x')) and Intoo(x') = Intoo(?). Then either

x' or ZTcoincides with ?. Proof. It is sufficient t o prove that if

134

ihterpolation spaces and interpolation functors

(2.2.22)

2

2*

and

then X i @ Int(f) for

?,

i = 0 or 1.

Let us consider two cases: (a)

A(x') is closed in each X i , i = 0 , l ;

(b)

A(x') is not closed at least in one o f them, say, in X o . According t o Corollary 2.2.31, in the case (a) Int(?) = {A(x'), C ( x ' ) } ,

and in view of (2.2.22), at least a single X ; does not belong t o Int(?). In

A(x') is not closed in X o that the norms of these spaces are not equivalent on A(r?). Indeed, if the converse were true, for some y > 0 and all 5 E A(x') we would have t h e case (b), it follows from the fact t h a t

11~11x15 7

ll4lxo

7

c A(2)

and from t h e fact that ( X ~ ) , , ~ N

is fundamental in X o it would

follow that this sequence is fundamental in XI. The limits of this sequence in X o and in X I are equal, so that i t s limit in X o belongs t o X o n X , = A ( 2 ) . Consequently, A ( 2 ) is closed in X o in spite o f the assumption. Since the norms of the above spaces are not equivalent, there exists a sequence (2,)

c A(X) such that

112nllxo 5 , 12 E . ll~nllxi= IIZnlla(2) and Il"nllc(~)I IlZnllxo SO that llznllA(R)

Then

(2.2.23)

and

=

IIZnllA(2)

=1

7

I l Z n l l q ~ )I l / n

7

ll~nllxl=

1

7

.

n E

Let us now suppose that X ; E Int(f). This leads us t o a contradiction. indeed, let us consider a functional

fn

E C ( g ) * such that for a given

2E

XI, we have f,,(5) = K(n,Z ; F) and

If,,(s)l 5 K ( n , z ; F) , z E C ( Z )

.

Since the function I + K ( t ,2 ; f )is positive, homogeneous and subadditive [see (2.2.13)], the existence o f

Let us consider the operator from (2.2.23). have

fn

follows from the Hahn-Banach theorem.

T,,:=

2, @

f,,,where (2,)

is the sequence

According t o relation (2.2.23) and the definition of

T,,, we

135

Intermediate and interpolation spaces

It should be recalled that (YO,Y1) := (A(T), C ( 2 ) ) . Consequently, llTnllp

5

1, and since

X1

is an interpolation space, we

obtain, taking into account (2.2.23)

Thus, we have established that

sup K ( t ,f ;

f-) < , 00

t>O

and in view of Lemma 2.2.21 and Proposition 2.2.20, we get +

i E Yt := A ( 2 ) ' = A(Z') + X i

.

As 5 E X1 is arbitrary, it follows hence that X1 L) X; and consequently C ( x ' ) := Xo+XI L-) Xi. Then X ; E C ( x ' ) , and in accordance with Proposition 2.2.17, C ( 2 ) coincides with Xo. Then A(x') = C(x') n X1 +

coincides with

X1as well, and hence the couple X

coincides with the couple

? despite (2.2.22). Thus, if (2.2.22) holds,

Xi # I n t ( f ) at

least for a single

i.

0

Remark 2.2.34. Thus, if A ( 2 ) is closed in C(x') or is not closed in any X i , i = 0,1, the +

spaces Xi, i = 0,1, are not interpolation spaces for the couple Y := differ from ?). (A(x'), C ( x ' ) ) (of course, provided that x' and However, one of the spaces X; can be an interpolation space in the couple ?. Let us show that Xi E Intm(f-) if and only if A ( d ) is dense in X; and closed in XI-;. Indeed, if A ( 2 ) is not closed in Xl-i, then, as was proved above, Xi # Intm(f-). If, however, X1-; is closed in C ( f ) , then in view of Lemma 2.2.29, X;= X:-; Xi = A ( 2 ) X; = Xi, i.e. Xi is also closed.

+

+

Interpolation spaces and interpolation functors

136

But since an interpolation space in Intw(Y) closed in

C(?) coincides either with C(?) (= C(x’)) or with C(?)O = yo+ qli = X p (see Corollary 2.2.30 and Proposition 2.2.12), in this case Xi = Xi”. Thus, the property of A(x’) being dense in Xi is a necessary condition, and, together with the property of A(@ being closed in X I - , , it is also a sufficient condition of the embedding Xi E Intoo(+?). 0

C . Let us briefly introduce “relative” interpolation. We call a triple a set

x ’ , X with X in I ( 2 ) .

Definition 2.2.35.

A triple 2 , X is called an interpolation triple relative to the triple for any operator (2.2.24)

?,Y

if

T E L ( z , ? ) we have

T ( X )c Y .

Using the closed graph theorem, we establish in this case the validity t o an “interpolation” inequality similar t o (2.2.3): (2.2.25)

llTlXIlX,Y

5 i ( X , Y )IITlln,p

*

Here the interpolation constant i ( X , Y ) i s equal t o the norm of the operator defined by the formula

TX

:

L(Z?,?)+ L ( X , Y ) .

Definition 2.2.36.

A triple 2 , X is called an ezact interpolation triple relative to the tripZe P, Y if i(x,Y >= 1. 0

Henceforth, we shall denote by Int,(x’, Definition 2.2.35, and by

?) t h e set of spaces { X ,Y}

Int(l?, ?),those in Definition 2.2.36.

in

Intermediate and interpolation spaces

137

Example 2.2.37. (a)

{A(Z), A(?)}, as well as { C ( f ) , C(?)} obviously belongs t o Int(z,?). ( { X a , Y a } ) a Ec~ Int(d,?), then denoting by X the intersections o f the Banach family ( X , > , , A and by Y those for the Banach family we have, as can be easily verified, { X , Y } E Int(d, P). Of course, a similar statement is valid for the sums as well.

(b) If, more generally,

(c) If { X , Y } E Int(d,?) and the spaces and

Y

X,

are such that

X A X

A p,then {x,p}E I n t ( d , p ) .

(d) In view of Theorem 1.5.1,

for 0 < 19 < 1 and pi

5 q;, i = 0 , l .

(e) Similarly, Theorem 1.10.5 yields

for 0 < 19 < 1.

In the latter two cases, the multiplicative interpolation inequalities are valid for the norm of an operator, which are more strict than (2.2.25).

By

analogy with Definition 2.2.8, for the relative situation also we shall consider only the concept o f interpolation property of 9 - t y p e (or for p o w e r 19-type for cp(t) := holds:

t'). In this case,

it is assumed that the following inequality

As regards 'p*, see Definition 2.2.8.

The spaces X and Y satisfying this condition form a subset Int['](~,?) o f t h e set Int(J?,f). If cp(t) = iff,we simply write Int'(J?,f). Thus, the left-hand side o f (2.2.26) belongs t o Inte(L,-,Lp'), while the left-hand side of (2.2.27) belongs t o IntL(Lp,LF).

138

Interpolation spaces and interpolation functors

Statements (b) and (c) of the example under consideration indicate that among the elements of Int(z,?), there exist “primary” elements t h a t can be used for determining the remaining ones. In order to describe the situation precisely, we introduce an order i n the set Int(2,

{x,Y} 5

{X,P}

if

p) by assuming that

x & X,P C: Y

.

Definition 2.2.38.

A triple 2 , X is called an optimal interpolation triple relative to { X , Y } is a maximal element of the set Int {T,?}.

?, Y

if

0

Thus, it is impossible t o increase

X and decrease Y without losing the

interpolation property. Remark 2.2.39. Henceforth, we will use the same term for a somewhat wider concept obtained from Definition 2.2.38 by replacing Int by Int,

and

1 L,

by

-.

It can always

be easily determined from the context what we are dealing with. In the next section (see Theorem 2.3.20) it will be established that for any element in Int(Z,?), there exists a maximal element majorizing it, so th at theoretically we can always confine ourselves t o the analysis of optimal triples.

I n the same section, some other properties of triples will also be

established. For example, if { X , Y } E Int(d,

X E Int(2) and Y E Int(?).

p) is a maximal element, then

It can be easily shown t h a t this may be

incorrect for those elements which are not maximal. Concluding this section, it is appropriate to give some illustrative examples o f optimal interpolation triples (in the sense o f Remark 2.2.39); see also Corollary 2.3.22. Example 2.2.40

(Calderdn, Dikarev-Matsaev). Suppose that for (2.2.26), p 5 q , where p := p ( 9 ) and q := q(9). Then the triple L , L , is an optimal interpolation triple relative to the triple L,, L ,

Intermediate and interpolation spaces

139

if and only if p = q. If p < q , the former triple is an optimal interpolation triple relative t o

L?, Lqp.

( A . Dmitraev-Semenov). On the other hand, if p; 2 qi, i = 0,1, the triple L,;Lp is an optimal interpolation triple relative t o the triple L,j, L,. Example 2.2.41

140

Interpolation spaces and interpolation functors

2.3. InterDolation Functors

A. The concept o f interpolation space has emerged as

a result of the

generalization of the situation involving classical interpolation theorems (see Chapter 1). The concept o f interpohtion functori4 refers t o a later, “constructive” trend of the theory aiming a t the construction and analysis of the methods in which each couple is associated with a fixed interpolation space. -4

Let us give exact definitions. Let B be the category of Banach couples and B be the category of Banach spaces (see Proposition 2.1.10 and Remark 2.1.11). Definition 2.3.1.

A (covariant) functor F : (a)

+ B is called an interpolation functor if

F ( 2 ) is an intermediate space for

(b) F ( T ) Since

:=

2;

T ) F ( y )for , every T E L ( 2 , f ) .

F ( T ) E L ( F ( z ) , F ( ? ) ) , according t o (b) we have

(2.3.1)

T(F(2)) c F ( f )

L ( 2 , f ) . Thus, the triple 2,F ( 2 ) is an interpolation ?,F(?). In particular, ~ ( 2 is an ) interpolation space of

for each operator T E triple relative t o the couple

x’. Therefore,

(2.3.2)

IITIF(~)IIF(~),F i F( (P- )f ,

where

it follows from (2.2.25) that

5

?) I I T I I ~ , ? 7

i ~ ( z , is ?the ) interpolation constant of the triples under considera-

tion. Definition 2.3.2. The interpolation functor F is called ezact if for all

i&?)

-?,? E 6

51.

14The fundamental concepts of category theory used in this section are considered in 2.7.1.

141

Interpolation functors

It is called bounded if

It will be shown later (see Corollary 2.3.25) that with the help of an F(r?),a bounded functor can be

appropriate renormalization of all spaces

converted into an exact functor. In most of our problems, such a procedure can also be used for any unbounded functor (see Theorem 2.3.30). Therefore, in theoretical analysis we can (and shall) consider only exact interpolation functors. For such a functor, the following inequality holds:

II%(d)llF(R),F(P) 5 IITllR,?

(2.3.3)

t o the category B1 (see

and hence it is also a functor from the category Proposition 2.1.10 and Remark 2.1.11).

Henceforth, we shall use everywhere the term “functor” instead of “ezact interpolation functor”. The class of such functors will be denoted by

JF.

Let us define a continuous embedding of a bounded interpolation functor F into a similar functor G t o mean that G L, G if F ( 2 ) L, G ( 2 ) for all

2 E 6. Proposition 2.3.3. If f

-+

G and i(2)is the norm of the operator of embedding F ( 2 ) -+

G ( z ) , then (2.3.4)

sup R

Proof.If (2.3.4)

i(2)< 00 , is not satisfied, there exists a sequence

such that i(zn) 2 n. Let us consider the couple (2.3.5)

F(r?)L, G ( 2 ) .

x’

( r ? n ) n Eof~couples,

:= $ l ( X n ) n E Then ~.

Interpolation spaces and interpolation functors

142

Further, suppose that I, :

d,

d

-t

and

P,

:

d

-+

d , are the canonical

injection and the canonical projection respectively. Then I, E

LI(2,2?,,),and

and P, E

Ll(Z,,Z)

hence for a certain constant 7 , which does not

depend on n, we have

InIqZ,) E LT(F(-fn),F(y)) 7

(2.3.6)

P,. Obviously, the same is true of the

and a similar embedding is valid for functor G as well. Choose now an element

Il~nIlc(~,) 2

2,

E

F(2,)

12 llxnIlq2,)

7

such that

n E N .

Since P,I, = lg,, in view of (2.3.6) this gives

llInznllq2) 2 YI II~nIlq~,)2 72n llznllF(B,) 2 73n llInxnllF(2) with certain constants "1; independent of n. Since n is arbitrary here, this is in contradiction t o (2.3.5). 0

Let us now define the equality and the equivalence of functors by putting

F=G*F&G

and

GLF,

(2.3.7)

F

G*F

L--)

G and G

-

F .

Corollary 2.3.4.

If F E

JF,there exist

A(d) for any couple

Proof. C :

x'

> 0 such that

F(2)uc)C ( x ' )

2.

: x' --f A ( 2 ) and C ( d ) are functors. Here, in view of Definition 2.3.l(a), we have

It can be easily verified that the maps A -t

A-F-C. 0

constants 6(F), a(F)

143

Interpolation functors Let us now define some operations on the set

3.F. Let

us start with

unary operations. For this purpose, we use the following obvious Definition 2.3.5. Let U be an operation associating t o each space X E Int(r?) a space U ( X ) which also belongs t o Int(2). The couple spondence U F :

r? is arbitrary here.

If the corre-

r? + U ( F ( d ) ) is an exact interpolation functor for any

F E J'F, then U is said t o be functorial operation. 0

Choosing for U t h e operation of closure, regularization or relative cornpletion, we obtain from a given functor

F :

2 +F@),

:

F € 3.F the functors

r? + F ( z ) ' ,

F" :

r? -+

F(2)".

One more unary operation, which will be encountered in the further analysis, associates with F the functor

t F , where t > 0 is fixed.15

Definition 2.3.6. The functor F is called regular if F = p ,and relatively complete if

F = F". 0

Let us now define the sum and intersection of functors. We directly

consider the case involving an admissible family

(Fo)aEAc J'F. This

means that there exist constants 6, ~7> 0 such that (2.3.8)

A(2

&

F,(Z)

4

C(2)

for all couples r? and all a. In the case of an admissible family, C ( F , ( 2 ) )

,€A

and A(F,(X')) are well-defined [see (2.1.20) and (2.1.21)]. Moreover, &A it follows from the properties o f sum and intersection of Banach spaces that the spaces constructed are intermediate for verify t h a t t h e maps of 151t should be recalled that

into

2.We leave it t o the reader t o

B generated in this way have the property

11 . lltx

:= t 11 . IIx.

Interpolation spaces and interpolation functors

144

of functors. Thus, the following definition is correct. Definition 2.3.7.

If (F,),€Ais an admissible family o f functors, i t s sum C a EF,~ is the functor x' + C(F,(x'))aEA,while the intersection A,€A F, is the functor 2 -+ A(F,(-f)),EA. 16 Example 2.3.8. (a) Suppose t h a t P, :

(b) Let

(ta)aEA

C

R

x'

+ X i ; obviously,

P; E J F ,i = 0 , l . Then

be such that

0 < inft, 5 s u p t ,

< 00.

Then it can be easily seen that

Let us suppose further that F , Fo and Fl are three functors. Since is contained in C ( x ' ) ,

(Fo(-f), Fl(-f))

F;(z)

is a Banach couple. Consequently,

F((Fo(-f),Fl(x')))is defined. It can be also easily verified that the correspondence x' -+ F ( ( F o ( z ) Fl(x'))) , is functorial. Thus, we have the space

Definition 2.3.9.

The functor F(Fo,Fl) defined by the formula

161t is worthwhile to note that in the definition of the sum, the left embedding in (2.3.8) is superfluous.

Interpolation funct o m

145

is called the superposition of the functors

F , Fo and F1.

0

Finally, we define the fundamental function of a functor F (denoted by ( P F ) by

the identity

(2.3.9)

F ( ( s R , t R ) ) = pF(s,t)R

( s , t > 0) .

Going into details, we see that any intermediate space of the couple

( s R , t R ) has the form r R for a certain r > 0. Consequently, the space on the left-hand side o f (2.3.9) also has the same form, and the corresponding r is denoted by p ~ ( s , t ) . ProDosition 2.3.10.

The function

pF is nonzero, positive homogeneous and non-decreasing.

Proof. Since the space 0 . R is not an intermediate space for (.El,tB)for 1 s , t > 0, we obtain p(s, t ) > 0 . Further, from the embedding (sR, tR) ( s ’ R , t ’ R ) ,where s’ 5 s and t’ 5 t , it follows that p~(s’,t‘)2 pF(s,t). Finally, t h e operator of multiplication by X > 0 transforms ( s R , t R ) into L)

( X s B , X t R ) and has norm equal t o unity. Hence

V F ( k

wI

XVF(S, t ) .

Applying this equality t o s’ := X-’s and t’ := X-lt, we obtain the opposite inequality. 13

We shall later see (from Corollary 2.3.27) that the converse is also true, i.e. any function satisfying the conditions of this proposition is also a fun-

damental function for a certain functor. Example 2.3.11. (a) It can be easily verified that

146

Interpolation spaces and interpolation functors

(b) Since (ppi(sg,sl) = si

i = O or 1 ,

,

it follows, in particular, t h a t

Remark 2.3.12. Henceforth, t h e function t + cp~(1,t)will also be called the fundamental function for

F

(with the same notation).

B. Let us now consider and important functor (“orbit”) which we will A’ and a E C(A’)\{O} are fixed. To be exact,

use later: Orb,(i; .), where we put (2.3.10)

Orb,(A’;

d ) :=

{ T u ;T E L(A’,Z)}.

Obviously, this defines a linear subspace in

E(d).In this space, we introduce

t h e norm by the formula

Proposition 2.3.13. The correspondence x’ + Orb,(A;

x’) defines a functor.

Proof. The positive homogeneity and the triangle inequality are obvious for equation (2.3.11); we shall verify its non-degeneracy. If z = T a , the functorial nature of C leads t o the inequality Taking inf in (2.3.12)

(Izllc(~)5

IITIIJ;RI l a l l c ( ~ ) .

T,we obtain

11~11c(a)5 Ilallc(A) I140rb.(A,a) .

Consequently, if the magnitude of the norm (2.3.11) turns t o zero, then 2

= 0. Thus, Orb

:=

Orb,(i;

completeness of this space; l e t

d ) is

xy

a normed space.

Let us verify the

llznllorb< 00. In view of (2.3.12) the

147

Interpolation functors

C

+

C ( X ) to some element z. Let us verify that this series has the same sum in Orb also. For this purpose, we choose for each n an operator T, such t h a t series

z, converges in

(2.3.13)

IITnllA,f

L

11znllOrb

In this case, the series

C T,

+ 2-"

and

converges in

zn

= Tna

*

L ( i , x ) t o an operator T , which

rnea ns that

z-c z,= m

1

c

m

T,u=

n>m+l

Hence, and from (2.3.13), we get

Consequently, x = C z, in Orb. Let us now verify that

Orb is an intermediate space for the couple

x'.

The embedding in C ( X ) follows a t once from (2.3.12); it remains only t o

A ( d ) ~t Orb. Let z E A ( 2 ) and l e t f E C ( x ) * be a linear functional for which f(a) = 1 . We consider the linear operator P, : z 4 f ( z ) z ; 1 and llfll =

verify that

~

Ilallqz)

since z belongs t o

A(Z), we have P, E L ( L ; 2)and, moreover, P,a = z .

Consequently,

IlzllOrb 5 IIP,Ih,R 5 llfli

llztlA(d)

)

and we obtain finally

Thus, relation (2.3.3) is satisfied for t h e orbit functor. 0

Interpolation spaces and interpolation functors

148 Example 2.3.14. (a) If

a E A(L)\{O}, then

Orb,(A; .)

(2.3.15)

A

Indeed, i n this case, T a

E A(x') for all T E L ( z ; 2). and the left-hand

side is embedded in the right-hand side. This embedding is a bijection since for each

2

E A ( 2 ) there exists an operator P, E L(A,x') which

transforms a into 2.

(b)

$ A(2). For example, let us veA(2) is not even dense i n the space For this purpose, we show t h a t for all b E A(X)

Relation (2.3.15) is not valid for a rify that in this case the set

Orb := O r b , ( i ;

A).

E Orb, we have a - b = T a for a certain T E C ( 2 ) . Let us TI,(A) in the space L ( A ( 2 ) ) is not less than 1. Since llTll~is not less than the norm of the trace, this would lead to (2.3.16) if we take into account (2.3.11). Since a - b

show that the norm o f the restriction

In order t o obtain the required estimate, let us suppose that, conversely,

IITIAcx)ll< 1.

Then the operator

S :=

llA(x)- TI,(J)

is reversible, and

it follows from the equality a - b = T a that a = S-'b E A(L)),which is contrary to the assumption.

A generalization o f the orbit functor is the functor O r b A ( A ; .), where A E I ( 2 ) . Let us now determine this functor. For this purpose, we consider the family of functors (IlallAOrb.(A'; .)),EA,jol. In view of (2.3.12) we have II2C(lC(X')

5 O A llallA

'

Il"llOrb,(A',f)

7

where oA is the constant corresponding t o the embedding of

A in C ( i ) .

Consequently, the sum of this family of functors is well-defined (see Def.

2.3.7). We put (2.3.17)

Orba(d; .) =

( ( a ( ( A O r b , ( i.) ; .EA\{O)

149

Interpolation functors and study the properties of this functor. Theorem 2.3.15 (Aronszajn- Gagliardo).

(a) The functor (2.3.17) is minimal (by inclusion) among all functors G for which

G(A).

A

Thus, for any such G, we have

OrbA(2; *) A G .

(2.3.18)

(b) If A E Int,(A),

OrbA(2, A'> coincides with the minimal exact interpolation

(c) The space space

A-

(d) OrbA(At;

then

(see Prop. 2.2.6). +

0

)

= OrbAmi,(A;.).

Proof. (a) Let

G be the functor indicated in the formulation. Then for z = T a , T E L ( 2 , z ) and a E A\{O}, we have

where

Taking inf over

all T , we obtain the inclusion

(b) If A is an interpolation space, then for a certain constant a E A, we have

As before, this gives

y A and every

Interpolation spaces and interpolation funct om

150

OrbA(2,i)

(2.3.19)

A .

On the other hand, 1lAa = a, and hence

Consequently, the norm of element a in the space IlallAOrbq(2,i) is equal t o

I l ~ l l From ~ . the definition o f sum, it follows hence that i t s norm

in the space orbA(i,A) does not exceed I l a l l ~ .Thus, the converse is a Iso true :

(c) From Definition (2.3.17). we have

Consequently, if

A E I n t ( i ) and satisfies condition (2.3.21),

we obtain

from this statement and (2.3.19), (2.3.20) (with 72 = 1) that

Thus, OrbA(2,

2)= Amin.

(d) In view of the minimal property of OrbA,,,(i;

.) and the equality

OrbA(A, A) = Ad,, proved in (c), we have

Since A -+

1

Amin, the inverse embedding is also true in view of (2.3.21).

Before applying the results obtained above, we introduce another functor, dual t o the orbit functor. In order t o make this duality explicity, we write t h e norm in OrbA(X;

x') as follows:

151

Interpolation functors

Recalling the definition of the Banach space T ( A ) [see

(2.1.11)],we find

that the right-hand side coincides with the norm of z in the sum of the Banach family (T(A))TEL(2;2). Thus,

which leads t o a new definition of the orbit functor. Apparently, the definition dual t o (2.3.23) will be the one in which the sum is replaced by intersection and the image by inverse image. The inverse image T-'(A) for

T EL(2,x)

is defined as usual: (2.3.24)

"-'(A) :=

{X E

E(d); T XE A } .

Here, as before, A E I ( 2 ) . Next, we put (2.3.25)

llzllT-l(A)

:= rnax { S A

llzllc(a),IITzIIA)

7

where S A is a constant corresponding t o t h e embedding of

A(2) in A , and

verify that T-'(A) is a Banach space. Indeed, if the sequence (z,,)

is fundamental in T-'(A), it is also funda-

+

C ( X ) and converges in it t o some element z. On t h e other hand, (Tz,)is fundamental in A and hence converges in it [or in C ( i ) ] t o an element y. Since T acts continuously from C ( 2 ) into E(L),y = T z and (2,) converge t o x in T-'(A). Let us also verify that T-'(A) E I ( 2 ) . Indeed, in view of (2.3.25). mental in

we have T - ' ( A )

62 L-$

C ( 2 ) . On the other hand, if 6 A is the constant

corresponding t o the embedding o f A ( 2 ) in A , we have

Consequently, we get (2.3.26)

A(2)

-MAT)

T-'(A)

C(d),

Interpolation spaces and interpolation functors

152 where 7 A ( T ) :=

SA

max {llTll~,x, I}.

Then (T-1(A)Tccl(2,~))forms a Banach family, and hence i t s intersection is well-defined. For this we use the notation

Proposition 2.3.16.

The correspondence

2

-+

CorbA(x',i) defines a functor (coorbit).

If

A(A'> # (01, we also get Proof.From (2.3.26) A(2)

%

and (2.3.27) it follows t h a t

CorbA(2;

i) c(2).

Hence c o r b A ( 2 ; i ) is an intermediate space for the couple

let us verify formula (2.3.28). Let z linear functional such that

llfll

E CorbA(2 ; A)and f

x'.

Next,

E C(x')* be a

= 1 and f(z) = 1 1 ~ 1 1 ~ ( ~ , Since . A(i)

and SA is t h e norm of the embedding operator A ( i ) in A, for a given there exists an element a with ~ ~ a =~ 1 ~ for A which ( ~11all~ ~ >

#

{0}

E

>0

6~- E .

Let us consider the operator P z := f(z)a. Since a E A(A), we have

P E L(x',J),and since

Consequently,

and hence

llfll

= 1, we have

153

Interpolation funct o m

It now remains t o verify t h e functorial nature of the coorbit. For this purpose, we confine ourselves t o t h e case A(X) # {0}, and note that the general In case can be obtained in a similar manner. Suppose t h a t S E .C(-f,?). view of (2.3.28), we have

5

~ ~ s x ~ ~ C o r b A ( ?= ,~)

~ ~ T S x ~ ~ A

TELi (?,i)

Let us now consider the statement for the coorbit functor dual t o Theorem 2.3.15. Theorem 2.3.17 (Aronszajn-Gagliardo).

COrbA(*; 2) is maximal in inclusion among all functors G for which G ( A ) A.

(a) The functor

Thus, for each functor, we have

(2.3.29) G

A corbA(.; A ) .

(b) If A E Int,(i),

then

A)coincides with the maximal exact interpolation

(c) The space CorbA(i, space A,,

(d) CorbA(. ;

(see Prop. 2.2.6).

A)= CorbA,,,

(. ; A ) .

Interpolation spaces and interpolation functors

154

Proof.We consider only the main case A(A) #

(0) and leave the conside-

ration of the general case to the reader. (a) Let

G be the functor indicated in the formulation and T E L l ( i , i ) ;

then

5 IITzlI~(L)5

llTsllA

IIsllG(,f)

.

Taking sup with respect t o T and considering (2.3.28), we obtain (2.3.29).

(b) If A E Int,(A),

then for a certain constant

?A

and all a E A, we have

Taking sup with respect to T E L l ( A ) ,we obtain the embedding (2.3.30)

A

2

CorbA(i,.i)

.

On the other hand, in view of (2.3.28), we get

Hence the following embedding i s also valid: (2.3.31) (c)

CorbA(/i,/i)

A

A.

-

It follows from (2.3.28) that the following statement is valid:

(2.3.32)

A

A

A

+ Corbn(.; 2)

1

CorbA(.; A ) .

Consequently, if A E Int(2) and satisfies condition (2.3.32), then this statement and the embeddings (2.3.30) (with yd = 1) and (2.3.31) give

A = Corbn(i, 2)

A

c o r b A ( i ,2)

A

Consequently, (d) The proof is similar t o that of Theorem 2.3.15(d). 0

A

.

Interpolation func t ors

155

Let us now apply Theorems 2.3.15 and 2.3.17 t o establish certain new properties of interpolations spaces. Namely, the following theorem is valid. Theorem 2.3.18 (ATonszajn- Gagliard 0 ) . (a) If A is an exact interpolation space for the couple Athere exists a functor

which is its generator. Among all such functors there exist a maximal and a minimal functor. (b) If the space A

E Int,(i), there exists an equivalent renormalization of it which makes A into an exact interpolation space.

(c) If A

+

E Int,(A) then A,,

= Adn.

(d) If A , B E I n t ( i ) , where A is embedded in B but not closed in it, there exists an infinite number o f different interpolation spaces between A and

B.

Proof. (a) It is sufficient t o take

OTbA(i;

0

)

or CorbA(.; 2) as such a functor. In

view of (2.3.18) and (2.3.29), any functor G for which

G(2)= A

lies

between these functors. (b) Since according t o the above proof

the norm in A can be transfered from one of these spaces. (c) In view of (b), we can assume A t o be exact.

A)

OrbA(&

In this case, A =

[see (2.3.19) and (2.3.20)], and similarly + +

A = CorbA(A, A) = A,

.

1

B and is not closed in B, there exists an element b E clos,(A)\A. Let [b] denote a one-dimensional subspace R b with the

(d) If A

~t

norm induced from

C ( A ) , and let X

:=

~

'lb'lB . In view of the choice

Ilbllc@) of X [ b ]

1 L)

B, this means that

Interpolation spaces and interpolation funct o m

156

A

&

A

:= X [ a ] + A

1

B

Hence in view o f (2.3.21) we obtain

A

Orbi(L,L) E Int(2) and lies between A and B . It was mentioned above that A # A because of the choice of b. It now remains t o verify that A # B . For this purpose, we proceed in the same way as in Example 2.3.14(b) and find that for any a E A Consequently,

:=

llbll~ := [lbllorba~x,~~ apparently coincides with the right-hand side of the above inequality, this means that the element b

Since in this case

does not belong t o the closure of

A

in this space in the topology

A. On

the other hand, b belongs t o the closure o f A in B in the topology of B . Thus,

A

; A ;

B ; moreover,

either

A is not closed in A or A is

B (otherwise A would be closed in B , which is contrary t o the assumption). For example, suppose that A is not closed in B ; replacing A, B by A,B , we construct another interpolation space lying between A and B . and so on. not closed in

U

Corollary 2.3.19. If A ( i ) is not closed in

C ( A ) ,the set Int(Z) is infinite.

0

Recall that if this is not true, Int(i) cannot contain more than four spaces [see Corollary 2.2.311. Using the above properties o f orbit and coorbit functors, let us prove

the properties of optimal interpolation triplets formulated in Sec. 2.2 (see Definition 2.2.38).

157

Interpolation functors Theorem 2.3.2Q. (a) The triple

triple

i , A is an optimal interpolation triple with respect t o the

g,B if and only if

In particular,

A E Intm(i), B E Int,(Z),

interpolation triple with respect t o the triple

and the triple

Z , B is an

2, A .

(b) If { A , B } E I n t ( i , g ) , there exist spaces A 3 A and

B c B , such t h a t

the triple x , A is an optimal interpolation triple with respect to the triple

2,B. &f.

& A be an optimal interpolation triple with respect to g , B . If T E &(i, g),then T ( A ) A B , where i = i ( A ,B ) ,is the interpolation constant. Consequently, in view of (2.3.23), we have

(a) Let

(2.3.34)

orbA(i;

2)=

T ( A ) L) B . T

1

A)

OrbA(li+; and hence the triple i, A is an interpolation triple with respect t o the triple g, OrbA(i,$). From this, equation (2.3.34) and the optimality condition (see Def. 2.2.38) we find that the second relation in (2.3.33) is valid. However, A

~f

The first relation is proved in a similar manner on the basis of the inclusion A

L)

T-'(B), where T E &(i,g),and equality (2.3.27).

Let us now suppose that the relations (2.3.33) are satisfied. Since A L) O r b A ( i , i ) , & A is an interpolation triple with respect t o the triple Z , B , because B OrbA(A',g). Let us verify that the optirnality condition is satisfied. Suppose that the triple triple with respect t o the triple

B\B #

0, then for

&A

g,B, where A

L,

is an interpolation

A

and B

L)

B . If

B\B there exists, in view of the second equality in (2.3.33) an element a E A and an operator T E L ( i , g ) , each b in

158

Interpolation spaces and interpolation functors such that T a = b.

However, in this case T ( A ) is not contained in

a, which i s contrary t o the assumption about the interpolation. assumption that

A\A

# 0 is also refuted in t h e same way

first relation in (2.3.33).

Consequently, A

2

A,B

The

by using the

”=

B

and the

optimality is proved.

It now follows from the formulas (2.3.33) that the spaces A and B are -+ interpolation spaces in the couples A and B’ respectively. We have t o prove that the triple B’,B i s an interpolation triple with respect t o the triple 2,A . For this purpose, we consider S E L l ( g , 2).Using the first relation in (2.3.33) the definition of the norm in the coorbit, and the interpolation property of

B

in the couple

B’, we write

Here, we have confined ourselves t o the main case A(B’)

#

(0); the

remaining cases are trivial. Thus, we have established t h a t t h e triple property with respect t o t h e triple

2,B

has t h e interpolation

i, A.

(b) Let us now suppose that the triple .&A has the interpolation property with respect t o the triple

B

:=

g , B. We put

OrbA(2,g) and

Since by assumption

A

:=

Corbg(i,s) .

T ( A ) & B for a certain constant y > 0 and all

T E L l ( i , # ) , we obtain B := Orb,(X; Further, A

B’)

:=

C T ( A ) L1, B .

& OrbA(i; i) and hence the triple i , A has the interpo-

lation property with respect t o the triple certain constant y > 0 and all and hence

2,B.This

means that for a

T E L l ( i , B ’ ) .we obtain T-’(*) b A ,

Interpolation functors

Thus,

A

~t

A and

159

L+

B. It only

(2.3.33) is satisfied for the triples

&A

remains t o verify that condition and

g,B. The second of these

conditions is satisfied by definition, and this leaves only the following condition t o be verified:

It follows directly from the embedding A

On t h e other hand,

(A,B)E

Int,(&g)

~t

that

since

A

:=

Corbg(2; 3)

and B = C o r b g ( i ; g).Consequently, T(A)A B for some constant y > 0 and all T E tl(&l?). As in the above case, this leads t o the em bedding

which is inverse t o the embedding proved above. Thus, it is established that the first condition i n (2.3.33) is satisfied.

Let us use the above result t o display some examples of optimal triples. For this purpose, we require Definition 2.3.21. The couple 3 is called a retract of the couple A if there exist linear operators

P :

i+ Zand I

The operators 0

P

: g+isuch

and

that

I are called

PI= 1,.

retractive mappings.

Interpolation spaces and interpolation functors

160

Example 2.3.22. (a)

If

B' is a complemented subcouple of t h e couple A',it is a retract of A'.

Indeed, the role of embedding of

P

is played by the projection of A'onto

B', while the

B' in A' can be treated as the operator I .

A' := el La(11 is the sum of the couples), any summand A'o will be a retract of A'. In this case,

(b) If

a

where ya := 0 for a (c) Let

# p and yp

:=

5.

LAW') := (Lpo(wO),Lp,(wl))be a weighted couple of Lebesgue

spaces defined on

R+, and l e t

be an analogous couple of weighted

spaces of bilateral sequences. Further, suppose that

We assume that each of the weights w; has the following property:

Then

ZAfj')

(2.3.36)

Indeed, let

will be the retract of

ijn :=

LJ

)

Gp(t)dt

LP(Z)if 1l P

(n E 23)

161

Interpolation functors and also suppose that

Then in view of Holder’s inequality and the conditions (2.3.35) and

(2.3.36), we get


l f W i l P d t) l I P

Consequently, P : LAW‘) +

I :

ldg

---t

ZAii).

(i = 0 , l )

.

It is even easier t o verify that

LAG). Since it follows from the definitions of P and I

that P I ( f ) = f , t h e above statements are proved.

Corollary 2.3.22’.

If I? is a retract o f $, and P, I are retractive mappings, the triple $,A with A E Int,(A) has the optimality property with respect t o the triple l?, P ( A ) .

Proof. For T E Ll(&d),we have

since the number o f elements of the type Ib with b E P(A) cannot be larger than the number of elements of

llPll

-

property of A , does not exceed

ZA

right-hand side is equal t o

A. However, P I = 18, and hence the JJIT~IIA and, in view of the interpolation

llpll llITllff11x11~5 i~ llpll l l I l l 1 1 Z l l ~ .

Interpolation spaces and interpolation funct o m

162

Consequently, the triple

A,A has the interpolation property with respect

t o the triple Z , P ( A ) . Consequently, T ( A )

A P ( A ) , where y

pendent of T . Hence, i n view of (2.3.23), OrbA(A’,J)

cf

is inde-

P ( A ) ;since the

inverse embedding is also true in view o f the same relation (2.3.23), we have

Identical arguments based on (2.3.27) show that A

CorbB(A’, g)

0

Let us now consider a “natural” transformation o f orbit functors resulting from a variation o f the triple defining them. Proposition 2.3.23. Let

B’ be a retract of the couple A’ and let P , I

be the retractive transfor-

mations. If the intermediate spaces A and B are such that P ( A ) coincides with

B and I ( B ) coincides with A , we get OrbA(i; *)

%

OrbB(3; -) .

The isomorphism constant does not exceed llPll

. 11111.

Proof. Since it follows from while

the equality P I f 118 that P is a surjection I is an injection, the above formulation follows from two statements:

(a) If P :

i--+ 2 is a surjection

and P ( A ) coincides with B , then

Indeed, according to the closed graph theorem, the operator P generates the isomorphism P : B and each

E

> 0, there

Consequently,

--+

A / k e r P . This means that for each b E B

exists a E A , for which

163

Interpolation functors

Since the inf on the right-hand side

2

&~

~ x ~ ~ owerobtain b A the ~ ~ ~ ~ ~ ,

required embedding. Further, we note that if

B’ is a retract o f 2,the operator P is equal t o

~ ( I I B where ), x : A

AjkerP is the canonical surjection. Hence the

--f

norm o f the embedding operator in (2.3.37) does not exceed llPll (b) If I :

11111.

2 -+ A’ is an injection and I ( B ) coincides with A , we have

Ile is a bijection of B onto A , there exists a continuous 1 := (IB)-l. Further, some o f the operators T E L ( 2 ;2)can be represented in the form T = S I , where S E L(A’; 2); Indeed, since

inverse operator

consequently,

which proves the embedding (2.3.38).

If 2 is a retract o f

A’,we obviously have 1= PIA,and hence llfll 5 IIPII.

Thus, the embedding constant on the left-hand side o f (2.3.38) does not exceed IlPll

. 11111.

Interpolation spaces and interpolation functors

164

This type of a result is also valid for coorbits. We leave it t o t h e reader t o formulate and prove this.

Let us now pass t o a generalization o f the Aronszajn-Gagliardo theorem, which we will find usefulat a later stage. To be more precise, let us consider a class

K

of triples and assume that for certain constants a,& > 0 and all

( 2 , A )E K , (2.3.39)

A(2)

&

A A

(A).

Under this assumption, t h e following theorem is true. Theorem 2.3.24. (a) There exist functors G and

(2.3.40) for all

A

& G(A)

H such t h a t

and H ( i

&

A

( 2 , ~E K. )

Moreover, G is minimal and

H

is maximal among all functors for which

t h e corresponding embedding in (2.3.40) is satisfied. (b) If t h e class (2.3.41)

K

is generated by a functor F , i.e. if

F ( i )= A

for all {&,A)

E

K ,

we also have (2.3.42)

G(2)= F ( 2 ) = H ( 2 )

for all { & A } E

K

Proof. (a)

In view of (2.3.39), (2.3.17) and (2.3.12), we have for { i , A } E

K

A similar relation is also valid for the coorbit. Consequently, relation (2.3.8) is satisfied and the following functors are well-defined:

165

Interpolation functors

A large number o f theoretical difficulties associated with the fact t h a t t h e sum and intersection are considered for a class and not a s e t can

be easily overcome with t h e help of the following arguments. For a fixed couple

x’, the

spaces

orbA(li; 2)and CorbA(x’;

i) are subsets of

C ( x ’ ) . Since the class of subsets o f a given set is a set, the pairwise different spaces o f the class (OrbA(ii; x’)){A,a}cK (and of the analogous class of coorbits) form a s e t

Similarly, we can define

Kf. In this case, we shall assume that

H(2).

Let us verify that G is the minimal among all functors F for which the first embedding in (2.3.40) is satisfied. Indeed, if A A F ( A ) , then from the rninimality property o f the orbit we obtain

it then follows from (2.3.43) that

G

F as well.

The statement for H is proved in a similar manner. (b) If relation (2.3.41) is satisfied, then in view of the interpolation property of A we have

for all

{A,,}E K. On the other hand, in view of the minimal property

of an orbit, we have

Interpolation spaces and interpolation functors

166

for an arbitrary triple

{ d , B }E R .

Consequently, taking into account

(2.3.43), we have

G(A)

:= A

+C

OrbB(d,A’> = A = F(A)

B#a’

H

The proof of (2.3.43) for

is obtained similarly.

U

Corollary 2.3.25.

If F is a bounded interpolation functor, there exists an equivalent renormalization after which this functor becomes exact.

Proof. Let us consider the class {Fi (, i ) }which contains all couples of the category 2, and let G be a functor constructed for this class with the help For this class, relation (2.3.39) is satisfied in view of

of formula (2.3.43).

Proposition 2.3.3. Let us verify that for any couple

G(A)

A,

F(X),

E

1

which will prove the statement. Since G ( 2 ) e-r

F ( A ) , we need only

t o establish the inverse embedding. In view o f the interpolation property

L(d,i).we have T(F(2)) 1 F(A), where y 8. However, it follows from here and relation (2.3.23)

of F, for any T E is independent of

that Orb,(fil(g; A )

F ( i ) , and hence the sum of the orbits, i.e.

G(A) & F(A). 0

Let us now suppose that F is a functor defined on a subcategory

c

5.

Corollary 2.3.26. There exists an extension o f F t o a functor which is defined over the entire category

2. Among all such extensions F, there

and a maximal one

F,

such that F-

Proof. I t suffices t o put F-n

:= G and

1 -+

F,

-

F

exist a minimal one 1

-+

:=

Fdn

FA,,.

X ,where G and H

are

167

Interpolation functors defined by t h e formulas (2.3.43) with

K:

:=

{A;F(A)}zEz.

0

We use the statement obtained here to derive the inverse proposition t o

2.3.10. Corollary 2.3.27.

If 'p :

R: + R+is positively

homogeneous, nondecreasing and nonzero,

there exists a functor for which cp is the fundamental function. d

Proof. Let F D 1 be a subcategory o f one-dimensional regular couples of the type ( s R , t R ) . We put F(s,R,tE2)

:= ' p ( s , t ) R .

-

Since ' p ( s 7 t ) # 0, this defines an intermediate space for the couple.

We show that F is a functor on F D 1 . For this purpose, we consider

T : ( s R , t R ) 4 ( s ' R , t ' R ) . Obviously, for some A E R, we have Tx = Ax ( x E R ) . Calculation of the norm of T leads t o the quantity 1x1 max ($ , while for the norm of the restriction T l q p ( S , , ) ~ , the operator

r),

we obtain the expression 1x1 cp(s" t'). However, in view of the monotonicity cp(s7t) and homogeneity o f the function considered above, we have

which means that the norm o f the restriction does not exceed the norm of T. Thus F is a functor on couples

Taking its extension t o the category of all

6, we obtain the required result.

0

Before concluding this subsection, we show that in most of the arguments, we can replace an arbitrary interpolation functor (which may be unbounded!) by an orbit functor. In order t o formulate the result, we use

Interpolation spaces and interpolation functors

168 Definition 2.3.28.

rf c 6

Th e subcategory

is called

small if there exists a subcategory

kl c k whose class of couples is a set and which is such t h a t for each isometrically isomorphic to it. couple i n Z? there exists a couple in

-

Example 2.3.29.

The subcategory D F consisting of finite-dimensional regular couples is small. Indeed, each such couple is isomorphic to a couple o f the type where v; is the norm on

(nZ:,

B".The class of such couples forms a set.

0

F

Let us now suppose that is a small subcategory of

-+

is an arbitrary interpolation functor and

6. Under these conditions,

K

the following theorem

is valid. Theorem 2.3.30. There exists a triple

F(A)

Proof. Let K the Zl-sum

I?

%

{c,C}

such that for any couple

Orb,(c;

2).

be a set of couples from Definition 2.3.28 and let

o f this set.

is a retract o f

F ( P 2 ) F ( I x )=

P

and

I

c' denote

In view o f Example 2.3.22(b), each couple

6;since F

A' in

is a covariant functor, we have F ( P 2 , I i ) =

lF(2),where PA

:

c' + L a n d 12

responding retractive mappings. Consequently, and if

A' E z, we have

+

:

A'--+C are the cor-

F ( x ) is a retract of F ( C ) ,

denote the retractive mappings in this case, then

isomorphic t o the complemented subspace

I ( F ( i ) ) in F(C).

&'(A') is

Using this

isomorphism, we replace the norm i n F ( 2 ) by the norm induced from and denote the space thus obtained by

F(C),

A.

U

Lemma 2.3.31. The class of triples

{A,A } , where A' E 3,satisfies condition (2.3.39), while

the minimal functor G constructed for this class has the property that for all

A' E 8 ,holds

169

Interpolation functors

F(X)

S

G(A).

k f . Let F ( 6 )

C(6). Then, in terms of the notation employed

above, we obtain for a E

A

llallA

Illallc(c?) .

= Illall~(c?) 2

Further, since a = P(1a) and

lI4l.E(i) 5 Iliallc@,

I l P i I l ~5, ~1, we get '

Together with the preceding equality, this gives the embedding A where

0

is independent of

C(i)),

i.

Thus, for the class of triples

{ i , A } , where

A' E 2,the right-hand em-

bedding (2.3.39) is satisfied. This means that the functor G constructed for

this class i s well-defined. Moreover, in view o f Theorem 2.3.24(b), we have G(A)= A F(A)for Z E 2.

=

0

Let us now suppose that for couples

C

is the Il-sum o f spaces A constructed above

A' isssn 3,.Then for the functor G constructed above the fol-

lowing lemma is valid. Lemma 2.3.32. G = Orbc(6; .).

Proof. From definition (2.3.43)

and the condition of small subcategory

2,

we have

G=

OrbA(2; * ) . A d 1

Using once again the retractive mappings

Pi and 1~ introduced

above, we

obtain in view of Proposition 2.3.23

OrbA(L;

a)

-

= OrbIx(A)(6; .)

Since in this case I l ( A )

1

C,the

. functor on the right-hand side is

embedded in Orbc(6; -). Together with the preceding equality, this gives

170

Interpolation spaces and interpolation functors

G

1 c-)

C

4

Orbc(C;

= OrbC(2; .)

a)

.

k, In order t o prove the inverse embedding, we make use o f the fact that by definition on the ll-sum, llxllC

=

c

llpAzllA

>

A d where only a countable number o f summands have nonzero values. Moreover, since PAIL= 112, we obtain for any operator

T E L(C,X)

It follows from these two relations that

where we have put xn,

:=

Pzmxnand T,,IA,, Since the right-hand side

of this inequality inequality is obviously not less than

inf

x=C sno,

~ ~ S n ~ ~llanllAn ~,,,f

:= 11x11~(2) 7

ndV

this means that OrbC(C; 2) A G ( 2 ) . Taken together, these two lemmas prove the theorem. 0

Remark 2.3.33.

A similar statement is also valid for coorbits.

C. Finally, let us consider a few examples illustrating the concepts and results described here. W e begin with a functor which was introduced i n an implicit form i n Chap. 1[see (1.10.17)].To be more precise, let us associate to each couple

d

a space z8p, assuming that

171

Interpolation functors

Here and below, LE denotes the weighted space Lp(t-') constructed on ( B , , d t / t ) [see (1.3.16)], while the quantity under the norm on the righthand side is the K-functional o f the element 2 E C ( X ) in the couple J? [see (1.10.16) and (2.2.13)]. Proposition 1.10.7, when applied t o the Banach space, gives the following inequality for T E L ( 2 ,?): (2.3.45)

ll%BpIlf,p,v,p I IITlxo ' ; ; 11

llTlxlll~l.

On the basis of this inequality, we can easily establish the following Proposition 2.3.34.

If 0 < 6 < 1and 1 5 p < 00, or 0 5 6 5 1 and p = 00, the correspondence

2 -+ 28, defines a functor. 0

We omit the proof of this proposition, since a similar result which is of much more general nature will be established in the next chapter. For time present, we just note t h a t the restriction 6

#

0 , l for p

< 00

is connected

with the fact that for such values of the parameters, there are no nonzero concave functions in the space

LE. Since the K-functional is a concave

function o f t (as the lower bound o f linear functions), the space

2fiP = (0)

cannot be an intermediate space for such values of 6 and p .

The second remark is associated with inequality (2.3.45). The functors

F satisfying this condition will be called functors of power t y p e 6 . Finally, we can also define functors of type 'p on the basis of the following inequality [see (2.2.28)]:

(2.3.46)

s ' p * ( l l T l x o l l x o , Y o , IITlXl llXl.Yl)

IlTIF(T)IlF(f),F(B)

As an example illustrating the calculation of the functor us refer t o the result (1.10.22), which gives

(2.3.47)

(Lporo, LpIr1 )

~ q

.

Lp(~)g

Another example is Lemma 2.2.21, according t o which

'

-.

-t

X + Xs,, l e t

172

Interpolation spaces and interpolation functors

Zi, = X f ,

2

€ (0,l)

.

Let us also consider the computation of of (the number)

x

(am,p B 2 ) d p . Since the K-functional

for this couple is equal t o

we get

( a mP

W S ,

= I1 min(a, tP>llL,. 1x1 *

In view of Definition 2.3.9, it follows from here that the fundamental function

of t h e functor considered above is equal t o c ( d , p ) ~ r ’ - ~ P ’where , c ( d , p ) :=

( O ( 1 - O)p)-l/P. We shall turn t o specific calculations in the following chapters; for the time present, we consider a somewhat unexpected connection between a “specific” functor and “abstract” orbit and coorbit functors (see, however, Theorem 2.3.30). In order t o formulate the statement in question, we define t h e couples +

-+

L , and L1, assuming that -+

(2.3.48)

L,

:=

(L,,LL) ,

Ll := (L1,L;) .

It should be recalled once again t h a t are constructed on

(R+, dtlt).

:=

Lp(t-’) and that all the spaces

In this case, the following relations are valid:

28,

OrbL;(z, ; I?) ,

2 d p S

Corbq(2 ;

(2.3.49)

L:

z,)

,

These relations will be proved in Chap. 3.

These relations, together with (2.3.47), provide the first meaningful example o f “calculation” of orbits and coorbits. In the general case, such a problem is obviously unsolvable in view of Theorem 2.3.30. This lends even more importance t o the few cases for which it has been possible t o carry out the computations so far. We shall describe two such results, and then

Interpolation functors

173

refer the reader t o the appropriate articles o f the authors listed below for the proof.

To formulate the first result, we consider the discrete analog of the couple of spaces (l:, 1:)

of bilateral sequences

x :=

z,

of

( x , , ) ~ € z . Here,

-#

Denoting this couple by 1,

and the element (gn9)nEzby

29,

we obtain

(Gagliardo)

Here, L,- :=

(Lm,Lpl)is a couple of complex spaces, and the isomorphism

constant tends to 1as q -+

1.

+

Let us now suppose that L' := ( L I L,,L1 n L,), where the spaces L, are again considered on R+with the measure dt/t. Further, let xp E C(z) be the function defined by the formula x p ( t ) := d*(t'/,-', t'lp'-' ), where 1 < p < 00 and, as usual, l / p + l/p' = 1. Then the following statement is valid (Ovchinnikov):

(2.3.51) Orb,,(z;

z)

S

L,,

n L,,, .

174

Interpolation spaces and interpolation functors

2.4. Duality

A. In view

0. the

significant role o f duality in Banach theory, it seems

natural t o develop a similar theory for the category of couples. W e shall describe some initial results obtained in this direction. Th e Banach theory serves as a model for constructing the corresponding theory for couples, although a complete analogy between the two does not exist. Therefore,

the tendency t o treat dual objects i n the two theories as nondistinguishable, which prevails in modern works on the theory o f interpolation spaces, may lead (and leads) t o serious errors. In this book, dual objects of the Banach theory are marked by asterisk, while dual objects in the category of couples are primed. For example, for the operator operator conjugate t o

T

T

E

L($,?), T* denotes

the

which is treated as an operator f r o m C ( 2 ) into

C(p), while T’ denotes an operator dual t o T in the category o f couples, which acts from the couple

?‘ dual to ? onto a similar

couple

2’.In the

X E I ( X ) , X’ denotes the space in the Banach theory conjugate t o X , while X’ denotes the space dual t o X in t h e category o f Banach couples. Here X‘ is not equal to X * and not even same way, for the intermediate space

always isometric to it.

Let us define t h e basic objects in the duality theory. W e start with the definition o f a dual intermediate space and a dual couple. Definition 2.4.1.

X’ dual t o the intermediate space X o f couple consists o f the linear continuous functionals z’ E A(Z)* for which the quantity

The space

is finite. Here and below, ( z ’ , ~ )denotes the value of the functional z’ in

A(Z)* on the element z E A(-?). 0

Obviously, the definition of

X’

depends on the couple for which

X is the

intermediate space. This dependence is not reflected in the notation since

Duality

175

it is always clear from the context which couple i s considered. It should also be noted that (XO)’ and X’ coincide since the norms of the spaces X and

X o coincide on the set A ( d ) . Definition 2.4.2.

2’ :=

(X&Xi).

0

It follows from what has been said above that the couple defined by the dual couple

x’ is uniquely

d’ only for a regular couple x’.

Let now X be an intermediate space for the couple

2.The consistency

of the definitions introduced above is assured by the following theorem. Theorem 2.4.3. The couple

x’l is a Banach couple, while X’is a relatively complete inter-

mediate space of this couple.

Proof. We denote for

each

2’

E

X’ by z*the extension

by continuity t o a

functional on X o . Then z*E ( X o ) * ,and the linear operator maps X’into

EX

: 2’ --t z*

(Xo)*.

0

Lemma 2.4.4.

The operator

E,

is an isometry o f

X’on ( X o ) * .

Proof. Since the operator z*+ z*la(g) is obviously erator cx is a bijection. Further, since coincide on the elements o f

Il~xz’ll(x~)* =

inverse t o

EX,

the op-

A(d)i s dense i n X o and z’and z’

A(x’), we can also write

SUP

{ ( 2 * , z ) x o; ll~llxI 1,

2

E A(2)) =

= s u P { ( ~ ’ , z ) ;11zIlx 51) . Here and below, (.,

.)A

denotes a canonical bilinear form on A* x A (recall

that the subscript is omitted when A := A(@). Thus,

ll&x~’lI(x0)* = ll4lx~7

176

Interpolation spaces and interpolation functors

X'

and

_N

(XO)'.

0

Therefore, X' is a Banach space. Let us prove that it is embedded into A(x')*. This follows from a more general fact given below. Lemma 2.4.5. If X , Y E 1(x')and

X

Proof. By hypothesis, we

Y , then Y'

& X'.

have

Since it is obvious t h a t (2.4.2)

A(x')' = A(x')*

,

the embedding

X'

~t

A(x')*

follows from the embedding

A(x')

X.

~ - t

Applying the facts proved above to the spaces X ; , z = 0,1, we establish that

x"

is a Banach couple.

Let us now show that X' is an intermediate space for the couple this we need the following important proposition. Proposition 2.4.6. (a)

~ ( =2~(2)'; )

(b) A(x") = C(x')'.

2'.For

177

Duality

Proof. (a) Since

X i , in view o f Lemma 2.4.3 we have X i

A(2)

1 ~f

A(x')*,

i = 0,1,whence (2.4.3)

A(x')* .

C(2')

Let us prove the inverse embedding. For this purpose, given an element

X O $,

E A(X')*, we define on the subspace

D

XI; 20 = 21)

a linear functional z#

2 '

o f the space Xo $ ,

X1

:=

{(q,,x1)

E

by the formula

The definition is consistent since the element

2;

obviously belongs t o

A(d).Moreover,

Consequently, it follows from t h e definition of I# that

Using the Hahn-Banach theorem we extend the functional entire space X o $ ,

X1with

x# on the

preservation of the norm. The extended

x# then belongs t o the space ( X o Xl)*, which can be identified with X: X y . Consequently, I* = (I:, x;), where zf E Xi.,

functional and

Moreover, for ( z 0 , r l )E

where I :=

If we put

20

Z: :=

(=

D

we also have

21).

zfla(2,, the last equality indicates that

178

Intelpolation spaces and intelpolation functors I’ = I;

+ z: .

Consequently, taking into account (2.4.1) and (2.4.4) we have

Thus, t h e embedding

is established.

Combining it with (2.4.2) and (2.4.3)

we prove the

statement.

A

C ( 2 ) . it follows from Lemma 2.4.5 that there exists the 1 embedding C(x’)’ L) X;l, i = 0 , l . Hence

(b) Since X

i

(2.4.5)

C(x’)’

&

A(x”)

,

and we only have t o prove the inverse embedding. For this we consider t h e functional z*

:=

Lemma 2.4.4. If z = zo

E ~ ( ~ ) ( I ’ ) ,where

t h e operator

E,

is defined in

+ z l , where x i E Xi”, then

If we consider the statement of Lemma 2.4.4, we obtain the following inequality:

Taking infimum for all representations o f

I

in the form

10

+

11,

I; E

X:, and considering that C(Z0) = C o ( z ) (see Proposition 2.2.12), we obtain

Duality

179

Consequently,' recalling the definition o f z* and using the statement of Lemma 2.4.4, we get

which establishes the embedding inverse t o (2.4.5).

Let us now show that X ' belongs t o I ( X ' ) . In view of Lemma 2.4.5 and Proposition 2.4.6, we have

= E(d)'

A(?) where 6 and o

X'

&

A(x')'= C(x"),

> 0 are the constants of the embeddings A ( 2 ) A X

C(2). Thus, X' E

I(X '), and

it remains t o show t h a t this space is relatively

complete in C(x'l). For this we must prove that the unit ball B ( X ' ) is closed

A(x')*. Suppose that ( ~ 6 c )B ( X~' ) and ~ ~that x' is the limit of this sequence in A(x')*. Then (zh,x) -t ( z ' , x ) for any z E A(x'), and hence in

Remark 2.4.7. It can be easily seen from (2.4.1) that the unit ball B ( X ' ) is *- weakly closed in the space

A(Z)*. +

Let us also note a connection between X ' , regarded as a generalized couple, and the generalized couple 2.1.27 for a regular

x'*

:=

( X : , X r ) defined in Example

2.The following proposition

holds.

Interpolation spaces and interpolation functors

180 Proposition 2.4.8.

If

d

is a regular couple, the map ~ ~ ((see 2 )Lemma 2.4.4) establishes a

---t

BY1 isomorphism o f the (generalized) couples x" and X * .

Proof.It follows from the definition of the operator ~

~ (that 2 its ) restriction

on X i coincides with exi. Hence in view of Lemma 2.4.4 and the fact that f

X is regular, this restriction gives an isometry of X l on X:, i = 0,1. Since, in addition, acc2, obviously commutes with the operator of identification of the couples x" and X * (see Examples 2.1.24-26)

it is ~ ~ (that 2 establishes )

4

the required BY1-isomorphism. U

Remark 2.4.9.

) henceIn connection with the statement proved above, the map ~ ~ (will2 be forth denoted by

€2.

Let us now consider the second dual couple I?'. In analogy with the

X there exists a canonical isometric X + X** defined by the formula

Banach case, where for every space embedding

KX

:

(Kx("),

"*)x* :=

(5*, ")X

7

we introduce the canonic isometric embedding ~2 :

2 + 2..

Here we

shall confine ourselves only t o regular couples. Thus, in the general case "2 will carry out the embedding of

3 '

into

3".

Definition 2.4.10. Let

x'

(2.4.6)

be a regular couple. Then :=

.

E > K ~ ( ~ )

0 +

Since €2 establishes the isometry of C(2)' onto C ( X ) * ,E> is an isometry of C ( d * * )onto (C(Z)')*. In view of Proposition 2.4.6, we have

(C(d)')* = A(x")* = C(x'l))

Duality so that

181

: C(x')** II C(x'").

E>

Since, in addition, ~ ~ (: 2C ()x ' ) + E(d)** is the canonical embedding,

it follows from (2.4.6) that

"2 : C ( 2 )

*

E(x''l).

Let us verify the following proposition. ProDosition 2.4.11. (a) KB is an injective embedding o f a regular couple

Besides, for z (2.4.7)

E A(x') and 2' E

( K ~ ( z )2 ,')

x'

into the couple

2".

A(d'),we have

= (z', 2 ) +

(b) If X is a regular intermediate space of couple X , the restriction of KT on X belongs t o

& ( X , X").

&f. (a) Let us start with the proof of (2.4.7). Since

A(2') = C(x')', in view of

(2.4.6) we have

(2.4.8)

(.a(z>,4 = ( E W ( Z ' ) , ")X(d)*

Since the right-hand side is equal t o ( z ' , ~ )for z E A(x'), (2.4.7) is proved. Let us now prove the first statement. The embedding ~ 2 ( X i ) X,!' follows from item (b). Further, if K W ( Z ) = 0, in view of Proposition 2.4.6 and Theorem 2.4.12, which will be proved later, we have

182

Interpolation spaces and interpolation functors Thus, the injectivity o f "2 is proved.

(b) In view of (2.4.8), for all 2 E

A(x') we have

According t o Lemma 2.4.4, 11z'llxl =

IIEx(~')IIx-

since X o = X . There-

fore, the right-hand side of the previous equality does not exceed

Thus, for z E

A(d)we obtain

Since X is regular, this is valid for any z in this space.

Using the canonical mapping, let

us establish an important relation be-

tween the relative completion of the intermediate space and its second dual space. Theorem 2.4.12 (Aronszajn-Gagliardo).

+

If X i s a regular intermediate space of the couple X , then

X' = K;il(X") . To be more precise, for z E X' we have

l l ~ T ( ~ ) I l= X 11~11X~ ~~ > and X" coincides with the inverse image K$'(X").

Proof.Since (2')''= Z", the regular space X

E I ( x o ) ,and X'?'

= X'"',

we may regard X as aregular couple without any loss of generality. Suppose now that z E then have

A(d)and z'

E

A(r?'). According t o (2.4.7). we

183

Duality

Taking the supremum for all

2 '

E A(2') with the norm IIz'llx, 5 1, we

arrive a t the following inequality:

Since X is regular, we can assume t h a t this inequality is valid for all

2

Let us now suppose that supJJs,JJx 5 1 and that the sequence

EX. (2,)

converges t o z in C ( x ' ) . Then llzllxc 5 1. Let us show that in this case I \ ~ ~ ( e ) l l x5 t t 1 as well. Indeed, it follows from (2.4.9) that the sequence

(nl(e,,)) belongs t o the unit ball B ( X " ) . Proposition 2.4.11 implies t h a t this sequence converges i n C ( X " ) t o the element KX(Z). Finally, according t o Theorem 2.4.3, the unit ball B ( X " ) is closed in C(2"), so t h a t

Il.y(4llx" 51. Thus, we have proved that

In order t o prove the inverse inequality, it i s sufficient t o show that if

20

$!

B ( X " ) , then

To verify that, we shall make use of the closeness and convexity of t h e unit ball B ( X " )in C ( x ' ) . Since in this case zo # B ( X " ) ,there exists a functional C* E C ( z ) * , which strictly separates zo from B ( X " ) .Thus,

From the second of these inequalities, for 11Z'llXt := sup { (2',z) i

2 '

:=

z * l a ( ~we ) obtain

IlZllX I 1) I

I SUp{(z*,")C(y);2 E B(2"))51 * Taking into account (2.4.6), we obtain from this inequality

Interpolation spaces and interpolation funct om

184

$0

!2 B ( X " ) =+ IIq(~0)llX~r> 1

9

and thus the inequality inverse to (2.4.10) is proved. 0

Corollarv 2.4.13.

If X is a reflexive Banach space and Y is an arbitrary Banach space containing X , the relative completion Xcqy of the space X in Y coincides with X .

Proof. Replacing, if necessary, Y

by the closure of X in this space, we can

Y . By X and XI := Y , we obtain a regular ordered couple such that XA = A(x')' = XG and X i N X ; . Let us prove the assume, without loss of generality, that X is densely embedded in

changing the notation Xo := following lemma. Lemma 2.4.14.

The space X ; is densely embedded in X;.

Proof. It is sufficient t o show t h a t

if cp E (XA)* (= X,") vanishes on X i , then cp = 0. Since X O is reflexive, cp is generated by the element z E X o according t o the following formula: cp(z') = (zI,z)xo ,

If y' E

X i , then y*

2'

:= EX,(Y')

E x; .

belongs t o X ; and (y',z)x,, = (y*,z)xl.

Consequently, in view of the choice of cp, we have

0 = cp(Y') = (Y*,2)X1

185

Duality Since this is valid for any y’ E

X : , and hence for any y* E X ; ,

2

= 0, and

therefore cp = 0 as well. 0

Thus, we have established t h a t the ordered couple

X r = (X(>’

x“

is regular. Con-

X;*, and hence the canonical embedding “2 : x’ t 2”is generated by the canonical embedding K X , : XI -t X;* [see (2.4.6)]. Here K ~ , ( X=~XG* ) because Xo is reflexive. Since it also

sequently,

follows from the fact that

N

2’is regular that X:

= (XG)’

N

X r , we have

On the other hand, by Theorem 2.4.12, the left-hand side is equal t o Xg 0

Remark 2.4.15 (Petunin).

The following converse of Corollary 2.4.13 is valid. If X = XCvyfor any enveloping Y , the space X is reflexive.

B. Let us now consider one o f the fundamental problems in t h e duality theory, viz. the stability o f the interpolation property relative t o dual objects.

To formulate the problem emerging in this connection, it is convenient t o use the following definition. Definition 2.4.16. -+

x’

A couple is called complete interpolation couple relative to Y if for any spaces X and Y such that { X ,Y } E Int(2, ?) the following statement is valid: (2.4.12)

If 2 and

{Y’,X’} E Int

? in this definition coincide, then 2 is called a

lation couple. 0

{?,z‘) complete interpo-

Interpolation spaces and interpolation funct o m

186

In some cases we shall use a wider definition where the right-hand side in (2.4.12) is replaced by I n t w ( p , z ' ) . In this more general case, we use the same term for couples

2 and f . Although all couples known t o us are

complete interpolation couples, the following theorem is valid. Theorem 2.4.17 (Krugljak). There exists a regular but not complete interpolation couple.

b f . Let

A' be a

regular ordered (A0

1 -+

A , ) couple that satisfies the

following condition: (2.4.13)

the norms

(=

on

I/ . ll~,,and 1) . I(A; A(A')) .

are not equivalent

An example of a couple satisfying the above conditions is indicated in Example 2.2.24 and Remark 2.2.25. Further, l e t B' be a regular relatively complete ordered couple (Bo -+

1

B,)

for which t h e following condition is satisfied: (2.4.14)

BI, # (BI,)'

(as sets)

.

A simple example o f such a couple will be given below. Finally, we put,

2

+

:=

i @ l

B,

and make sure that this couple is not a complete interpolation couple. For this we consider a space X := A: (2.4.15)

X E Int(2) but

BOand prove that

$1

X' @ Intw(r?') .

The first statement can be easily established. Indeed, i n view of Proposition 2.2.20, we have

Ac(x') = A(x") = A ( 2 )

A(@) = 4

@1

Bo

X = Ac(z), and since A' is a functor Proposition 2.3.5 and Proposition 2.2.17), X E Int(2).

since (see

8 is relatively complete.

$1

Thus,

Let us verify the second statement of (2.4.15). Suppose that the opposite is true. Then for any

T E L(-%?'), we have

Duality (2.4.16)

187

T(X’) c X’

.

On the other hand, each operator

T in L($)

acts on an element (a’,b’) in

C(2‘) by the formula

T(a’, b‘) = (Twa’ + T’lb’, TI’U’ where

+ Tllb’) ,

z),To, E L(A‘,B’) and Tll E L($).

TooE L(z), To, E L($,

Therefore, in view of (2.4.16), for an element z’ := (0, b’) E X’, where

b’ E B;\(B;)’ 4

[see (2.4.14)] and an operator

Tol := (2.4.17)

? E L($,z),

T E L ( X ‘ ) with Too = 2-10 = TI1 = 0 and

we have

T(O,b‘) := (Pb’, 0) E X’ = (A;)’

BA .

In view of the choice of b’ and Lemma 2.2.14, for any a’ E A; there exists

T E L(l?’,&

for which a’ =

Pb‘. Hence we obtain from (2.4.17) the

following embedding:

Since & L) A;, the inverse embedding (A;)’ ~f Ah is also valid. Thus, the isomorphism (2.4.18)

4

2

(4)’

is established. We shall show that it leads t o a contradiction.

Indeed, in view of the Hahn-Banch theorem and the fact that (A;)’ and

(A;)” are isometric for a E A , we have

Similarly, for the same a E A’, we have

Interpolation spaces and interpolation functors

188

In view of (2.4.18),the last two suprema are equivalent, which implies that

11 . ll~,, and 11 . I(A;

are equivalent on Ao. However, this statement contradicts

(2.4.18). I t remains t o give an example of a couple

B’

that satisfies condition

(2.4.14).We put Bo := l1 and B1 : Z,(W), with the weight ~ ( n := ) n-’, 1 n E PV. Then Bo ~ - tB1 and A ( 3 ) = Z1, and since finite sequences are dense in Zl(w), A(Z) is also dense in Zl(w). Consequently, the couple B’ is regular. The relative completeness o f this couple follows from the Fatou theorem (see Theorem 1.3.2). Indeed, this theorem implies that unit balls of spaces

Z1 and Zl(w)

are closed with respect t o pointwise convergence, and

even more, with respect t o the convergence in the space C(Z) = l l ( w ) . +

Finally,

B’ = (Z,,Z,(l/u)),

and hence A(@) = Im(l/w). However, this

space is obviously not dense in

= 1, so that (B;)O # B;.

(7

Before formulating several sufFicient conditions for complete interpolation couples, let us first introduce and analyze t h e concept of a dual operator, which is important in itself. Definition 2.4.18. The operator

T‘ := (TI@))* is called a dual operator with respect t o

TE

L(Y,?).

U

Proposition 2.4.19. Suppose that T E ments are valid.

(b) (ST)’ = T’S’.

L ( z , ? ) and S E L(?,Z). Then the following state-

189

Duality (c) If T is reversible, T‘ is reversible as well:

(T’)-1 = (2-1)’

.

(d) For the duality of an operator

RE

,C(?‘,z’)it is necessary (and suffi-

2 is regular) that R be *-weakly continuous as an operator from A(P)*into ~ ( x ’ ) * .

cient if

Proof. TI,($)

A ( z ) + A@), we have T’ := (Tl,(a))* E L(A(P)*,A(z)*). Here, according t o Proposition 2.4.6, A(X‘*) = E(d’) so that T’ : E(?’) + C(2’). Let us suppose that y’ E y,’ and z E A(x’). Then from t h e definition o f a dual operator we get

(a) Since

:

Taking the supremum for all x E

x’

llzllxi

5 1, we obtain

T‘(Y,‘) c X l , i = 0,1, and inequality (2.4.19)

Hence it follows that satisfied. Let now

A(d)with

is

be a regular couple. Then

A(Z), Tx

A(@. Moreover, since y’ runs over the unit the Hahn-Banach theorem the inner (second) supremum is equal t o IITzlly,. In view o f the fact that A(Z) is dense in X i . we have Since z E ball

y’

N

E

(yo)*,by

190

Interpolation spaces and interpolation functors This means that in the case under consideration, inequality (2.4.19) becomes an equality.

(b) Since (ST)IA(y)= (SIA(q)(TIA(y,), the problem is reduced t o the corresponding property for Banach couples. (c) If

T

is invertible, then

TT-' = lp.

Therefore,

T-'

maps

A(?) on

A(x'> and (TIA(X))-' = ( T - ' ) l A ( ? ) . Now, in view of the well-known property of the Banach conjugation (see, for example, Dunford-Schwartz

[l],Lemma V1.2.7), we have

(d) We need a statement whose proof unfortunately is not given in the available books on functional analysis.

Lemma 2.4.20

If an operator T E L(Y*,X*), then T = 9 for some S E L ( X , Y ) iff T is *-weakly continuous.

Proof.Suppose that T = S' U := {x* E X ' ;

and that

sup

I(Z*,Zk)I < € }

lsksn is one of the neighbourhoods of zero t h a t determine the *-weak topology in X ' .

Let us show that T-'U

is a neighbourhood of zero in the *-weak

topology of t h e space Y * . Indeed,

T-' =

{y*,Ty* E U } = {g*;

where Yk := S x k . Thus, Conversely, l e t and put

T

sup I ( T y * , q ) l 1Sksn

<e}

=

is *-weakly continuous.

T be *-weakly

continuous. We consider

T* : X"

-+

Y**

191

Duality

S := K ~ * T * K, x

(2.4.20) where

S :

(2.4.21)

X + X"* is the canonical embedding. Let us show that

:

ICX

X

+ Y, for which the following embedding has t o be established:

T'Kx(X) C t~y(Y).

For this purpose, we consider t h e linear functional :

fi

Y* -+

(T*Kx(~),Y*)Y*

E X . Since t h e quantity in the right-hand side is equal t o ( K X ( Z ) , Ty*)x. = ( T y * , x ) ~and T is *-weakly continuous, the functional

for a fixed x

fi is *-weakly continuous. But all such functionals are generated on Y' by elements in

Y

according t o the formula y* + (y*,y)y (see, for example,

Dunford-Schwartz [l], Theorem V.3.9).

Consequently, for a certain y E Y

= (y*,y)y = (~y(y),y*)y.. Then from the definition of fz we obtain P K X (= Z~)y ( y E ) KY(Y),which proves embedding (2.4.21). we have f,(y*)

Embedding (2.4.21) and equality (2.4.20) imply that S is a linear operator mapping X into

Y. It remains t o verify

t h a t s' = T

Considering that

.

is an isometric isomorphism o f Y on the image KY(Y)and that (K?)-',

(KT')*

KY

=

we obtain from (2.4.20)

S* = K;T**(&)-'

= ( K ; , K ~ * ) ( I E X ~ T ~ * K E ; ) - ' ( K E ; K. ~ .

It is well known, however (see, for example, Dunford-Schwartz [l],Lemma V1.2.6), that K ~ ! T * * K= E T~ .. Hence it remains t o show that K > K ~= l X . . For this we take arbitrary elements x E X and z* E X * , and taking into account the definition of the canonical mapping, write (GKX.(Z*),

4x= =

Thus the equality

K>KX.

=

(KEx.(Z*), K X ( Z ) ) X * * (KX(Z),Z*)X.

=

=(2*,x)x

.

lx. is established.

0

@',z')

Let us return to the proof o f item (d). Let R E be such that R = T' for a certain T E L ( 2 ,f ) .Then, in view of the definition of T ' , we

Interpolation spaces and interpolation func t o m

192

obtain from the lemma t h a t T’ is *-weakly continuous as an operator from

A(?)* into A(@*. Conversely, suppose t h a t

T’, which is regarded as an operator from A(?)*

into A ( X ) * , is *-weakly continuous and t h a t the couple according t o the lemma, there exists an operator S which

S’ = T .

2 is regular. Then,

E L(A(@,A(?))

for

Consequently,

(Ty’,z) = (Y‘,SZ) for y’ E A(?)* = C(?) and z E

A ( d ) .Since Sz E A(?) and y;‘

II

(yo)*,

by the Hahn-Banach theorem we have

IlSzlly, = SUP IVY’, 4 ; llY’llY/

I 11 I IITIb,a,l1~11x, *

Since A ( d ) is dense in X i , the operator S thus extends by continuity t o an operator Si acting from Xi into

x,i = O , 1 .

So, Sl define an operator

d

s’

:

+

Here SolA(y)= SIIA(y)SO that

?. Thus,

:= (SJ,(y))* = s* = T

.

U

Corollary 2.4.21.

If a couple

x’

is regular and the couple

then any operator from

f

.C(?’,z’)is dual.

is such that A(?) is reflexive,

Since T is continuous as an operator from A(P)*into ~ ( x ’ ) * , it is weakly continuous (see, for example, Dunford-Schwartz [l], Theorem

roof.

V.3.15). In this case, the set U which is open in the *-weak topology of the space A ( d ) * is open in the weak topology as well so that

T-’(U)

is open in

the weak topology of the space A@)*. But this space is reflexive, and hence the weak and *-weak topologies coincide in it. Consequently, T-’(U) open in the *-weak topology, and hence the operator T : A(?)*

is

+ A(-?)*

is *-weakly continuous. Thus, in view of Proposition 2.4.19, there exists

SE 0

,C(d,?)for which T = S’.

Duality

193

We can now describe the main result representing a wide class of complete interpolation couples. We need the following Definition 2.4.22.

d possesses the approximation property if for any E > 0 and 2 E A(d)there exists a linear finite-range operator P := P,,, : C(x') + A couple

A(x') for which

Recall that a linear operator

T has finite

range if dim(1mT)

< 00.

Examde 2.4.23.

The couple (L,(wo), L p 1 ( q ) ) possesses the approximation property for pi < 00, i = 0 , l . Indeed, let f E L,(wo) n Lp,(wl) and E > 0 be given. Since the set of simple functions is dense in L p ( d p )with p < 00 (see Theorem 1.3.2) there exists a function

k

= 1,..., n, such that

than

E.

f

:=

fc

differs from

fe

(YkXAk,

where p ( & )

< 00,

in the intersection norm by not more

We put

Then P is a finite-rank operator for which inequality (2.4.22) obviously is satisfied. It is not difficult t o verify the validity of (2.4.23), and we leave it t o the reader as an exercise [see the proof o f inequality (1.5.5)]. Theorem 2.4.24. The couple

x'

is a complete interpolation couple relative t o

following conditions is satisfied: (a)

d

is regular and

(b)

x'

possesses the approximation property.

A(?) is reflexive;

? if one of the

Interpolation spaces and interpolation funct om

194

Proof.We shall require Lemma 2.4.25.

The couple

2

is a complete interpolation couple relative t o

T E Lc,(?‘,~?’),y’ E C(?)

and z E

? if for

any

A(r?) t h e following inequality is

satisfied : (2.4.24)

I(Ty’,z)l I s u p { ( y ‘ , S z ) ; S E

Proof. Using the definition of X‘

&(r?,?)} .

and (2.4.24), we have

If now { X , Y } E Int(d,?), we have

Using this inequality together with the previous one, we get

Thus,

{ Y ’ , X ’ } E Int(?’,Z’).

0

We can now easily prove t h e theorem for the case (a).

Indeed, in view

of Corollary 2.4.21, the left-hand side o f (2.4.24) is equal t o (y’, Sz) for a certain S E

Ll(d,?) so that

inequality (2.4.24) is satisfied.

Passing t o the case (b), we take C(?’) and specify E

> 0. If P

:=

T E Ll(?’,d’), 2 E A ( 2 ) and y’ E

P,,,is an operator from Definition 2.4.22,

we have

+ (Ty’,

(Ty’, z) = (TY’,Pz)

whence in view of (2.4.22) we obtain

(Ty‘, z) = (P’Ty’,z) + O(&).

- Pz) ,

Duality

195

Further, denoting by

f a finite-dimensional

regular subcouple

2‘containing

P’(Z’),in view o f (2.4.23) we have

IIlPlla ll~IlP~,a, I1 + E

IlPt%J~,V

*

It follows from this and preceding inequalities that it is sufficient t o prove + Lemma 2.4.25 for the case when X is a finite-dimensional regular couple. Then, however, the problem is reduced t o the following

-.

Lemma 2.4.26.

If X is a finite-dimensional regular couple, t h e set

B

(2.4.25)

:=

{T‘;T E Li(Z,q)}

is dense in the ball

Ll(?’,?)

in the topology determined by the family of

neighbourhoods of zero: (2.4.26) Here E

U,(y’,z)

:=

> 0, y’ E C(f’)

{T E L(?‘,*); I(Ty’,z)l < E } and

.

x E A(2).

The proof of this lemma in the framework of functional analysis is cumbersome. We shall give it on a later stage (see 2.7.3) using a generalization of the concept of couple, which will lead t o the definition of tensor product in t h e extended category o f couples obtained in this way. 0

If

T,determined

by the conditions of Lemma 2.4.26, now belongs t o

&(?’,z’),then, in accordance with this lemma, there exists a generalized sequence

(Sa)aE~ c Ll(z,f) such that

and Lemma 2.4.25 is proved. Remark 2.4.27.

The only way o f verification of inequality (2.4.24) known t o us consists in the proof of the .r-density of the set of operators {S’, S E L 1 ( 2 , f ) } in the

Interpolation spaces and interpolation functors

196 unit ball

Ll(?’,

r?’), However, t h e majorizing property expressed by inequa-

lity (2.4.24) is essentially more general than the property of T-density. This will be shown in 2.7.3 even for the category of Banach spaces. In this case, the analog of t h e majorizing property holds true for any Banach spaces

X,

Y, while the analog of the property of the .r-density does not hold even for “nice” Banach spaces. Remark 2.4.28.

If the approximation property is taken in a weaker form involving the substitution of the inequality

for (2.4.23), where

y > 1, the statement o f Theorem 2.4.24 has t o be

modified as follows.

If {X,Y} E I n t ( g , p ) , then (Y‘,X’) E Int,(?,i’)

and the following

inequality is valid for the interpolation constant:

i(Y’, X’)

5y

C . Let us now consider the dual interpolation functor. The following definition would be the most natural. Suppose that F E J F ;we consider a mapping (2.4.28)

F‘ :

of the category

2’4F ( 2 ) ’ 2’ o f dual

couples into the category

B

of Banach spaces.

Since F ( X ) ’ is an intermediate space for the couple X’, for T E

L(-?’,?‘)

we can also put

F’(T) := TIF(g),. Unfortunately, F’ is generally not an interpolation functor even if we confine it t o the subcategory

-0

( B )’ consisting of couples that

are dual t o

regular couples. This follows from Theorem 2.4.17 in which we essentially -0

proved that for F := Ac, the map F‘ is not a functor on ( B )’.

Duality

197

This circumstance forces us t o use a more complex definition. This choice

is dictated by our wish t o retain as many properties that are “natural” for a dual functor as possible. Thus, we can expect that the dual functor coin-

F’ when the

cides with

l a t t e r is a functor, that the spaces generated by it

are relatively complete, and that the second dual functor coincides with for “nice”

F

F . All these properties are inherent i n the object described by t h e

following definition. Definition 2.4.29.

The functor D F is called dual t o a given functor F E among those G E JF for which

G(2’) for all regular couplex

JF

if it is maximal

F(Z)’,

2.

0

D F is the maximal functor determined by the class of triples {?, F ( 2 ) ’ ; 2 E go}(see Theorem 2.3.24). Since in this case the fact that F is an interpolation functor implies that condition (2.3.39) is satisfied, the existence of D F follows from this theorem. This definition immediately leads t o the identity D F = F’ for the case when F’ is a functor. We have the following proposition. Thus,

Proposition 2.4.30. The space DF(r?) is relatively complete in the couple

Proof. We

must show that the closed unit ball

B

3.

of the space D F ( 2 ) is

closed in C ( 2 ) . By definition [see (2.3.43)], 11211DF(Y) :=

{IITz[lF(p)t

;

E

,cl(27

?),

E Bo}

*

Thus, the unit ball o f

D F ( 2 ) is the intersection of the family of sets {Z’-’(B(F(?)’)) ; T E Lc,(x’,p)}.But each ball B(F(f)’) is closed in C(?) since F(f)’is relatively complete in the couple f’ (see Theorem 2.4.3). In view o f the continuity of

T as an operator from C ( 2 ) into C(f‘),

Interpolation spaces and interpolation functors

198

the inverse image closed in

T-’(B(F(Y)’)) is closed

in C ( 2 ) . The ball

B

is also

C ( i ) being the intersection of these inverse images.

In the next section, we shall isolate a class of functors for which D2F coincides with

F‘ (it is natural t o call such functors reflezive). Here, we

prove another similar fact, which establishes an unexpected relation between

D F ( 2 ) and F’(2‘’)

for an arbitrary

F.

Theorem 2.4.31 (Aizenstein).

If x’ is a regular couple, then

D F ( 2 ) = fcpyd”). To be more precise, for any

5

E D F ( 2 ) the following equality is valid:

llKx(4llFy2~~) = ll4lDF(X, and

2

D F ( 2 ) coincides with the above inverse image F’(2”) as

a linear

space. Proof. For a regular couple,

KY

belongs t o

C,(2,d”)(see

Proposition

2.4.11). Hence, for the functor D F we have IIKPIIDF(B**)

The couple

5

-

Il~llDF(a)

2‘’ is dual with

respect t o the regular couple ( z ’ ) o s o that, by

the definition of D F ,

llWIlF’(R”) I Il“Y”IIDF(2’~) . Together with the preceding inequality, this gives (2.4.29)

h 4 1 F ’ ( 2 ” ) I ll4lDF(d)

for o E D F ( 2 ) . Let us prove the inverse inequality. For this we require the identity [see (2.4.8)) (2.4.30)

(~22,= ~ ’( E ) Y ~ “’ ), c ( d )7

Duality

199

C ( 2 ) and z1 E A(X‘). Using this identity, we shall show that

where z E

the formula

(2.4.31)

:=

IlG(2)

11

“20’

llF((2t)o)f

defines a functor.

G ( 2 ) = G(x’O), it is sufficient t o consider regular couples. Let and ? be regular couples and l e t z E C(x’) and T E L(x’,?) be

Since now

r?

such that

IIzIIc(2) 5 1 and

(2.4.32)

IITII?,? L 1 .

We must prove t h a t (2.4.33)

IITzllG(?)

5

in this case as well. In view o f (2.4.30), for fixed z E

A(x’) and y’ E A ( F )

subject t o the condition II~’llF((gt)o)5 1 9

(2.4.34)

we have the following inequality:

(“?%Y?

= (EgY’,wc(?) =

= (T’qJY’, &(2) = (ERT’Y’, &(R)

T‘Y’) = (K.2z,

5

IlnnzllF((2t)O)l~

=

~ T ’ d ~ ~ F (’ ( ~ ~ ) o )

Since F is a functor, it follows from (2.4.32-4) that the right-hand side does not exceed 11z11G(2)

!lT’ll?~2~ Ily‘llF((nf)o)I IITlld,? 5

.

Thus, we have established that SUP {(K.pTz,YO

;

IlV’llF((?~)O) I 11 5 1 .

On the other hand, in view o f (2.4.31), the left-hand side is equal t o

IITz~~~(~)

so t h a t inequality (2.4.33) is proved.

Thus, G is a functor. It also follows from (2.4.29) that for regular couples

2 the embedding

Interpolation spaces and interpolation functors

200

(2.4.35)

DF(2)

G(2)

F'(x'") = F((X')')']. Let us now prove that if is regular, then

is valid [since

x'

(2.4.36)

G(d')

A F(2)' .

If this is established, we obtain from the maximal property of D F (see Definition 2.4.29)

G(2)

DF(2).

Together with (2.4.35) and (2.4.31), this proves the theorem. 0

Thus, it remains to establish the embedding (2.4.36).

First of all, it

should be noted that since 62 embeds A ( 2 ) into A(2") isometrically and A(X) into X i , i = 0,1, densely, "2 : + (d")c. Therefore, it follows 4

x'

from the interpolation inequality for F that (2.4.37)

IlnaxllF((att)") 5 IIzIIF(a) .

Suppose now that x E A ( 2 ) and z' E A(x"). Then in view of Proposition 2.4.11, we have (2.4.38)

( ~ ' ~= 2 )( K ~ L 5,')

= (IE(~,)~ "2") z', . I ' E E'(2') and z E A ( 2 ) . Indeed, ( z ) ~ ) ~c ~A(X') N which converges to

Let us verify that this equality. holds for

-s

in this case there exists a sequence 3'

in E0(x''). Applying (2.4.38) to zk,passing to the limit as n + 00 and

considering that continuously maps C'(2') into C ( i ' " ) , we establish the validity of (2.4.38) in the case under consideration as well. From (2.4.38) and (2.4.37) we now obtain

Since the bilinear form on the right-hand side does not exceed

201

Duality

we have

ll4lF(2~)5 l l 4 l G ( 2 ~ ) 7

and embedding (2.4.36) is proved. To conclude the section, we consider two examples of calculation of the functor D F . These results confirm t o a certain extent the intuitive idea about the dual nature of the functors o f an orbit and coorbit.

Let

A' be a regular couple and let A be regular intermediate space.

We

Put

F

:=

G := CorbA,(.;

OrbA(i; ; a)

2).

Then the following theorem is valid. Theorem 2.4.32 (Bwdnyt').

The norms of spaces D F ( 2 ) and G ( 2 ) coincide on A ( 2 ) . If, in particular, (2.4.39)

G(2)

(G(2)')" ,

the equality

D F ( Z ) = G(Z) holds.

Proof.Obviously, it is sufficient t o establish that Go(Z)

DF(2)

&

G(2).

Indeed, this relation leads t o the first statement o f the theorem. Further, from the relative completeness of D F ( 2 ) (see Proposition 2.4.30) and the left embedding, we obtain (G(X)')"

D F ( 2 ) . Combined with the first

embedding and Proposition (2.4.39)

this proves the second statement of

the theorem. Let us begin with the proof of the rightmost of the embeddings written above. In view o f the definition of D F , we have

Interpolation spaces and interpolation functors

202

DF(2)

L: P ( Z ) ’ ,

A (see Theorem 2.3.15), and hence F ( 2 ) := OrbA(2,L)A) 1 F ( 2 ) ’ + A’. Together with the previous embedding, this gives

where

&

DF(& Since

G

A‘ .

2)is the maximum of the functors for which the

:= CorbA,(.;

above embedding is valid (see Theorem 2.3.17). we have

DF

(2.4.40)

L: G .

In order t o prove the inverse embedding, we observe t h a t in view of (2.3.22),

11211~(2 = )

(2.4.41)

{c

IITnII,T,Y

IlanllA}

7

where the infinum is taken for all representations of series

C,

~ , a ,convergent in

x

in the form of the

~(2).

Since A(2) is dense in A, we can (and shall) assume, without any loss of

A(2). Let now x’ E A(-?) and x E A(-f). We represent x in the form of a series En T,a, convergent in C ( X ) ,where generality, t h a t all a, belong t o

T, E L ( 2 , z ) and a, 2.4.6,

A(*)

E A(A), n E

Hv.

Since, according t o Proposition

= C(x’)‘, x’ extends t o a continuous linear functional x* E

C(x’)*. Here ( ~ * , x ) ~= ~ (( x2 ’), ~ and ) the series

C, T,a,

converges in

~(2 since ) all ~ , a ,E ~ ( 2Consequently, ). (XI,x)

=

c

(z’,

T,a,) =

n

c

(“XU,,

T y ).

The last identity follows from Proposition 2.4.11 if we observe that TAX’ belongs t o

A ( 2 ) . Let us estimate each term on the right-hand side. We

have (“#&a,

TAX’)5 IIT~x’llqR) lI“XanllG(&)t

In view of the definition of

G(2)

G

and in view of Theorem 2.3.17, the embedding

A’ holds. Consequently, A”

1

G(2)’. Hence, taking into

account Proposition 2.4.11, we obtain (2.4.42)

((“,T%(IG(A’)’

‘

5 lI“,TanllA” 5 IlanllA

*

203

Duality Moreover, the interpolation inequality for

IITAx’llG(2)

G gives

5 I I ~ A I I ~ ~Ilx’IlG(2~~ , . z ~ 5 IITnlli.2 II4lG(2$) -

The last three estimates lead t o the inequality

). 5

c

IITnllx,2

llanllA

n

llx’ll~(2~) .

Taking here the infimum for all representations of

En Tnanand taking into account x, 5

(2.4.41). we get

11’11F(m) l l x ’ l l G ( 2 ~ )7

whence it follows that for I’ E 11Z’IIF(2)l

x in the form of a sum

5

A(Z’),we have

11Z’llG(21)

*

This leads t o the embedding Go(?)

&

F ( 2 ) ‘ . In view of the maximal

property of D F , we obtain from this embedding

Go

&

DF,

which together with the embedding (2.4.40) proves the theorem. 0

CorolI ary 2.4.33.

If the space G ( 2 ) is regular, then

~(2 = DF(X-). )

I t is now natural t o consider the calculation of a functor dual to a coorbit. Unfortunately, the functor obtained strongly differs from the orbit even on the intersection. The situation becomes simpler for a coorbit of a dual couple. This circumstance was noted for the first time by Janson whose theorem will be given somewhat later. We begin with the following general result, where in the formulation we put, as before

204

Interpolation spaces and interpolation functors

We assume that

A’ is a regular couple and the intermediate space A E I ( i )

satisfies t h e condition (2.4.43)

A

&

(A’)”

whose meaning will be clarified later. Theorem 2.3.34 (Brudny6Krugljak). For a given regular couple (2.4.44)

x’, the equality

G(x’)‘ = F ( 2 )

is satisfied iff t h e closed unit ball o f the right-hand space is

*- weakly closed

in the space A(x’)*.

Proof.We shall require Lemma 2.4.35. Suppose t h a t the space X E

J’(d’). For this space t o be dual, it is necessary

(and if the condition (2.4.45)

X

Xoc

is satisfied, also sufficient) t h a t the closed unit ball

in the space

B ( X ) be *-weakly closed

A(Z)*.

Proof. Let X = Y’for a certain space Y E

B ( X ) :=

{z’ E

where V := B ( Y )n

B(X)=

g ( 2 ) .Then

A(Z)*; I(x’,z)I 5 1, z E V } ,

A ( 2 ) . Consequently,

n

{z’E A@)* ; I(Z’,Z)I

I 1) .

ZEV

Since each set in the braces is *-weakly closed, the necessity is proved. In order t o prove the sufficiency, we define Y with the help of the norm

Duality

205

where z E Co(z). In view of t h e properties of the map defines an intermediate space for the couple (2.4.47)

B ( X )n ~

Indeed, if z E

( c3B(Y) )

IE,

formula (2.4.46)

go.Let us verify that

I

A(X'), norm (2.4.46) coincides with the quantity

In view of this inequality,

Ilz'll~j := so that z' belongs t o that

Xo

SUP{(^',^); z E

B(Y')as well.

A(-f),llzll~ 51)5 1 ,

This proves (2.4.47). It follows hence

Y'. Since Y' is relatively complete, (XO)"

(Y')"= Y'.

Taking into account (2.4.45), we obtain the embedding

x L: Y'. Let us verify that here indeed we have an equality. If the opposite is true, there exists an element y' belonging t o B(Y')\B(X). Since B ( X ) is closed in the *-weak topology of the space A(X')*, it follows t h a t there exists a linear functional

F

which is continuous in this topology and which strictly

separates y' from B ( X ) . Thus,

F(y')

> sup{F(z');

But each such functional

F( 2') = ( 2 ' 7

2 '

E B(X)).

F has the form

ZF) 9

where z~ is a certain element from

A ( 2 ) (see Dunford and Schwartz, [l],

Theorem V.3.9). Consequently, the previous inequality can be written in the form

Interpolation spaces and interpolation functors

206

By (2.4.48) the right-hand side is equal t o I I I F J I Y , while the left-hand side does not exceed Ily'llyt I l z ~ l l y5 I ~ I F I I Y . We have arrived a t a contradiction. Thus,

X

= Y' and the sufficient condition is proved.

0

Lemma 2.4.36. Let the space A E J(i) satisfy condition (2.4.43). Then, putting X := OrbA(li; (2.4.49)

p),where ? is an arbitrary couple, we have X

Proof. Let

A (X')" .

A,,, E Int(2) be a minimal interpolation space containing A .

Then, in view of (2.4.43), we have (2.4.50)

A

A (A')" A

.

(A:)'

Furthermore, in view of Theorem 2.3.15, (2.4.51)

+

OrbA(A;

a)

= OrbA,(d;

.)

.

Since in this case A$ E I n t ( i ) , by the same theorem we obtain

We put

F(.)

:=

O r b A k ( x ; .)" .

Then F is a functor, and in view of (2.4.50), A the minimal property of the orbit that (2.4.52)

X := OrbA(A'; ?)

A F(?)

&

F ( x ) . It follows from

:= orbAk(A';

?)' .

Since OrbAL(2; .) A OrbA,(x; .)' in view of the minimal property of the orbit, it follows from (2.4.52) and (2.4.51) that

x & 0

(OrbA(li;

p)')"

:=

(x')".

207

Duality Lemma 2.4.37.

In the hypothesis o f the theorem, equality (2.4.44) holds for a given regular couple

I? iff OrbA(A',x')

is a dual space.

Proof.The necessity of the condition is obvious. Suppose that for a certain regular space X E (2.4.53)

Let us verify i t s sufficiency.

J ( 2 we ) have

X' = OrbA(i; 2').

-

L e t us verify the validity of the embedding

(2.4.54)

x'

1

(COrbAt(3;

2))' .

For this purpose, we first establish that

F

A

DG.

In view of the minimal property of the orbit, we only have t o prove that (2.4.55)

A

A

DG(A')

.

But according t o Theorem 2.4.31, we have

DG(A') = ~ ~ ' [ ( C o r b ~ t ( 2 , k ? )A ' ] Ki'(A'')

.

By Theorem 2.4.12, the right-hand side is equal t o (A')" and hence contains

A in view of condition (2.4.43).

and hence the required

Thus (2.4.55),

embedding of the functors, is established.

But this embedding and the

definition of the dual functor lead t o

X' := F ( 2 )

A

DG(2)

A

G(2)'.

Taking into account the definition o f G, we obtain (2.4.54). In order t o prove the inverse embedding, it is sufficient t o show t h a t (2.4.56)

x

1

(CorbA'(2; 2))':= G'(2)

L)

.

Let z E A ( 2 ) . Then from the definition o f the coorbit, we have

IIzllq2) =

SUP

:=

SUP

{ ( T r , a ); llQllA 5 1, IITll2,At 5

{(T'K,&'(a),z); 11allA

5 1,

=

a E A ( x ) , llTll2,2 5 1)

*

208

Interpolation spaces and interpolation functors

Here we take into account the fact t h a t A’is regular and make use of (2.4.7). Since the element

( T ‘ K A ) (E~A) ( P ) and the couple d is regular, we obtain

In view of Proposition 2.4.19, ~ ~ T ’ =~IITll23. ~ ~ l ,Moreover, ~ , according to Theorem 2.4.12 we have

Combined with the previous inequalities, this gives

Using the regularity of X, we obtain (2.4.56). Let us now prove the theorem. If condition (2.4.43) is satisfied, by Lemma 2.4.36 the space X

X

1

v

(XO))‘.

:= OibA(ki;

2’)(:=

F(r?‘))satisfies the condition

In accordance with Lemma 2.4.35, this embedding and

the condition of the *-weak closedness imply that X is a dual space. It remains t o apply Lemma 2.4.37 to X and obtain equality (2.4.44). Thus, the sufficient condition is proved. Let now equality (2.4.44) be satisfied.

Then OrbA(2;

2’)is

a dual

space, and in view of Lemma 2.4.35, the unit ball of this space is *-weakly closed in A ( X > * . This proves the necessity. 0

Duality

209

Remark 2.4.38. Condition (2.4.43) has a simple meaning. Namely, it singles out an interme-

diate space A of the couple

i, for which K A A )ct A".

Finally, l e t us consider another similar result which was mentioned before Theorem 2.4.34 was formulated. intermediate spaces (2.4.57)

Let the couple

A' be regular and let the

B E I ( 2 ) and A E I ( 2 ) be such that

.

K ~ ( A=)B'

Further, we assume that

OrbA(i;

a)

is generated by a single element a

E A.

Thus,

Then the following theorem holds. Theorem 2.4.39 (Janaon).

For any regular couple

2,the equality

(CorbB(2,z))' = OrbA(2;

2)

holds. 0

The reader can find the proof of this theorem in Janson's paper quoted in Sec. 2.7, item A. Here we shall only clarify the role of condition (2.4.58). For this we note that the unit ball of the space O r b a ( i ; X ' ) is the image of the a unit ball

L , ( i ; 2')for the map

cp :

T

4

Ta

(we assume that

11~1= 1 ~ 1, which obviously does not lead t o any loss of generality).

It can be

( i ; 3 )is a conjugate space (see Proposition 8 in Section 2.7.2). Consequently, the compactness of the ball L I ( 2 ; 2')in the verified that the space L

*-weak topology follows from t h e Banach-Alaoglu theorem. On the other hand, the map cp is obviously continuous in the *-weak topology, and hence the unit ball of the space Orb,(i; of the space

2)is compact

in the *-weak topology

A(x')*. Thus, (2.4.58) implies that the unit ball of the space

Interpolation spaces and interpolation functors

210 Orb,(A;

it) is *-weakly

closed (cf. the corresponding condition of Theo-

rem 2.3.34).

Remark 2.4.40. It would be interesting to check whether Theorem 2.4.39 is a corollary of Theorem 2.4.34.

Minimal and computable functors

211

2.5. Minimal and Computable Functors

A. An arbitrary interpolation functor does not have a wide range of useful properties. The functors which will be introduced and investigated in

this section are much richer in this respect. This is due t o the fact t h a t these functors are completely determined by their values in the subcategory -.

F D of finite-dimensional regular couples. Most of the results considered in this section are based only on the properties of this subcategory which are described in the following proposition. Proposition 2.5.1. +

--t

(a) F D contains a subcategory

FD1 of all one-dimensional regular couples.

+

(b) The subcategory (c) For any couple

F D contains, along with any two couples, their Il-sum.

2, the set

F D ( x - ) :=

( 2 E F Z l ; A c: x-}

is directed by inclusion. (d) For any operator T E

L(z,z), where 2EF%

and

2 E 6,the couple

T ( 2 )belongs t o @(d). (e)

-

F D is a small subcategory of

Proof.Properties (a)

6.

and (b) are obvious, while property (e) was established

in Example 2.3.28. Let us prove (c). Let

A+ B'.

A' and B' belong t o F% (2) and

c' EF> (2)and 2,B' established t h a t F> (2)is a directed set. c'

:=

Then

Finally, property (d) follows from the fact that A0 a regular finite-dimensional couple. Therefore, T(A0)

c'. S

Thus, we have

A1 since A' is T(A1) as well,

so that the finite-dimensional couple T(A)is regular. In view o f Definition 2.2.16(b), T ( 2 ) A x', i.e. T ( 2 )EF% ( X ) . 0

Interpolation spaces and interpolation functors

212 Remark 2.5.2.

A subcategory I? c

6 possessing the properties (a)-(d)

o f the above pro-

position will be called factorizing. Using similar arguments, most o f the results considered below can also 4

be proved if we replace F D by an arbitrary factorizing subcategory. It can 4

be easily seen that in this case F D is the minimal factorizing subcategory

2 is the maximal subcategory.

and

Another example is the subcategory R

of the couples formed by reflexive spaces. Let us now describe the first o f the classes o f functors analyzed in this section. Definition 2.5.3. The functor F is called a minimal functor if it coincides with the minimal Aronszajn-Gagliardo extension o f its restriction FI 4

-

FD

to the subcategory

FD. W e denote the class o f minimal functors by Min. 0

Recall that the construction o f minimal extension is described in Theorem

+

2.3.24. In the case under consideration we regard the class

{2, F ( i ); A’ EFD

} as the class o f triples K appearing in this theorem. Consequently, the fact that F is a minimal functor is equivalent t o the possibility of representing the norm o f the space F ( 2 ) i n the form

Here we take the infimum over all representations of z i n the form z =

C

~,a,

(convergence in

~(2))

n

4

and

(/in)nEm runs over the sequences from F D . d

The existence of additional properties o f the subcategory F D allows us t o simplify formula (2.5.1) considerably. Indeed, the following proposition

213

Minimal and computable functors holds. Proposition 2.5.4.

The functor F E Min iff for any

x'

t h e norm in the space

F ( 2 ) can be

represented in the form

x in the form

where infimum is take over all representation o f the element

z =

C a,

(convergence in

~(2))

n

+

+

(in),,== runs over the sequences from F D ( X ) . Proof. The necessity follows immediately from (2.5.1). and

Let us prove the

sufficiency. The infimum in (2.5.2) coincides with the norm of t h e element z in the sum of the Banach family

(F(A'))

PGFD(2)

(see 2.1.34-2.1.36).

Consequently, the right-hand side is the norm o f t h e Banach space

Let us show t h a t the map G :

+ B is a functor. We shall first prove

that G ( x ' ) is an intermediate space of

2,generated

A' L x' for A' EF% (x'),

F ( X ) L) C ( x ' ) .

G(x')

Further, suppose that z E A ( 2 ) and of

Since

F(r?), and from the definition of G it follows that

we have F ( i )

(2.5.3)

x'.

x".]

by this element. Then

(2.5.1) and t h e definition o f

11x11G(2)5

is a one-dimensional subcouple -.

x'rz1 E F D (2).and

in view of

G , we have

~ ~ z ~ ~ 5 F 7 ( ~ llzllA(nbl) f ~ l )

where 7 is the constant of embedding of

=

llxllA(a)

7

A ( 2 ) in F(.J?).

I ( x ' ) , and it remains t o prove the interpolation inequality. For this we take T E ,Cl(x',?) and z from the unit sphere of G ( X ) . Then Thus, G(x'7)E

we only have t o prove the inequality

hterpolation spaces and interpolation functors

214

'

5 .

IITzllG(?)

For this purpose, for a given E

> 0 we take a representation of z in t h e form

of the sum

C, a,

(2.5.4)

C llanllqJn) 51 + E

such that

This is possible since operator

T,

:=

IIzllG(~)=

(2, EF% (Z), R E N ). 1. In view of Corollary 2.1.17, for the

TIE(~n) we have

A P

Tn(2n)

IITnIIin,q,in) 5 IITII~,? 5 1*

9

+

Since the couple

7

f

B,

:=

T,(&)

belongs t o

FD

(2)(see

2.5.1), taking into account (2.5.2), (2.5.4) and the identity

Proposition

Tx = C T,a,,

we have

5

IITxllG(?)

5

c

5

'

~ ~ T ~ a ~ ~ ~ F ( & ~, )~ a ~ ~ ~ F ( +&&)

Hence, G is a functor in view of the arbitrariness of

'

E.

- for couples from F D , d

Since the functor G obviously coincides with FI

FD

the minimal property o f F leads t o the embedding F

1 L)

G. Together with

embedding (2.5.3), this gives the equality F = G. 0

Corollary 2.5.5. A minimal functor is regular.

Proof. If z

+

2 = C, a,, where a, E F ( X , ) , + F (2) ~ and C, ~ ~ u , ~<~00.~ Since ( J ~for ) a couple A' in F D (2) we have C ( 2 ) E A(A'>,it follows that C ( i ) L) A ( 2 ) . Consequently, each summand a, belongs t o A(x'). Let now N be such that C n > l l~& l l F ( ~ n ) < E . Then in view of (2.5.1), for the element 6, := C n l N a, E A(X) we

A, E

have

E F ( X ) , in view of (2.5.1)

Minimal and computable functors

215

In order t o define the other subclass of functors under consideration, we require some preliminary analysis.

X be a directed family of Banach spaces. Hence for a certain Banach space W we have Thus, l e t

X

(2.5.5)

4

W

,

X EX

and, moreover

X , Y E X =+ 3 2 E X ,

(2.5.6)

Further, l e t

UX

X,Y

2

.

denote the union of the sets of the family. For z E U X

we put

Let us show that

U X is a linear space and that formula (2.5.7)

defines a

norm on it. Indeed, since i n view of (2.5.5) we have

Ilzllux = 0 iff2 = 0. Therefore, it is sufficient t o verify only the triangle inequality. Suppose that 2 = 2 1

-

((2i((xi E

for a given

+

E

E X are such that Ilzillux 2 > 0, while the space 2 is such that X i 2, 22

and spaces Xi

i = 0 , l . Then

llzllux

+ 4 l z I 112111x1+ I I l ~ l l l O X+ Il.2llux + 2 E , I

and the required statement is proved as Definition

I

ll~211xz

1121

E

4

0.

2.5.6.

The limit of a directed family of Banach spaces X is the (abstract) completion of the normed space

UX.

We denote this completion by lim X . Thus, (2.5.9) 0

lim X = (UK)"

Interpolation spaces and interpolation functors

216

We now have everything t o formulate the main definition. Definition 2.5.7.

A functor (2.5.10)

F

is called computable if for any couple

F ( 2 ) = lim F ( 6

2 we have

(2)) .

Here we put (2.5.11)

F(fi

(2)):=

{F(A ' ); xEF%

(2).

We denote the set o f all computable functors by Comp. 0

Remark 2.5.8. (a) In view of statement (c) of Proposition 2.5.1 and the embedding F ( 2 )

F ( 2 ) ,which is valid for any A'in F D (2), the set (2.5.11)

is a directed

Banach family. Thus, Definition 2.5.7 is consistent. --.+

(b) The limit in (2.5.10) can be taken only for the directed family FDo (2) of those El% (2) which are subcouples of 2.Indeed, each couple

A' A' EF% (2) can be replaced by its image I ( A ) ,where I

:=

A'

x',

by taking in the space I ( A , ) the norm induced from X i , i = 0 , l . Here

(1 . [ ( ~ p5,(1). I ( A ~ ,

( .

i = 0,1,and hence the norm in U F ( F D 0

(a))does

(a)).Since 6 0 (a)C F D (a), the inverse inequality also holds so that U F ( G 0 (a))coincides with not exceed the norm in

UF(F3

U F ( F G (2)). Let us now establish the relation between the classes of functors introduced in this section. ProDosition 2.5.9. Comp

c

Min.

Proof.We shall require

Minimal and computable; functors

217

Lemma 2.5.1Q.

A functor F E Min iff the norm in F ( 2 ) can be written i n the form

where the infimum is taken over all sequences which are fundamental in the space

(2)) and converging t o 5 in C ( x ' ) .

UF(F%

Proof.

Recalling t h e definition o f the Cauchy completion (see Definition

2.2.26), we see t h a t the right-hand side o f (2.5.12) is a norm in the space

(2)))". Further, l e t Co denote the algebraic sum of the family of spaces F(F% (2)) supplied with t h e norm (UF(F%

(2.5.13)

I I ~ I := I ~ i~d ( C

IIanIIqA,,)}

*

Here the lower bound is taken over all representations of z in the form of finite sums: z = C a,, where an E

F(2,)

and

2, EF% (2). The right-

hand side of (2.5.2) is, in view o f the same Definition 2.2.26, a norm i n the Cauchy completion of the space Co. Thus, t o prove (2.5.12) we have only t o establish that the normed spaces

(2)) coincide.

Co and UF(%

But if

x belongs t o the union, in view o f (2.5.13) we have

llzllu so that

U -+

1

:= inf { I I X l l q ~ ,;

A' E F G l

(m1

11~11C0

E

>0

M

that

Co. Conversely, suppose that z E Co, and for a given

we have x = Cr an and N

c

IlanllF(An)

5 llzllCo

+E

*

1

Using the fact that

A', A

i f o r 15 n

( 2 ) s directed, we choose

5N.

A' EF% (2)

Then the left-hand side of the above inequality

is not less than N

C Ib,lIp(,~,,) 1

N

2C

Thus, for e + 0 we obtain

1

IIanIIqi) 2 IIzIIF(i) 2 IIzIb .

Interpolation spaces and interpolation functors

218

In the further analysis, we shall require the following lemma. Lemma 2.5.11. 4

If F E Comp, then every sequence fundamental in the space U F ( F D

(3))

and converging t o zero in C ( x ' ) converges t o zero in the former space as well.

Proof. Indeed, in view of

Definition 2.5.7 and the identity (2.5.9), we con-

clude that the (absolute) completion of the space UF(F%

(2)) is con-

tained in the same space C ( 2 ) as all the spaces of the Banach family

F(F%

(a)).Since the Cauchy completion is unique (see Definition 2.2.26),

we can write the following equality:

(2))y.

(UF(F3(2)))" = (UF(FD

However, according t o Proposition 2.2.27, for this equality t o hold it is necessary and sufficient that the condition in the statement of the lemma be satisfied. 0

Let us finally prove the proposition. For this we denote by G ( 2 ) the space in which the norm is determined by the right-hand side of (2.5.12). Let us show t h a t

UF($

(2)) is isometrically inclosed in G(Z) if F

is a

computable functor. Since the union is obviously dense in G(L?), it follows that G ( 2 ) is isometric t o the (abstract) completion of the union, i.e. is equal t o

F ( 2 ) [see (2.5.10)]. Thus, the norm in F ( 2 ) can be represented

in the form (2.5.12), and this means that F is a minimal functor in view of Lemma 2.5.10. In order to prove the above isometric inclusion. we only have t o show that the norm of

2

in

G ( 2 ) coincides with its norm in the union for all

z E

Minimal and computable functors

UF(F?D

219

(a)).Otherwise, there would exist a sequence ( a , ) in U F ( F D ) .

(i)), which is fundamental i n this space, converges t o C(x') and such that (2.5.14)

nlim -m

Ilanl)u < Ilzll~.

(z)),converges t o zero in E(d)and does not converge t o zero in U F ( F-D f

Let us show t h a t then the sequence (z - a,) is fundamental in U F ( F D

(2)). For this purpose, we choose for E > 0 a number which

is less than

t h e difference between the right- and left-hand sides of (2.5.14). If (z - a),

converges t o zero in the union, we have

for all n >_ N ( e ) . Passing t o the limit as n + 00, we arrive at a contradiction. Thus, (2.5.14) would lead t o t h e existence of a sequence fundamental .+

---?

in

U F ( F D (2)), converging t o zero in C ( X ) and not converging t o zero

in

UF(Fi)

(a)).This, however, is in contradiction t o the statement of

Lemma 2.5.11. 0

The above proof shows that for a minimal functor F we have

~(2 = (lim ) F(F> where

N

(@))IN ,

:= N ( 2 ) is the subspace o f lim F(F%

(a))generated by

fundamental sequences converging t o zero in C(x'). Thus, the computability of F is equivalent t o the equality N ( 2 ) = (0) for all

2.Although it follows

from general considerations that this equality is not always satisfied, examples of minimal but uncomputable functors are unknown t o us. Let us formulate a convenient criterion for membership of a functor t o the classes M in and Comp. Theorem 2.5.12 (Aizenstein-Brudnyi). (a) A functor

F is minimal iff

Interpolation spaces and interpolation functors

220

F = OrbA(A; *) ,

(2.5.15)

A’ possesses the approximation

where

property, while the intermediate

space A is regular. (b) A functor F is computable iff the conditions in (a) are satisfied as well as the following condition. For any couple

r? and any element I E A(?)

we have

(2.5.16)

where infimum is taken over all (ak)keI

C

finite families

(Tk)&I

C

L(.&?)

and

A(A) for which

Proof.

-

the necessity. Since 3 is a small category (see Definition 2.3.28), there exists a s e t of couples 9 c F D such that each couple F D

(a) Let us prove

---t

is Gl-isornorphic to a certain couple

9 (see Example 2.3.29).

Then, in

accordance with what has been proved in Lemma 2.3.32, the functor F can be represented in the form (2.5.15) where

Here

F(9)

:=

{ F ( r ? ) ;x’ E ?}.

A’ satisfies the conditions of Definition and that A E p(i).Since A’ is an ll-sum, the set of elements

It only remains t o show that 2.4.22

with a finite support is dense in each A;. Each such element has the form

Cap

U&J,

where

9 0

c9

for

6

=

g).

Since each couple

B’ and C’ # I? and 1

is a finite subset, ag E

63 are the basic delta functions (i.e. S,(C’)

= 0 for

B’ is finite-dirnensional

A ( 2 ) 2 C(I?), and hence every element in contained in @I A($) C A(A).

and regular,

A’ with a finite support is

Minimal and computable functors

221

A ( 2 ) is dense in A;, i = O,l, so that A ' i s a regular couple. The fact that the space A is regular is proved in a similar

Thus, we have proved that way.

+

It remains t o verify that A possesses the approximation property. Supand pose that a E A(2); then a :=

Consequently, for a given 9 0

E

> 0 there

exists a finite-dimensional subset

c 9 for which

(2.5.17)

C

i = 0,l .

I ( a g ( (<~E~ ,

bC?\?o

Let us now define a finite-rank operator

P

:=

Pa,, with the help of the

formula

Then in view o f

(2.5.17),we have

and from the definition o f

P

we obtain

so that both conditions o f Definition

(b) Let us prove the sufficiency. Let

F

2.4.22are satisfied.

be given by formula (2.5.15), where

A' possesses the approximation property and A E I"(2).We shall show that the functor F E Min, for which we first prove that it is regular. By the definition o f orbit, we have

Interpolation spaces and interpolation functors

222

Therefore, by the minimal property of the functor on the left-hand side and the regularity o f

2,it follows t h a t

Since t h e inverse embedding is obvious, equality holds in this case. Thus,

F is regular. Further, l e t G be the minimal Aronszajn-Gagliardo extension of the functor FJ

-.

In other words, G is determined by the family of triples

FF (5,F ( B ) ) 8 E F o .Then, according t o Theorem 2.3.18, we have

G

F .

If the inverse embedding is established, we shall thus prove that F coincides with the minimal extension of its trace FJ i.e. F € Min.

-

DF'

In order t o prove the embedding F and

E

>

0 and suppose that P

A

G, we take an arbitrary a E A(A)

:=

in the approximation condition. Then

,C1+,(A,,P(A)) (see

Corollary 2.1.17).

Pa,, is the

finite-rank operator

P E Ll+,(@ and hence P E Here P ( A ) c A(A) and has a

finite rank. Consequently, there exists a regular finite-dimensional couple

B'in F D (2)for which P ( i ) & B'. Then P E L1+,(i,B'), and hence (1 + €1lbIlF(X) 2 IIPallF(B) ' It follows from the definition of G and from the choice of of (2.5.2) t h e right-hand side is not less than

B' that in view

IIPallG(a,which in turn

is not less than

Here 6 is the constant o f the embedding A account the choice of

P

:=

-

Pa,, [see (2.5.22)]

G. Finally, taking into we finally obtain

Minimal and computable functors

223

As E + 0, we obtain the inequality

which can be extended over all a E

F ( 2 ) by using the regularity of F ,

established above. Thus, F

&

G, and hence F E Min.

(b) Let us prove the necessity o f the conditions of the theorem. Suppose t h a t F is computable. Then in view of Proposition 2.5.9 it is minimal.

By what has been proved in (a), F is then representable in the form (2.5.15) with L a n d A specified in Theorem 2.5.12. It remains for us t o verify the validity o f condition (2.5.16). For this we consider an arbitrary couple of

2 and an arbitrary element 5 E A ( 2 ) .

F , we find for a given

IIxIIF(ir) < (1

>

E

Using the computability

B' EF% (2)such that

0 a couple

+ €1 II"IIF[X,.

In view of (2.5.15) and (2.5.1), in this case there exists a representation

x = EEl

+

Tkak

(2.5.18)

[convergence in C ( B ) ]for which

llTkllL,ir ! l a k l l A

< (l -/-

IIxII~(Q .

Since A is regular, we can assume that all

ak

belong t o

A(A).

B'

It follows from the fact that is finite-dimensional and regular that ~ ( 2z) A(@; consequently, the series T k a k also converges in A(@, and hence in A ( 2 ) as well. Therefore, for any 6 > 0 there exists

c

n :=

126

for which

We choose an arbitrary element 6 in A(A) and assume that

f E C(A)*

is a functional such that (f,ii) = 1. Further, we define an operator ' i !

by the formula f U

:=

(f,U)

(5

-

2

k=l

TkUk)

,

aE

c(2) .

Interpolation spaces and interpolation functors

224 Then

so that

f' E L(2,Z). By choosing S sufficiently small,

we can ensure

that the inequality

is satisfied. Thus,

and in view o f (2.5.18), we have

In view of the embedding B'

4 2,we also have the inequality JITII,-,g 2

IITllff,~. Consequently, the right-hand side of (2.5.16) does not exceed the left-hand side o f (2.5.19).

Since, on the other hand, the inverse

embedding is also valid in view o f (2.5.15) and (2.5.1) (2.5.16) is proved. Let us prove the sufficiency. Let F be representable in the form (2.5.15), where

A' has the approximation

property and A E I"(A). According t o

what has been proved in (a), the functor F then belongs t o Min. It remains t o verify, with the help of condition (2.5.16) putable. It follows from this condition that if

x E A(x'), and

E

that it is com-

2 is an arbitrary couple,

> 0 is specified, there exists elements a k 5 k 5 n, such that

E

A(Z) and

operators T k E L ( i , , r ? ) ,1 n

(2.5.20)

=

Tkak 1

and

c

)(Tk(l,.T,j? b k l l A

< (l + &)

Using the approximation condition, we can find finite-rank operators P k and couples i i k

EF%

(A)such that

.

ilZII~(j?)

Minimal and computable functors

225

F D (2) is directed (see Proposition 2.5.1), we find B’ EF% (2)for which T k ( B k ) B’, 1 5 k 5 n. Then,

Using the fact that a couple

obviously,

Without loss of generality, we can also assume that the following inequality holds for the couple

B’:

Indeed, otherwise we can replace

B’ by a larger couple B’ + 2 [ Y l ,

where

z[YI is a one-dimensional couple generated by the elements y := XI=, “‘(ah - Pkak) (see t h e proof of Proposition 2.5.4). Then the -*

new couple is also contained in F D

(2) and inequalities (2.5.22)

and

(2.5.23) have already been satisfied for it. Using now (2.5.21) and (2.5.23), we obtain

In view of the definition of

F

[see (2.5.1) and (2.5.15)], we also have

Interpolation spaces and interpolation functors

226

Since in view of (2.5.21) IIPkJlz,gk5 1

+E

we obtain the majorant (1

+ E , using (2.5.22)

) IIxIIFc2, ~ for

and (2.5.23)

the right-hand side of this

inequality. Together with the preceding inequality, this gives

Hence it follows that

Since F(I?)

1 ~t

F ( - f ) ,the inverse inequality is obvious. Thus, we have

strict equality in (2.5.24), which means that the norms of the spaces

F ( 2 ) and U F ( $

(a))coincide on the subset A ( 2 ) [see (2.5.7) and

F is minimal implies that it is regular A(@ is dense in F ( - f ) . Obviously, A ( 2 ) as well, and hence in lim F(F> also.

(2.5.11)]. However, the fact t h a t (see Corollary 2.5.5), so that is dense in

(a))

UF($

(a))

This means that F ( 2 ) = lim F(F%

(2)). and F

is computable.

B. I t is expedient t o note for the further analysis that all concepts and results given above permit localization. In particular, we say that a functor

F is minimal on a couple 2 if F ( 2 ) coincides with the value of the minimal extension of the trace

FI

Min( 2).

-

FD

on

2.The set o f such functors is denoted by

Similarly, using the equality (2.5.25)

F ( 2 ) = lim F ( f i

(2)) ,

we can define a functor computable on a couple is denoted

2.The set of such functors

by Comp(2).

An analysis of the proof of the preceding item leads t o the following useful fact. ProDosition 2.5.13. The statements o f 2.5.4, 2.5.5, 2.5.8, 2.5.9 and 2.5.12 are valid for the

Minimal and computable functors classes Mi*(-?)

227

and Cornp(x’) if we replace the expression “any couple

by “a fixed couple

2‘

3’.

0

Let us introduce t h e following definition. Definition 2.5.14.

A couple x’ is called universal if every functor regular on on this couple.

x’

is computable

0

The existence o f universal couples is assured by Proposition 2.5.15. +

If a couple X possesses t h e approximation property, it is universal.

Proof. Just

as in the proof o f sufficiency in Theorem 2.5.12(b), we only

have t o prove inequality (2.5.24). Since the functor

F is regular on i,it is

A(X) only. To this end we choose for a given E > 0 a finite-rank operator P := P,,, which satisfies the conditions of Definition 2.4.22. Then P : C(x‘) + A($), and hence there exists a

sufficient t o establish (2.5.24) for z E

finite-dimensional subcouple

B’ of the couple x’, for which

Generalizing 2 if necessary [see the corresponding arguments in the proof of inequality (2.5.24)], we can also assume that

11%

- PzllA(8)

= 1Iz - p z l l A ( f )

’

Consequently, denoting by 6 the constant of embedding of A(@ in F ( a ) , we obtain from the preceding equality

ll41F(B) 5 6 IIz - P 4 l A ( B ) + IlPzllF(8) 5

<

115

- PzllA(a)

+ IIPzIIF(P(f)) .

In view of inequality (2.5.23) and Corollary 2.1.17, the operator

PE

,Cl+e(z, P ( 2 ) ) . Therefore, from this inequality and (2.5.22) we obtain

Interpolation spaces and interpolation functors

228

Il4lF(s, I + (1+ €111~11F(a) . Hence it follows that for all

2

E A(-?), we have

B’

11~11F(W) = inf {ll4lF(B’);

(x’))

*

The analysis is completed by the same arguments as at the end of the proof of Theorem 2.5.12. 0

C . Let us now prove that unlike the general situation (see Theorem 2.4.17), the map

F’ o f the computable functor F is a functor. Namely, the

following theorem is valid. Theorem 2.5.16 (Azrenstek-Brudnyi).

If F E Comp, then (2.5.26)

DF

= F’

.

Proof.We shall first show that F’

is a functor. Recall that F‘ is defined on

-I

the category B o f conjugate couples by the formula

F’(P) := F(x’)’ . First of all we have, in view of Theorem 2.4.24(b), for any couple any finite-dimensional regular couple (2.5.27)

x‘

and

B’

T ’ ~ F (EYL,(F(P)’, ~ F(B)’)

T E L*(-?,?). Next, let B’ EF% (2); since B’

for any

A x’, the trace operator Rs,given by

the formula

R ~ x ’:= ~ ’ ( ~ ,( g )2’ E C(x“) , is well-defined.

Let us verify that (2.5.28)

Rg E &(I?’,$)

.

229

Minimal and computable functors Indeed, from Bi

1

+

Xi,

i = 0,1, and the definition o f R g , we

I 1) 5 L sup { (2’,2); Il~llx.I 1 7 2 E A@))

IIRgX’IIB: =

SUP {(2’7b) ; IlbllB,

Let us now suppose that operator

have

d

and

? are arbitrary

= ll~‘Ilx;.

couples and that the

T belongs t o .C,(?,d‘). To prove t h e functoriality o f F’, we have

just t o prove the interpolation inequality

(2.5.29)

llTY’IlF(2)fI lIY‘IIF(P)~7

For this purpose, we also take b E

Y’ E

*

A(@, where B’ EF%

( d ) and , with the

help o f (2.5.27) and (2.5.28), write

(TY‘9

b, 5 IIbIIF(g)IIY’llF(P)#

for b E A(@ and y’ E C ( Y ’ ) . In this inequality, we take the i d over all couples

-

B’ EFD (2) for which

b E A(@. Then, using the computability o f F , we get

(2)). However, A(d)(see the proof o f Proposition 2.5.4). Hence, taking the sup in the last inequality over all b E A ( 2 ) with llbllF(d)I 1, we Here, b is an arbitrary element from the union UF(F%

this union coincides with

obtain from it inequality (2.5.29). Thus, F‘ is a functor. It now remains t o prove that F’ =

D F . We

shall establish a more general result which will be found useful later. Let us consider the “intermediate” functor

D- F FD

maximal among all functors G for which

G(2‘)

A

F( 2) ’

(= F‘(2‘))

dual t o F and defined as the

Interpolation spaces and interpolation functors

230 on all couples

2 E&.

Our aim is to prove the formula (2.5.30)

F’ = D F = D

-F .

FD

To begin with, we note that

DF

is defined in the same way as

.-.-+

F D is replaced by a larger category DF

1 L+

D

- F , but

FD

6. Hence we obtain directly

D-F. FD

In view of this, equality (2.5.30) follows from the two embeddings

(2.5.31)

DFiDF

&

F’

DF

which we shall establish. The second embedding follows directly from the fact that F’ is a functor, and also from the definition of D F as the maximal among all functors Gfor F‘(2‘) for regular 2.In order t o prove the first embedding

which G ( 2 )

in (2.5.31), we take

x’ E C ( 2 ) such that

1 1 ~ ~1 1 F~ ( 2 ’ ) = 1. In this case

(2.5.31) follows from the inequality (2.5.32)

FD

llx’Ilq~)t51

which we shall prove. By the definition of the maximal extension [see Theorem 2.3.24(a)], we have

-

11xf11~

= SUP { IITz’~~~(J), ; T E &(?,

~(21)

FD

f?), B’ EF%} .

Hence, in view of the choice of x’,we obtain (2.5.33)

T E Ll(z’,g),B’ EF% I I T z ’ ~ ~5~1~, ~ ,where , +

.

(z),

in F D and let Rg be the above-mentioned operator Let us now take R d . In view of (2.5.28) and (2.5.33), we obtain for b E A(@ (2’7

b) = (Ril”’, b) I IIRs~’IIF(ir,~ IIbllF(8) I llbllF(Ef) .

Taking the inf over all B’ from F % (2) for which b E A ( 5 ) and using the computability of F as in the proof of (2.5.29), we obtain the inequality

Minimal and computable functors

(.‘,a)

I Ilbllqw,

7

231

b E A(Z)

7

which is equivalent t o inequality (2.5.32). 0

Remark 2.5.17. If we j u s t assume that F E Cornp(2) for a fixed couple leads t o the following statement: (2.5.34)

2,the above proof

F(Z)’ = D F ( Z ’ ) = D F Z F ( 2 ’ )

In particular, it follows from here and Proposition 2.5.15 that equation

(2.5.34) is valid for any couple satisfying the approximation condition, and for any regular functor on it.

By way of a corollary, we obtain from the above result the following important statement: Theorem 2.5.18 (Janson). If the functor F := orbA(/i; .) is computable, then

Proof. Using the embedding in Theorem 2.4.32, (2.5.35)

corbA,(.; A)’

A

DF

A

we obtain

corb,p(.; 2).

Since F’ = D F in view of (2.5.30), we get

F’ A CorbA,(. ; 2). Let us prove the inverse embedding. In view of relation (2.5.30), F’ = F . It then follows from the definition of D F that it is sufficient to

-

-

FD

DFD

B’

+

prove the inverse embedding for the couples 3, where E F D ,only. Thus, the whole problem is reduced to proving the embedding (2.5.36)

CorbAt(9; 2)A F(B’)’,

B’EF%

.

Interpolation spaces and inteIpolation functors

232 Since the couple

9 is regular and finite-dimensional,

A($)

S

C(l?),

and hence the space on the left-hand side of (2.5.36) is regular in the couple

g.Accordingly,

the application of the left embedding in (2.5.36)

and (2.5.30) gives

CorbA,(i?, 2)= CorbA,(i?,/?)'

D F ( 9 )=F'(2) .

This proves (2.5.36). 0

Remark 2.5.19.

As mentioned in Remark 2.5.17, this result can also be localized. In particular,

(2.5.37)

OrbA(A',@' = corbA'(?,,?)

for regular x a n d A and any couple x' satisfying the approximation condition. Indeed, in view of Theorem 2.5.12, the orbit in this case belongs t o Min and therefore is regular. All that remains now is t o use Proposition 2.5.15.

D. We now show that the functors

in the above-mentioned classes are

invariant under the action of linear operators, as well as certain nonlinear operators. In order to formulate this result, we recall that for given Banach spaces X and Y, Lip(X,Y) is the space of Lzpschitz mappings of X into

Y . Thus, T E Lip(X,Y), if T : X -+ Y and

Also, we assume that (2.5.39)

T ( 0 )= 0

.17

In this case, (2.5.38) is a Banach norm. Definition 2.5.2Q. The space Lip(2, p) consists of continuous mappings

Minimal and computable functors

T

233

: C(x')+C(?)

for which

T ( x ,E Lip(X;,K)

, i = 0,l .

The maximum o f the Lipschitz norms of the operators Tlx, is taken as the norm in this space. 0

?),one encounters in the applications the subz f i o m the couple x' to the couple space Lip(")(x',?) (strong ~ i p s c h i t maps ?). This subspace is described in Besides the space Lip(x',

Definition 2.5.21. The space Lip(")(x',?) consists o f the mappings T

:

C ( 2 ) + C(?),

having the following properties: (a) T ( 0 )= 0; (b) if 21 - zz E X i , then T(z1) - T Y ( z 2 )E Y,, and

The norm in Lip(")(x', ?) is denoted by maxi,o,l

M;.

0

In order t o verify that Lip(")(T,?) is a closed subspace o f L i p ( 2 , ?), it is obviously sufficient t o prove the continuity o f T E Lip(")(x',?) which is considered as a map from proposition is indeed true.

E(d)t o C(?).

We shall prove that the following

Interpolation spaces and interpolation functors

234 for all z1 and x 2 from

E(d).

Proof. If I = zo + zl,where

z; E

X ; , then

in view of the conditions of

Definition 2.5.21,

+

T(ZO

11)

- T(z1) E Yo and T(z1)= T(z1)- T ( 0 )E Yi .

Hence, taking into consideration the definition of C(?), we have

IIT(~>llc(P) IIIT(z0 + 4 - T ( ~ l > l l Y o+ ll~(~1)IlYl I

I II~IILi,(B,P)(ll"ollx~+ Il~1IIX1). Taking the inf over all z presented in the form z o

+ z1,we obtain

lIT(z)lJZ(P) 5 IITIILip(R,P) 11~11Z(a)*

(2.5.41)

Let us apply this inequality t o I := 12

- 21,

+

where z; E C ( X ) , as well as

t o the operator T defined by the equality F ( z ) := T ( z

Obviously, choice of

+ z1)- T ( z 1 ).

? also belongs t o Lip(")(g,?).

T and I,we obtain

Hence for the above-mentioned

inequality (2.5.40) from equality (2.5.41).

0

-

Although the classes Lip and Lip(") are identical for some couples (say, for

x'

from

FD),

the first class is generally much larger than the sec-

If1

ond. Thus, the simple operator

T

2

= [0,1], belonging obviously t o Lip(x'),

:= ( L m ( I ) , C ( I ) ) where ,

I

:

f

+

does not belong t o Lip(")(x'). Indeed, if zl(t)

1

+ zl(t), we obtain x 2 - z1 E

C ( I ) , but

considered in the couple := sin(l/t) and z 2 ( t ) =

11~1

- lzll

is a discontinuous

function.

We now describe the main result o f the Lip-invariance of computable and minimum f unctors . Theorem 2.5.23. (a) (Aizenstein-Bmdnyi) If a functor F i s computable, then the following Lip-interpolation inequality is valid for any

T E Lip(2, ?):

Minimal and computable functors

235

Here zl and z2 are arbitrary elements in F ( 2 ) . (b) (Krugljuk) The same inequality is also valid in the case when imal, but T

F is min-

E Lip(8)(g,f).

b f . Let us begin with the proof of (2.5.42) for the case when x' is a finitedimensional regular couple. Since in this case Lip(")(z,

p) = L i p ( 2 , ?),

both statements o f the theorem will be valid in this case.

To begin with, let T E L i p ( d , f ) be a continuously differentiable (in Frschet's sense) function on the space C(x'). We denote the derivative of this function by dT. Then d T ( z ) E C ( C ( d ) , C ( ? ) ) , and dT(z) depends continuously on z. In view of the finite dimensionality of

2,all

norms on

C ( 2 ) are equivalent. Hence dT coincides with the F r k h e t derivative of the function T, even when the norm on

C(d)is

replaced by any other norm.

Taking into consideration this remark and the identity Xi

C ( x ) ,i = 0,1, we obtain the following inequality in view o f the definition of dT and the Lipschitz operator T:

Taking the least upper bound in h, we can prove that

d T ( z ) belongs t o

L ( 2 , ?), as well as t h e inequality

Il"(z)ll~,~ 5 Il'IlLip~~,~~ Applying the interpolation inequality t o the linear operator d T ( z ) ,we obtain from the above

Interpolation spaces and interpolation functors

236

- "1IIF(a,)*

II I T I I L i p ( ~ , p )21.

This proves inequality (2.5.42) for this case. Let us get rid o f the assumption concerning the differentiability of T.For this purpose, we identify C ( x ' ) with a suitable IR" and consider the function cp E

CF(ERn)such that c p2 0

Since

T

J

and

cpdx=l.

is a continuous mapping from

~ ( 2t o )c(Q,

the vector-valued

Riemann integral

J

T,(z):=

v(Y)[T(. + W ) - T ( E Y ) I ~ Y

wo exists. In the present case,

-+

X; Z A(X), such that T maps

~ ( 2into)A(?>.

Hence we can write llTc(.2)

5

J

I

- Tc(.l)IIY,

cp(Y>

IIT(Z~

III'IILip(a,p)

+ EY) - ~ ( z+i w ) I I Y , ~ Y I - ZlIIXi

21.

>

= 07 1 *

Thus, we have proved that

Since

T,is obviously

continuously differentiable on C ( x ' ) , we see from this

and inequality (2.5.42) proved for such operators that llTc(.2>

5

- '(.1)11F(p)

IITIILip(f,?)

lIz2 - ZlllF(R)

*

It now remains t o prove that (2.5.43)

/~T(Z )TE(z)llA(p, +0

Proceeding t o the limit for

E

4

a~ E ---t

0

.

0 in the preceding inequality and considering

that llT(z) - T E ( z ) ~ ~I Fa F( ~ llT(z) ) - T E ( x ) l l A (we ~ ~obtain , the required resuIt . In view of the definition of

T, and the choice of cp,

we have

Minimal and computable functors

237

This leads t o (2.5.43). Thus, inequality (2.5.42) has been proved for the case

if

x'

from

2 ~2%.

Now

is an arbitrary couple, we apply inequality (2.5.42) t o the couple

F %

22 - z1

(2)and

take in this inequality the inf over all

B' for

B'

which

E C(B'). This gives

Assuming that F E Min and

T E Lip(")(z,P),we obtain from (2.5.44)

the

following inequality:

From here, inequality (2.5.42) is obtained via a transition from T t o the operator T (see the proof of Proposition 2.5.22). Thus, t o prove the theorem in case (b), we just have t o prove (2.5.45). For this purpose, we make use of the fact that for

F

E Min,

+

where the inf is taken over all sequences that are fundamental in U F ( F D

(2)) and converge t o z in C ( 2 ) (see Lemma 2.5.10). In view of (2.5.44), the sequence T(z,) is fundamental in F(f),and hence has a limit in C ( f ) . Let us denote this limit by y.

Since

(5,)

also converges in C ( 2 ) and

T := C ( x ' ) -+ C ( f ) is continuous (see Proposition 2.5.22). T(z)= y. In view of (2.5.46) and (2.5.45), we then have

Interpolation spaces and interpolation functors

238

This proves (2.5.45).

The proof for the case (a) is based on the same inequality (2.5.44). In this case, T E L i p ( 2 , P ) and F E Comp. In view of the computability of

F , the space F ( 2 ) coincides with the (absolute) completion of the union UF(F% (2)). Hence for arbitrary z1,x2 E F ( T ) ,we can find sequences (2:)

and

(2;) in

t h e union, for which +

(2.5.47)

lim n+w

2’

= 2; in F ( X ) ,

z = 1,2

Since the union is isometrically embedded in

.

F(@,

we get

It follows from this and the previous relation that for a given

E

> 0 there

exists n, such t h a t

for a l l n

> n,.

From this inequality and (2.5.44), we obtain for n (2.5.48)

> n,

llT(G) - T(4ll,(P) I (1 + €1 IITIILiP(2,P) 11x2 - Z l I l F ( 2 ) .

In view of (2.5.47) and the continuity of

T

as an operator from C ( 2 )

into C(?), we obtain

lim ~(z;) = ~ ( 2 ;in) ~

n-ca

( 9, ) i = 0,1.

Moreover, the fundamentality o f (zy)in the union and the continuity of (2.5.44) lead t o the fundamentality of (T(z1)) in F@). Finally, the fundamentality of this sequence, the computability of F and the above limiting relation give

Minimal and computable functors

239

Consequently, we can proceed t o the limit in (2.5.48). This proves (2.5.42) for this case also. 0

Remark 2.5.24. In fact, we have proved a more rigorous statement.

To wit, instead of

F E Min in part (b) of Theorem 2.5.23, we can assume that F E Min(2). For part (a), it is sufficient t o assume that F E the condition

Comp(2) n Camp(?). Taking into account this remark and Proposition 2.5.15, we arrive a t CorolI ary 2.5.25. (a)

If the couple

2 has the approximation

and for any functor lPY.1)

Here,

F

-

property, then for any couple Y

which is regular on

2,we have

- T(zZ>llF(p)IllTllLip(X,?)

1 1 .

- 5211F(X)

*

T E Lip("l(2,P) and zl,z2 E F ( 2 ) .

(b) If, moreover, the couple ? also satisfies the approximation condition and the functor F is regular on this couple as well, then the above inequality is also satisfied for T in Lip(2, ?).

E. Finally, we shall also show that under isotropic conditions, the computability of a couple is stable under superposition of functors. This follows from Theorem 2.5.26 (Aizenstein-Brudny;). Suppose that the functor F E Comp, and the functors

P

:=

Go

and

2.Moreover, let A ( 2 ) be dense in A(?), (GO(Z),(Gl(Z)).

computalbe on a couple

G1

are

where

Interpolation spaces and interpolation functors

240

The functor F(Go,G , ) is then computable on

i.

Proof.The proof 0s this theorem is based on some auxiliary statements.

We

begin with Lemma

2.5.27.

2,and let B be A(x') equipped + with the > 0, there exists a couple c' E F D (2),

Suppose t h a t a functor G is computable on a couple a finite-dimensional linear subspace of the space

norm

11

- IlG(2).Then for a given E

such that

Proof. In view of the computability o f G on 2, we can find for and b E

B a couple

(2.5.49)

IlbllG(@

any

E

>0

B' := Bb,r in F% ( d ) such , that

5 (l

IlbllG(a)

*

If B does not belong t o G ( B ) ,the couple couple obtained by adding the couple

B' can be replaced by a larger

B' and the couple -f[e*l,

15 i

5

B (for definition of the one-dimensional couple XC1, see Proposition 2.5.4). Then t h e new couple do contains B' and B' A 20,so that inequality (2.5.49)is satisfied for it. + + in Thus, for given b E B and E > 0, we have found a couple B := I% (x'),for which inequality (2.5.49)and the embedding

n, where (e;),<;<,,

(2.5.50) B

-

is a basis in

G(B')

are valid. In view of inequality

(2.5.49)and the triangle inequality, we obtain

for I E B 11211G(@

5 llz - bll,(@

+ (l + E , llz - bllG(g)+ (l

Since the space B is finite-dimensional, the norms

llZllG(,?)

.

11 . IIG(gj) and )I . IlG(2)are

equivalent on it, and the last inequality shows that for some neighbourhood

U,(b) c B of the point b,

Minimal and computable functors

241

Using the compactness o f the unit ball in the space from a cover of this ball by neighbourhoods {U,(b),

15 k

{U,(b,), couple

5

( B ,11 . IIqx,), we select b E B } a finite subcover

m } . To each o f these neighbourhoods, we associate a

2, EF% (I?) for

which the conditions (2.5.50) and (2.5.51) are +

(x'), we choose a couple c', 1 5 k 5 m. Then inequality (2.5.51) is satisfied for all b E B for the couple c'. Thus, satisfied. Using the directionality of the set

FD

c' EF> (2). for which l?k

II~llC(C,5 (1+ E l

11~11G(m, 7

2

EB

7

and since the right-hand side is t h e norm o f the space

B , it proves that

B '.$' G(6). El

Lemma 2.5.28.

GI are computable on a couple 2,and r-) A(&. Then for a given couple 6 EF> ( X ) , such that

Suppose that the functors Go and

B' EG(F) be such that C ( 2 )

le t a couple E

> 0, there exists a

l? ts"

(Go(6),G1(c')) .

Proof. Applying the

last lemma twice, we can find couples

6 ( X ) ,for which B; % Gi(C;),

is directed, we find in this set a couple Then

B;

%

60and c'1 in

i = 0 , l . Using the fact that

c' such that

e; A

5(I?)

6, i

= 0,l.

G;(6;).

0

Lemma 2.5.29. Suppose that, under the conditions of Lemma 2.5.28, we also assume that

A(I?) is dense in A(F). Then for any 6 @(p) and an operator (a>

C (&)

-+

(b) llTEIlmr,S, (c>

>

0, there exists a couple

T,E L(d71?,) such that

A(2);

< 1+ E ;

IIZ - TCZllA(P) I

E

IIZllC(P)~2 E CtB,).

Interpolation spaces and interpolation functors

242

Proof.

+

Let (ej)lsjsn be a basis in

C ( B ) and let (e;)lljgn be the basis

dual t o it. We make use o f the equivalence of the norms

llzlll :=

Cy='=,Iej*(x)I on

I( . Ilc(p,

and

the finite-dimensional space C(g). In view of

this equivalence, we can find a 6 := 6 ( ~ I;?) such that

Using the density of (2.5.53)

lie,

A(x') in A(?), we find elements a, E A(x'), for which

- a,Ilc,(q < 6 ,

&,

Let us now define the couple

15 j 5 n , i = 0,l

.

assuming that Be,* := ( L , 11

. IIG,cz)),

i = 0,1, where L is t h e linear envelope of the set ( U ~ ) ~ S ,Then ~ ~ . SEis a regular finite-dimensional subcouple of t h e couple ?, i.e. EFD (?), and moreover, C(6,) L, A(d),and thus condition (a) is satisfied. Further, 4

we define the operator

T,

:

3 -+ I?e, putting Tc(eJ)

:= a,, 1 5 j

5 n.

Then, in view of (2.5.52) and (2.5.53), we have for any x = C e;(z)e,

in

C(2) IITEzllG,(B)

llxllc,(n,

11 c e;(z>(e, - 'J)llG',(81 5 ll"IlC,(B) I l+- 6'12111 < 1 + € , 2 = 0 , l . II IIc(R) +

Thus, the condition (b) is satisfied for

T,.

Finally, in view of the same

inequalities (2.5.52) and (2.5.53), we obtain for x E

C(&)

C Iej'(x)I IIej - %llG,(R)

IIx - ~ c 2 I I q p )I

<

i=OJ

and the condition (c) is also proved. 0

Let us now prove the theorem. Since F is computable, the space F(?)

is regular in the couple

? (see 2.5.5 and 2.5.9).

Hence the density of

in A(?) leads t o the denseness of this set in F(?). If we prove that

A(2)

Minimal and computable functors

(2.5.54)

243

ll~lltlIl l ~ l l ~ ( pfor ) z E A(-f) ,

where we have put

U := U (F(Go,Gl)(c');

c' EF% (Z)} ,

then in view of the above-mentioned denseness of be also valid for all

I from

A(Z), this inequality will

A(?) = U. Hence, passing t o the completion,

we obtain the embedding

F(?)

:= F(Go,G1)(2)

Since it is obvious that

A

U"

U & F(G0,Gl)(Z),

. the inverse embedding is also

true, and hence the functor F(G0, GI) is computable in

2.

Thus, it is now left for us t o prove (2.5.54). For this purpose, we use the computability of F t o determine for given condition ( ( ~ ( ( ~=( p 1a) couple (2.5.55)

11~11~(g) <1+E

E

> 0 and

T, E Ll+,(l?,&)

.

2,in 5(?)

and an

having the properties (a)-(c) mentioned in this

lemma. Moreover, if the element y := we replace the couple

A(Z) with the

B' from F% (f)for which

Next, we use Lemma 2.5.29 and find a couple operator

I E

- T,x

I

does not belong t o C(B',),

g, by the larger couple 2,+ Y[v].Since y E A(Z),

the properties (a)-(c) o f this lemma remain valid after this substitution as well. Taking into account the choice of

I.

- T ~ 4 1 A ( E l a= )

&, we now get from property (c)

Ib - TS41,(P)

< E 1141C(P) IE n 114lF(9) = E n

where o i s the constant o f embedding of F into C. Since, in view of property (b),

IIT,lls,a < 1+ E ,

(2.5.55)

l l w l F ( 3 e ) < (1 + E l 2

.

From the last two estimates, we obtain

we also obtain from

?

Interpolation spaces and interpolation funct o m

244

I 6 ))z- TCZJJA(&) + (1+ e)2 < 6ae + (1+ e)2 + 1+ O(E) , where 6 is the constant of embedding o f A in F . Thus, we have established that

ll4lF(&) = 1 + O(E) for z E A(x') under the condition JJsJJF(p) = 1.

zcand find a couple 3 EFD -t

Next, we apply Lemma 2.5.28 t o t h e couple

zc 't'

(2) for which

(Go(e),Gl(e)).

In this case, we obtain from the

last inequality II~IIF(G,,,G~)(C)

5 (1 + E ) IIZIIF(~*) = (1 + o(E)) I I ~ I I F (. ~ ~ )

Taking the inf in this inequality over all we obtain for

E

6 EF% (X)for which z E C(c'),

+0

ll~lluL 11~11F(p)

E A(@

for

*

This proves inequality (2.5.54). 0

Corollarv 2.5.30.

Let the functors F , Go,GI and couple

H

be computable, and suppose t h a t for any

B' in FD F(G0, G I ) ( Z )= H ( 8 ) .

This equality is then satisfied for any couple

2 for which A ( 2 ) is dense in

A(?). Here, ? := (Go(@,Gl(Z)). h f . The computable functors on

2 are uniquely defined by their values

d

F D . Hence it is sufficient t o mention that, in view of Theorem 2.5.26, the functor F(Go,G I )is computable on 2.

on the couples in 0

Remark 2.5.31. We leave it t o the reader t o verify that Theorem 2.5.26 is also valid for the case when

F is minimal. In this case, it is claimed that F(G0, GI) is minimal

on the couple

2.

Interpolation methods

245

2.6. Interpolation Methods

A. In the following chapters, we shall investigate families of interpolation functors that are stable under superposition. T h e aim o f this section is t o introduce the basic concepts and to consider certain examples. Definition 2.6.1.

A family o f functors F := ( F a ) is called an i n t e r p o l a t i o n m e t h o d o n t h e - . + subcategory K C B,if for any three functors Fa,Fa, and Fa, of this family there exists a functor Fp E F for which (2.6.1)

F,(Fa,, Fa,)(@ 2 Fp(J?) ,

In the case of

k

=

6, the family F

2 E I?

.

is called an i n t e r p o l a t i o n m e t h o d

(or simply m e t h o d ) . 0

Remark 2.6.2.

If the equality i n (2.6.1) is replaced by the isomorphism Z,we call

F

a

u n i f o r m interpolation m e t h o d in the case when the isomorphism constants are independent of the parameters a, ai and

p.

Otherwise,

F

is called a

n o n u n i f o r m interpolation m e t h o d .

The knowledge of the function

simplifies considerably the computation o f interpolation spaces generated by the method

F. Indeed,

by finding the spaces

we can compute such spaces for any couple of (2.6.1) and the function

F a ( T )for a fixed couple

2,

(Fao(T), Fml(z)) w i t h the help

RF.This is one of the

reasons behind our desire

t o use methods instead o f isolated functors. On the other hand, most of the interpolation constructors used in analysis generate, for reasons t h a t are as yet unknown, families o f functors stable t o superposition. We shall consider some basic examples below.

Interpolation spaces and interpolation functors

246

When we are given a method family

03

:=

(DF,).If DF

F

:=

(F,), it is natural t o consider the

is a method, we call it dual to

F

if Flc F.

We shall now describe the basic interpolation methods. For this purpose, we shall use Definition 2.6.3. A Banach space X o f measurable (classes of) functions defined on the measurable space (Q,dp) is called a Banach function latticela if its norm has the following property:

If1 I

(2.6.2)

191 a*e. 7 9 E

x * f E x llfllx 5 llsllx 7

*

Obviously, most of the function spaces considered in Chap. 1( L p ,L p ( w ) ,

L,, and their discrete analogs) are lattices. We can now describe the first o f t h e interpolation methods used in this book. Example 2.6.4 (the K-method). Let

a be a

lattice over a measurable space ( R + , d t / t ) , satisfying the con-

dition (2.6.3)

min(1,t) E

.

Generalizing definition (2.3.44), we introduce the Banach space K*(-,f)with the help of the norm

Apparently, the space 3 a p , introduced with the help of formula (2.3.44), is obtained from (2.6.4) for 0 :=

LB (:= Lp(t-')).

We shall also denote it by K d p ( z ) .It will be shown in Chap. 3 that Kip is a functor and the family ( K e ) , where 0 runs through the lattices with condition

(2.6.3),is a method.

laWe shall henceforth use just the term lattice, since no other Banach lattices except function lattices will be considered in this book.

Interpolation methods

247

The subfamily (2.6.5)

n

:= (KBp)o
is not a method, although condition (2.6.1) is satisfied for almost all cases.

It will be shown in Chap. 3 that the following relation (Lions-Peetre) is valid: (2.6.6)

K~p(K~opo, K~lp1)2

Kqp

.

Here, for do # d l , the parameter p is arbitrary, and (2.6.7)

77 := (1 -9)90+6191

.

For 90 = &, the parameter 7 = 9 and (2.6.8)

1 P

1-19 Po

+ Pl-9.

-=-

If, however, p is not defined by equation (2.6.8). the left-hand side of (2.6.6) will no longer coincide with any of the functors in the family n. Hence n i s not a method. However, in view of the above arguments, the subfamily ( K ~ ~ ) o <with d<~ a fixed p is submethod of the method K.

Example 2.6.5

(the J-method).

In order t o provide a natural motivation for t h e definition of the functor

Ja given here, l e t us consider the following problem. Let i = 0,l.Here,

<

be a couple

consisting of the spaces l i ,

11x11,; :=

c

12-"iZnl

,

i = 0,l .

nE Z +

Let a be an element in

E(Zl), such that a > 0. Our aim is t o calculate the

orbit (2.6.9)

Orb,(< ; 2)= { T a ;a E L ( < , Z ) } .

If ( e , ) , , € z is the standard basis in l I ( Z )then , e, operator T E ,C(<,T)can be written in the form TX

=

C nE Z

z,y,

(condergence in

E A(l',). Hence any

~ ( 2 ,~ ) )

Interpolation spaces and interpolation funct o m

248

c A(X).This

where yn := T e n . Here, (yn),,,=z

Consequently, we obtain for yn :=

means t h a t

Ten t h e equality

Let us write this equality in a somewhat different form by using the concept of the J-functional of the element z

J

:

R+x A ( 2 ) + R+defined

E A(2).By this we mean the function

by the formula

Then, in accordance with the above equality and the definition o f the norm in an orbit [see

(2.3.11)],we obtain

=

inf

z=Czn

sup n

J(2",xn; 2) Qn

Finally, we obtain

Thus, the quantity on the right-hand side defines an interpolation functor, -+

which we denote by

@ a

Jl,(a-l)(X). Replacing lm(~-') by

an arbitrary lattice

c C(<), we obtain a family of interpolation functors j := (Ja),which is method closely related t o the K-method (we shall prove this in Chap. 3).

However, we prefer t o use for most cases the continuous analog of this 4

definition, where the couple Here, as before,

is replaced by the couple

Lf := Ll(t-' ; d t / t ) .

L1

:=

(L:,L;).

249

Interpolation methods

j oconsisting

The subfamily

o f regular functors Ja forms, as will be

shown below, a submethod of the method

t o which corresponds the dual

03'. It is found that 03' is a submethod of the K-method. The K- and j-m e th o d s are combined because of their close connection.

method

They form the so-called real-interpolation method. Chapters 3 and 4 will be devoted t o t h e investigation o f the properties of this method. Example 2.6.6 (the complez interpolation method). Let us now consider Banach spaces over the field C . Any Banach space X over

R

can be canonically transformed into a Banach space Xc over

C by

putting

XC

:=

x

8

Let us now suppose that

of functions

f

c.

x' is a couple of complex spaces and 'FI(x')consists

with values in C(@, which are analytic in the open strip

S := { z ; 0

< Rez < 1)

and bounded and continuous in its closure. We equip 'If(.,?) with a norm by putting

Next, we define the Banach space

11~11c,(a)

Cs,0 < .9 < 1, by

:= inf {IlfllX(2, ;

f(4 =XI

putting for

2

E C(Z)

.

C8 is a functor, and the family C := (C8)0<8<1is a method (lower complez method). It turns out that DC is also a method (upper complez method). Example 2.6.7 (the cp-method). We shall confine the consideration o f this method just t o the subcategory of lattices having the Fatou property (the closed unit ball is closed with respect t o convergence in measure, cf. Theorem 1.3.2). Thus, l e t

2 be a couple consisting o f Banach lattices X , defined on a

measurable space

(a,+), and l e t cp

asing function. We put

:

R+4 R+be a concave nondecre-

Interpolation spaces and interpolation functors

250

@ ( s , t ) := scp(t/s)

and define the space

@(z) with the help of the norm If1

where inf is taken over all representations of

If1 = G(lf01,Ifil)

7

fi

E Xi

7

i n the form

= O,1

For example (see Proposition 2.3.8), if cp(t) := t 9 , 0 @ ( s , t ) = s1-’t9

< I9 <

1, then

and

where

1

Po

..-

I9 + . Po Pl 1-19

I t will be shown in Chap, 4 that the map functor on the subcategory L,,(sZ)

2 -+ G(2)

of lattices on

is an interpolation

sZ having the

Fatou pro-

perty, while the collection o f these functors is a method on this subcategory.

The extension of the functors of this family by minimality (or maximality) and their subsequent regularization leads t o a family of functors that are defined on all Banach couples.

B. An important problem of the theory is t o describe all interpolation spaces of a given couple (the “basic problem” in the theory of interpolation spaces).

In such a general formulation, this problem is not likely t o be

solved in the foreseeable future. In this sense, it shares the fate of all ”basic problems”, for example the problem o f classification of topological spaces in algebraic topology, or o f algebraic manifolds in algebraic geometry. However, attempts t o solve the “basic problem” for some specific couples (or classes of couples) and the few results obtained so far in this respect provide a powerful stimulus for the development of the theory. One possible approach t o the solution o f the “basic problem” for a given couple x’ is t o construct the method which generates all interpolation spaces

Interpolation methods

251

2.The use o f the method itself rather than certain families o f functors is justified by the obvious fact that Int(p)

c Int(2) for

spaces are interpolation spaces of the couple

any couple

? whose

2. In this connection,

it is

appropriate t o consider Definition 2.6.8.

A couple 2 is called F-adequate with respect to a couple f if for any spaces X and Y , for which the triple x’,X is an interpolation triple with respect t o triple f,Y , there exists a functor F E F such that (2.6.12)

& F ( d ) , F(?)

X

1 c-t

Y

.

0

If 2 :=

I’in this definition, the couple d is called F-adequate.

In this

case, in view o f (2.6.12) the F-method functors generate all interpolation spaces o f

2,and the

“basic problem” for the couple

2 can be considered

solved. Since the set o f interpolation spaces of a couple is very large in a “typical” case, we must use “prolific” interpolation methods. Among the methods described above, only t h e real method can be treated as a prolific method. The application o f this method lead t o the first example of solution of t h e “basic problem” for a specific couple. The following classical result played an important role in this solution. Theorem 2.6.9 (Calderdn). The necessary and sufficient condition for X to be an interpolation space for the couple

(L,,L,)

is that it be K-monotone. This means that the

following condition is satisfied.

If 2 E X and y E C ( L l , L , ) are such that

K ( * , y ;(L&O)) then y belongs t o

2 , and

llyllx I ll4lx .

I K ( * , z ;(Ll,Lo))

7

Interpolation spaces and interpolation functors

252

A more general result of this type will be proved in Chap. 4. For the present, we confine ourselves t o the remark that the Theorem 4.4.5 and the Calder6n theorem directly lead t o Corollarv 2.6.10.

(Ll, L,)

The couple

is Gadequate.

0

We shall confine ourselves to just this example of the solution o f the “basic problem” for a specific case.

A somewhat generalized form o f the definition 2.6.8 is Definition 2.6.11. The subcategory any couple

K

of a category of Banach spaces is called 3-stable if for

2 satisfying the condition X i E K , i = 0,1,

and for any functor

F E 3, we have

F ( 2 ) E K: 0

If 3 consists o f all interpolation functors (3 := J F ) , the subcategory

Ic

is called interpolation-stable. Let us consider t w o important examples o f

interpolation-stable subcategories. Examde 2.6.12. The subcategory

BL(S2) o f

Banach lattices defined on a measurable space

(a,d p ) is interpolation-stable. Indeed, if

X,

E

B L ( Q ) ,i = 0,1, and X

:=

spaces o f this couple, we have to prove that if If1

F ( 2 ) are

5

interpolation

191 almost everywhere

X , then f E X and llfllx 5 Ilgllx. For this purpose, we f consider the operator T ( z ) := h z , where h := - (at points where the

and if g E

9

denominator is equal to zero, we put since

h equal to zero). Then T E L I ( Z ) ,

llTllx 5 vraisup Ihl 5 1. According t o the property of interpolation,

Interpolation methods

253

f = T g E X and

0

To consider the second example, we make use of an important Definition 2.6.13.

A Banach lattice X on a measurable space ( Q , p ) is called a symmetric space'' if the following condition is satisfied. If z E X and the function y is equimeasurable with the function 2 , then y E X and 11x11~= \lyllx (see Proposition 1.9.2for the definition of equimeasurability). 0

It was shown i n Sec. 1.9 that the spaces L, and L,,

as well as their

discrete analogs, are symmetric. However, it can be easily verified that the space Lp(t8) is not symmetric. Example 2.6.14. The subcategory Sym(i2) of symmetric spaces defined on a measurable space

( Q , d p ) is interpolation-stable. For the proof o f this fact, see Kre'in, Petunin, Semenov

[I].

'gAlso called a rearrangemeni-invariant space.

Interpolation spaces and interpolation functors

254

2.7.Comments and Additional Remarks A. References Sec. 2.1. The basic concepts of the theory (Banach couple, linear op-

erator acting in couples, intermediate and interpolation spaces, and so on) were already defined a t the initial stage o f i t s evolution in the work by Lions, Gagliardo, Calder6n and Aronszajn (see the survey by Brudnyi', K r d n and Semenov

[2]concerning these references). In the fundamental paper of

Aronszajn and Gagliardo

[l], a detailed account of these concepts was given.

In particular, the concept of generalized couple in Sec.

C

was introduced

and analyzed by the authors. A further generalization of the notion of a couple was proposed in the paper by Kaijser and Pelletier connection with Example

[l] (see 2.7.2). In

2.1.22suggested by Yu.A. Brudny'i, t h e paper by

Peetre [ll]is worth mentioning. There a sufficient condition for a subcouple is formulated, which ensures the validity of an analog of the Hahn-Banach

theorem. To complete the "concrete" realization of this example (see Remark

2.1.23),it is necessary t o prove that B M O

is nonisornorphic t o L,.

In

addition t o the above-mentioned theorems by D. Newman, see for example Rudin [l],

BMO

-

55.9, and Fefferman, see Fefferman-Stein [l], the isomorphism B M O A has also be used, where the right-hand side contains

the space of functions which are analytic in the unit ball and have boundary values belonging to B M O almost everywhere. According t o a communication kindly delivered by 0.1. Reinov, actually B M O is not even an &,-space (i.e. (BMO)" cannot be embedded as a complemented subspace in any

LW(dC1). Sec. 2.2. Almost all results of this section were taken over from the paper

by Aronszajn and Gagliardo [l]. The concept of relative completion was introduced and analyzed by Gagliardo [4],Proposition by Behrens [l], while Proposition put forth Conjecture

2.2.21was established

2.2.22 by Yu.A. Brudny'i. The latter also

2.2.32.

The optimality of concrete interpolation triples mentioned in Examples

2.2.40and 2.2.41was established in the works by Caldercin [3],Dikarev and

Comments and additional remarks

255

Matsaev [l],A. Dmitriev and Semenov 111.

Sec. 2.3. The main results o f this section are due t o Aronszajn and Gagliardo [l]. The remaining results were taken from the works by Dmitriev, KreYn and Ovchinnikov [l] (the concepts o f fundamental function of a functor and Propositions 2.3.10 and 2.3.27), Janson [2] (Proposition 2.3.3). and Peetre [12] (the concept o f retract of a couple and Propositions 2.3.22 and 2.3.23). Theorem 2.3.30 was proved by Yu.A. Brudny'i. The equalities (2.3.49) are special cases of more general results established in Chap. 3 (see Theorems 3.3.4 and 3.4.3). The statement which is contained in equation (2.3.50) was obtained by Gagliardo [6] and is historically the first interesting example of "calculating" an orbit (see also Peetre [13], where absolutely summing operators are used for this purpose). A further development of this trend was obtained in a number of important works by Ovchinnikov (see his review [7]). In particular, this author formulated statement (2.3.51) (see Ovchinnikov [6]). Sec. 2.4. The concepts o f dual space and dual couple considered in this section differ from those used in earlier works on the theory of interpolation spaces (see, for example, Bergh and Lofstrom regular couple apparently

d*should

[l],where in the case of

a

be regarded as the generalized couple

(XG,X:)). The concepts used in this book are more convenient for a theoretical analysis since the regularity condition can be omitted. In particular,

it becomes possible t o consider

x'., z"', and so on.

A more "cardinal"

evolution of the duality theory, involving a further extension of the concept of couple, was proposed in the work by Kaijser and Pelletier [l](see 2.7.2 in this connection). It should be emphasized, however, that most of the results obtained by other authors and considered in this book are brought in line with the approach used here. A detailed analysis of duality in the category of Banach couples was started in the work by Aronszajn and Gagliardo [l]. In particular, Theorems

2.4.3, 2.4.12 and 2.4.24(a), Propositions 2.4.6 and 2.4.19 and Corollaries 2.4.13 and 2.4.21 were formulated by these authors. As regards Proposition 2.4.6, see also Lions and Peetre [1,2]. The mapping KT was introduced by Yu.A. BrudnyY, who also proved Theorems 2.4.24(b) and 2.4.32 and Propo-

256

Interpolation spaces and interpolation functors

2.4.11. The statement in Remark 2.4.15is due t o Yu.1. Petunin (see [l]). The example which proves Theorem 2.4.17was proposed by N.Ya Krugljak. In the review by Pelletier [l] there is the statement (without proof) that Theorem 2.4.17can be obtained from general considerations. Theorem 2.4.34was obtained by the authors from an analysis of the proof presented in Janson's paper [2].The important definition 2.4.22 sition

Kre'in-Petunin

(with supplementary condition of regularity of the couple) was introduced

[2]. The concept of dual functor was introduced by Aizenste'in [l], who also proved Theorem 2.4.31. This author carries out the analysis on the basis of the concept of dual functor (in Mitjagin-Schwartz [l] sense) in the category of D-diagrams (see 2.7.2below). A direct proof of Theorem 2.4.31was obtained independently by Yu.A. Brudny'i and M.H. Aizenste'in. by Janson

Sec. 2.5. A major role in the development of concepts considered in this section was played by the fundamental work by Janson lar, t h e important Theorem

[2].In particu-

2.5.18 on duality was established there with a

different approach and under the additional assumption about the regularity of the couple

A' and space A .

An attempt t o generalize this theorem for

the category of D-diagrams was made in the paper Kaijser and Pelletier

[l].

Although there are some gaps in the proof provided by these authors, it is essential that the concept of computable interpolation functor has appeared for the first time in interpolation theory in this theorem. This concept was introduced earlier in the Banach situation in a paper by Herz and Pelletier

(11. The concept o f minimal functor was introduced in the paper Aizenste'in and Brudny'i

[2],where the properties of minimal and computable functors

were analyzed for the first time. In particular, these authors proposed Theo-

2.5.12,2.5.16,2.5.23(a) [item (b) was independently proved by N.Ya. Krugljak], and 2.5.26,Propositions 2.5.4,2.5.9 and 2.5.15 and Corollaries 2.5.5and 2.5.25.The important role of Browder's [l] approach t o interpolation of Lipschitz operators should be mentioned. Corollary 2.5.25(b) can be

rems

regarded as a far-reaching generalization of this result. The idea of approximating the Lipschitz operator by a C'-smooth operator was also borrowed from the work by Browder.

257

Comments and additional remarks

Sec. 2.6. The first version o f the K-method o f interpolation was proposed by Gagliardo [1,3-51 who based his approach on the concept of the diagram” (or Gagliardo diagram) (the set Gx(z)

c Ri

“Y-

is a domain over

the graph o f the function t + E ( t , z ; 2)). The modern version o f this method was proposed by Peetre [l]who used an important concept o f the K-functional (see also Oklander [l]).The reiteration theorem (2.6.6) was proved in an equivalent form by Lions and Peetre (21, while the supplement t o this theorem (for

290

= 291) was established by Peetre [2]. These results

were later generalized by many authors. The second component of the real method (the ,?-method) was preceded by the average method developed by Lions and Peetre [2]. Many important achievements o f these authors (theorems on reiteration, duality and density) were based on earlier works by J.L. Lions on the method o f traces (see, in particular, Lions [1,3,5-81). The 3-method in its general form was introduced by Peetre [7]. The complex interpolation method was proposed almost

[l]and Lions [4] and is thoroughly investigated in Calder6n’s fundamental paper [2]. A construction close t o the complex method was analyzed by Kre’in [2], while the 9-method was introduced by Calder6n (21 for t h e power case. This author established its relation with the complex interpolation method. Lozanovskii [l-31 generalized this method for the case 9 E Conv and gave a detailed account o f the

simultaneously in work by Calder6n

general situation. For this reason, this method will henceforth be called the Calder6n-Lozanovskii construction. The fundamental Theorem 2.6.9 was established by Calder6n [3]; independently Mitjagin [l] gave an alternative description o f the set Int(Ll,L,) (see Sec. 4.7.A, comments t o 54.4, and 3.7.2 for the details). The interpolation stability o f the category o f symmetric spaces mentioned in Example 2.6.14 was established by Semenov

[l].

Its proof is carried out

in Sec. 2.1 of the book Kre‘in, Petunin and Semenov [l].

258

Interpolation spaces and interpolation functors

B. Additional Remarks 2.7.1. Category Language The theory of categories is known to form a developed branch o f mathematics, involving its own problems and methods of their solution. In the theory o f interpolation spaces, however, only the language of this theory is used, which sets a natural framework for representing a number o f "mass" concepts and results. T h e reason behind the employment of this language on ever growing scale in the fields o f mathematics which are.quite far form

the theory of categories is clearly revealed i n the following remark by Yu.1. Manin: "Category language embodies a "sociological"

approach to a ma-

thematical object: a group or space is regarded not as a set with its own structure but rather as a member of society of equals". For the sake of convenience for the readers, we shall describe here some concepts o f the theory o f categories. A part of them was used i n the first six sections of Chapter 2, while the other part will be used i n these additional remarks. The readers who are interested in a more detailed description o f the theory o f categories can find it i n the book by MacLane [l]. Definition 1. A category C is said to be specified if (a) a class Ob(C), whole elements are called objects, is given; (b) a class Mor(C), whose elements are called morphisms, is given, and (c) a l a w of composition is defined.

In this case, the following axioms have to be valid.

Mor(C) is the union Mor(X) = UC(X, Y )

Comments and additional remarks

259

C ( X , Y ) which do not intersect pairwise and which are defined for each ordered couple ( X , Y ) of objects. The law of composition is defined for each ordered triple X,Y,2 o f obo f sets

jects with the help o f a certain map:

C ( X ,Y ) x C ( Y ,2 ) ---f C ( X ,2 ) . The composition o f morphisms f :

X

-+

Y ,g : Y

-+

2

is denoted

by g o f and satisfies the following conditions:

f o ( 9 o h ) = (f o g ) o h For each object

for any f : Y

(associativity).

X ,there exists a single morphism

-+

X,g

:

X

-+

llx

E C ( X , X ) such that

Y (existence of identity).

Exa mples. (a) The class o f all sets and all possible maps o f these sets forms the category Set.

(b) The category B has as objects the class o f all Banach spaces, while linear continuous maps of these spaces serve as morphisms. In Chapter 2, we used the notation

L ( X , Y ) for the set of linear continuous maps o f X

into Y (in the sense of the previous definition, this object should be denoted by B ( X , Y ) ) . (c) Replacing in the previous example the Banach space ball

L ( X , Y ) by its unit

L I ( X ,Y ) , we obtain the category B1.

(d) Similar categories of couples

5

and

gl,considered

in Sec. 2.1, are

categories in the sense of the above definition. In this case,

"The notation f : X

Y is equivalent to the notation f E C ( X , Y ) .

Interpolation spaces and interpolation functors

260

d

(e) T h e same can be stated about the category o f generalized couples Y B (see Sec. 2.1.C).

Definition 2. The category 2, is said to be a subcategory o f t h e category C if the following conditions are satisfied: (a) Ob(D) C (b) Mor(D)

Ob(C);

c Mor( C);

(c) D ( X , Y ) c C ( X ,Y ) for all X , Y E V ; (d) if

f

E

D ( X , Y ) and g E D ( Y , Z ) , the composition g

o

f i n Mor(D)

is equal to the composition of g and f considered as elements from

Mor( C); (e) if z E D ,then the C-identity on X belongs to Mor(D) (and is equal t o the D-identity on X ) .

T h e subcategory

D

is called complete if D ( X , Y ) = C ( X , Y ) for all

X , Y E Ob(D). The category

C

is called small if Ob(C) is a set (or, which is the same,

Mor(C) is a set). Since

B1

is an incomplete subcategory of the category B , while

complete subcategory of the category

6 is a

Yk,none of the above categories is

small. Definition 3.

A morphism f E C ( X , Y ) which permits the inverse morphism f-' E

C ( Y , X ) ,i.e. such that

Comments and additional remarks

261

is called an isomorphism, while the objects

X and Y are termed isomor-

phic. 0

Remark. In the case when

X

and

Y are also the objects of other categories, we shall

X and Y are C-isomorphic in order t o avoid confusion. Thus, according t o t h e closed graph theorem, the Banach spaces X , Y are Bisomorphic if there exists an operator T E L ( X , Y ) which is a bijection. They are B1-isomorphic if X is isometric t o Y.

say t h a t

Along with C-isomorphism, we shall deal with the concept of C-mono-

morphism and C-epimorphism. They are described in Definition 4.

The morphism u E C ( A ,B ) is called a monomorphism if for any X E Ob(C) the mapping C ( X , A ) + C ( X , B ) defined by the formula v

--t

u o TJ is

injective.

This morphism is known as an epimorphism if for any X E Ob(C) the mapping C ( B ,X ) -+ C ( A ,X ) defined by the formula v --t v o u is injective. 0

Let us now describe one more fundamental concept in the theory of categories.

Definltlon. A map F of the category C into the category D , which preserves objects and morphisms, is called a (covariant) functor if

F(f 0

0

9) = W

)0 F ( g ) >

F( 1x1 =

lIF(X)

.

Interpolation spaces and interpolation functors

262 Example.

An exact interpolation functor (see Definition 2.3.1) is a functor in the sense of the above definition, which acts from the category of couples

B1. Here, t o a

the category of Banach spaces

couple

al into

x’ there corresponds

the exact interpolation space F ( 2 ) of this couple, and t o a linear operator

T E L l ( 2 ,p),the linear operator F(T) := Tl,(a,

+

acting from F(X) into

F(?) and having a norm which does not exceed unity. Another important example o f a functor is the so-called Teflector whose description will be given below. Let

D be a subcategory of the category C .

Definition 6. A morphism

t E C(X,Y)

morphism for the object z E

D

is called a D-universal (or simply universal)

x’

and for each morphism

if Y belongs t o D and if for each object

f E C ( X ,2) there exists a unique

morphism

g E D(Y, 2 ) such that

Thus, in this case there exists a natural bijection between the morphisms

C(X, 2 ) and the morphisms D(Y, 2 ) (for any 2). 0

Examdes. (a) Let

N be a category of normed spaces and linear continuous maps. Clearly, B is a complete subcategory of N . Then the canonical injection t o f the normed space X into its (abstract) completion X ” is, as can be easily verified, a universal morphism.

(b) Let us consider the couple (Xo,Xl)produced from the generalized couple

zTin Proposition 2.1.32. Then the “natural”

map of

ZTinto (&,,*I)

-t

(which is considered t o be an object in Y B l )is, as can be easily verified, a $-universal morphism.

Comments and additional remarks

263

Let us now suppose that for each X

E Ob(C) there exists a D-universal morphism t x : X + r ( X ) E Ob(D). If f E C ( X ,2). then t Z o f : X + r(2)is a C-morphism with t h e domain of definition Ob(D). Consequently, there exists a unique D-morphism g : r ( X ) + r(2)such that

In other words, the diagram

(3)

is commutative.

We put (4) The following proposition can be verified in a trivial way. Proposition 6'. is a (covariant) functor from the category C into the subcategory D . 0

I?

C + D constructed in this way is called a reflector, while the subcategory D in this case is termed reflective. The functor

:

Examples.

X ( " ) ,where X E Ob(N), we construct a reflector Cmpl : N + B , which associates t o the normed space i t s completion and t o the map f E L ( X , Y ) its extension by continuity f'"' E L(X("),Y'"').

(a) Using the family t x

:

X

-+

+

(b) We leave it t o t h e reader t o verify that the functor constructed in Proposition 2.1.32 is also a reflector.

R

:

YB1+

gl

264

Interpolation spaces and interpolation functors

L e t us now consider the concept of natural transformation (or functor

morphism ). Definition 7.

A natural transformation of the functor F t o the functor G, where F,G : C + D , is the term applied t o the family o f t morphisms (tx : F(X) -+

-

G(X))xEC such that for any morphism

F(X) tx

1

G(X)

F(f 1

f

E C(X,Y), the diagram

F(Y) Ify

G(Y)

is commutative.

A natural transformation t is called an equavalence if t x is an isomorphism for all X E C . Examples. (a) The family

6 o f embedding maps (6,

:

A ( 2 ) -+

F(2))xE~, where

F is an interpolation functor, is obviously a natural transformation of the functor A into the functor F. (b) Similarly, t h e family o of embedding maps (“1 :

F ( 2 ) L) C(x’)) is a

natural transformation o f the functor F into the functor C. Finally, let us consider the important concept of the limit in the category of “diagrams”. Namely, let us suppose that 6 is a small category and

r

: 6 + C is a covariant functor.

Definition 8. The object X E C is called the limit o f the “diagram” (a) there exists a &index family of maps

I?

is

265

Comments and additional remarks such that if g E S ( i , j ) , then

(b) X is universal relative to this property, i.e. if

X' E C and (7;

:

X' +

I'(i))iEffalso satisfy condition (a), there exists a unique map 7 : X'+X such that

^fi

o

7 = 7:for all i E 29.

The concept of colimit is obtained from t h e concept of limit by applying the category principle of duality (the process of "arrow inversion"). Thus, in t h e item (a) of the definition, there appears the d-indexed family of maps

7' :

r(i)+ X , i E 19, such that if g E d ( i , j ) , then 7' = 7'

o

r(g), and

so on.

Remark.

It is worthwhile noting that the limit and colimit are defined up to a Cisomorphism. Examples.

d be a directed set. It can be regarded as a small category by assuming that d ( i , j ) consists just of one morphism for i 5 j and is empty in the opposite case. We shall consider the functor r : d -+C whose domain of definition is the family of objects ( X i := I ' ( i ) ) i E f f .

(a) Let

The colimit X of the functor I? is called the direct limit of this family. We can write for it

X = lim X ; s' Thus, for i

5j

there exists a morphism ? r i E C ( X i , X , ) (the image of

the single morphism from d ( i , j ) for the map

7' : X i

t

r) such that for the maps

X by the definition of the colimit we have

Interpolation spaces and interpolation functors

266

where

X

is universal relative t o this property.

A special case of a similar construction (in the category B of Banach spaces) was used in Definition 2.5.6. (b) Another example which will be used in the next item refers t o the category (graph)

where the dots stand for the objects and arrows for morphisms. Let us consider t h e functor

r

which acts from this category into the

category C . Its image is t h e diagram

X (5)

If Y

4

Z

9

Then P E Ob(C) is called the pulZback of this diagram if it i s the limit of the functor

r.

Thus, there must exist morphisms u : P + X and v : P

-t

Y such

that the diagram

is commutative. If (P',u',v') is a set with the similar properties, there exists a unique morphism w : P' is commutative:

--+

P such that the following diagram

Comments and additional remarks

267

(c) The object Q, which is known as the p u h o u t o f the diagram

is defined in a dual way (by arrow inversion). (d) Let us find out the form of the pullback and pushout for the category B of Banach spaces. Thus, l e t us suppose that we have a diagram of the form (5) in which

X , Y , 2 are Banach spaces while f E L ( X ,Y ) and g E L(Y, 2 ) are continuous linear maps. We claim that the pullback of this diagram is a closed linear subspace (with induced norm)

taken together with the maps u E ,C(P,X) and v E L ( P , Y ) defined by the formulas

(10)

u(.,y)

:=

5 ,

?J(.,y)

:= y .

Since the commutativity of diagram (6) in this case is obvious, we have only t o show that for a set

(P, u', d)with similar properties there exists

a unique linear operator w E L(P',P) such that diagram (7) is commutative. For p' E

P',we

put

Interpolation spaces and interpolation functors

268

Since i n view o f the conditions imposed on u‘ and

f

u‘=g

0

0

w’ we

have

v’,

the right-hand side of (11) is an element o f

P. Thus,

w E L(P‘,P).

If w’is another map similar t o w, then from the commutativity of the diagram o f form (7) for w’ we obtain

= (4P’))

U‘(P’)

= ?J ( 4 P ’ ) )

7

7

which together with (10) gives

4 P ’ ) = ( 4 P ’ ) , “/(PO) = 4

4.

Thus, formulas (9) and (10) actually define a pullback of diagram (5) i n the category

B.

We leave t o the reader t o verify that the pushout of diagram (8) i n the category

B of Banach spaces is given by the formulas

(12)

Q

(13)

u ( z ) := 7rw(z,O),

:=

(X $1 Y)/wo := x w ( 0 , y ) .

v(y)

Here w is the closure o f the subspace { ( j ( z ) - g ( z ) ) ; z E Z } in the space

Y

X

and x,

:

X

@1

Y

--+

(X

@I

X ) / W is a canonical surjection.

Definition 9.

A D-diagrum21 in a category C is the term applied t o a commutative diagram o f the form

i-” U

(14)

v

Y-Q

If

9

21The termg “pushout-pullback diagram” and ”doolittle diagram” are

also

used.

Comments and additional remarks

269

of objects and morphisms of this category, where

P

is a pullback and Q is a

pushout. 0

The class of t h e D-diagrams is obviously nonempty and forms the class of objects of the new category C whose morphisms are defined as follows. Let

D

be a D-diagram of the form (14) and D’ is a similar diagram all

of whose objects are denoted as in (14) but primed. Then the couple ( h ,k)

is called a morphism in the category C if (a) h

E C ( X , X ’ ) , k E C(Y,Y’)

(b) the diagram

is commutative. The composition of morphisms ( h , k) E C ( b ,b‘)and ( I , m ) E

c(D’,D”)

is defined in a natural way:

(I,m) 0 ( h , k ) := ( I

0

h,m

0

k).

The fulfilment of the associativity axiom can be easily verified. In this case, I D :=

(1x7 a,>

+

Finally, it should be noted that C can be regarded as a complete subcategory of the category C if it is identified with the image of the functor q : C

4

C which associates the space X with the 2)-diagram:

X-X

lx

Interpolation spaces and interpolation functors

270 and the morphism

f E C ( X , Y ) with the

morphism q ( f )

:=

(f,f)E

C(q(x),q(y)).

2.7.2.Further Extension o f the Concept of Couple It was noted in Sec. 2.1 that the category

2 of couples is not

closed

relative t o a number of Banach constructions (conjugation, completion, fac-

The desire t o include the object ( X , " , X ; ) , where x' is a regular couple, in the framework of the theory under consideration led Aronszajn and Gagliardo [l] t o the concept of generalized Banach couple (see Sec. 2.1C). However, the category Y B of generalized couples is not closed either relative to the constructions mentioned above. A

torization, tnesor product, and so on).

d

radical solution of the problem can be obtained by extending of the category

fi of couples t o t h e category B of 2)-diagrams of Banach spaces, proposed in the papers of Kaijser, Pelletier 11-31 (see also the review by Pelletier

[l]).

The possibilities provided by such an extension in the theory of interpolation spaces were analyzed in these papers and the dissertation of Aizenste'in (see also Aizensteyn [2]). The review given below is based on the manuscript kindly put by him a t our disposal.

It will be more convenient to change the notation and denote henceforth a 2)-diagram in the category

B as follows:

Thus, A(X) denotes the pullback and C ( X ) the p u s h o u t of the 2)-diagram of

x. x

The set of morphisms T := (To,Tl) : + Y will be denoted by C ( X ,?). Thus, T;E L(X;,l'i), i = O , l , and the following equality holds (see Definition 9 from Section 2.7.1):

271

Comments and additional remarks

The set L ( X , Y ) is a Banach space with the norm IITllX,P := m=

llTiIlX,,Y,

.

i=O,l

The morphism T E L ( 8 , Y ) generates two continuous linear operators

A(X) + A(Y) and Tx : C ( X ) -+ C ( Y ) . The definition of operators is based on the formulas

TA

:

these

The consistency of these definitions follows from (16) and the definitions of pullback and pushout.

It also follows from the definitions t h a t

(19)

llTA1l, llTCII

5 IITllX,P

*

Henceforth, it will be more convenient for us t o deal with the subcategory

Bl of the category B .

It should be recalled that in these cases the elements

L l ( 8 , Y ) := {T E L ( 8 , P); IITllm,y 5 1) &-isomorphism such that

are morphisms. Therefore,

coincides with the couple o f isometries

(T'l,T
(To, Tl) E L ( X , Y)

For the verification o f the latter

condition, the following proposition is useful. Proposition 1.

(TO, 2'1) E L ( X ,Y )is a couple of isometries such that Tc C ( y ) is an injective map, then T is a B1-isomorphism. If T :=

:

C ( X )+

0

It should be noted that all propositions of this section have comparatively simple although sometimes cumbersome proofs. The reader may try and either reconstruct these proofs independently or with the help o f references given a t the beginning of this section. A detailed description o f the theory under consideration is contained in the manuscript by Aizenste'in [2].

Interpolation spaces and interpolation functors

272

It is essential that each couple 2 E 6 is canonically associated with the D-diagram

4

. ,

If the D-diagrams of X,Y are generated here in the way indicated above by the couples 2,?, t h e linear map T E L ( X , Y )generates the B1-morphism T := (To,Tl), where T; := Tlx,. Here TC= T and TA = Tl,[d). Thus, glcan be assumed t o be a complete subcategoryof Bl. Moreover, it turns out t o be a reflective subcategory. Namely, we shall associate with each D-diagram of

(20)

X E B, the couple

Z ( X ) := (a,X(Xo),aP(X1))E

2 1

.

Having identified this couple with the diagram generated by it, we shall consider the B1-morphism ax

:= (a,f,aF) acting from

X

Then the morphism ax is B1-universal for the D-diagram of In order t o prove this, l e t us take an arbitrary couple

into

X

Z(X).

[see (l)].

? and suppose that

T E B l ( X , Y ) . Then according t o the definition of Tc [see (18)] and equality (19), this operator belongs t o ,Cl(Z(rf), ?) and

It also follows from the definition o f Tc and ax that Tx is a single

g1-

morphism which satisfies this equality. Thus, i n the given situation condition

(1) in the definition o f universal morphism is satisfied. Using further the construction defined by formulas (2)-(4) (t,),,~

where D ' :=

g1, C

:= ( a x ) x , ~ , .Then we obtain the reflector

Zl

:= :

which is defined by formula (20) for objects and by the formula

on the B1-morphisms.

Bl

B, and -+

&

Comments and additional remarks

273

The reflector a' ensures a simple method of extension of interpolation functors specified on the subcategory

g1over

Namely, we associate with each such functor acting from

-

B1 into B1.By definition, H

(23)

m'(A(X))

(24)

H ( T ) = Tc

(25)

HIS, = F .

Here X and

Y

t h e entire category

F the functor H

:=

Bl.22 F

o

a'

has the following properties:

H ( X ) L) C ( X ) ,

,

are objects of the category

Bl and T E &(X,Y).

For many interpolation functors which are important for applications, the proposition described above is unique in the sense that if a functor

H

:

B1 t B

satisfies conditions (23)-(25),

then H =

particular, the uniqueness property is inherent in the functors t h e complex method and the functors

F

o

a'. In

C, and ' C

of

Jq of the real method (see Aizenste'in

121). As was mentioned above, most of Banach constructions can be transferred t o the category o f D-diagram. In such a process, the following two methods of constructing D-diagram are used throughout. Let us first suppose t h a t we have t h e diagram

xo

with objects and morphisms of the category of Banach spaces complete this diagram by the pullback

(P, uo,ul) and

B. We shall

then consider the

diagram

"Pay attention to the absence of the "natural" concept of interpolation functor on the category of 2)- diagrams. As regards the possibilities available here, see the works mentioned in this section.

Interpolation spaces and interpolation functors

274

Having completed it by a pushout

we obtain, as can be easily

(Q,wO,w1),

verified, the D-diagram uo

P-Xo

x

:=

u11

.

1.0

Xi A

Q

u1

We shall change the notation for t h e objects and rnorphisms of this diagram by putting P :=

A(X), Q := C ( X ) , u; := S?, w; := a?, i = 0,1,

and henceforth call the D-diagram obtained the pullback of diagram ( 2 6 ) (or pullback of the maps

00, a1).

Similarly, starting from the diagram

x1 X ,which is the pushout of the maps SO,61).

we construct a D-diagram for

called the pushout of diagram

(27) (or We can now define the fundamental Banach constructions. Let X be a D-diagram and Y , be a closed subspace of X ; , i = 0 , l . Definition 2. The subdiagTam Y of a diagram

X

is the pullback of the maps

a"yi

E

c1(K, C ( X ) ) , i = 0 , l . U

It will not be difficult for the reader t o verify the validity of the following statement. Proposition 3.

-

(a) The following embeddings are valid:

(28)

A(Y)

~ ( x, ) q Y ) q X ), Lf

Comments and additional remarks where

-

275

A(Y) is a closed subspace o f A(X).

(b) The embeddings

x.

1

Xi, i = 0, 1, induce the B1-morphisms o f

Y

in

It should be noted that Definition 1is “natural” in the sense that if a Ddiagram is generated by a couple, the concept o f subdiagram for it coincides with the concept o f subcouple (see Example 2.1.3). Let us now suppose that P is a subdiagram o f X and that 7 r i : Xi -+ are canonic surjections, i = 0 , l .

X;/x

Definition 4. The factor-diagram xi

0

X/Y

is the pushout o f the maps

Sf E Ll(A(X),Xi/K) 7

i = 0,1 .

Proposition 5.

-

E(X/Y) = E(X)/E(Y), where the bar denotes the closure in E(X)

(a)

[see (28)].

(b) The surjections Xi + Xi/E;, i = O , l , induce the B1-epimorphism. U

Remar k. Although

(29)

A(X/F) = A(X)/A(Y)

in a number o f special cases, in the general case this equality does not hold true. We shall associate each D-diagram rization o f

x with the subdiagram X o (regula-

X). It is described as follows:

276

Interpolation spaces and interpolation functors

where Xp is a closure of the space Im6f in X i , and E(X)' is the closure of the space af(Xo) n of(Xl) in C ( X ) . Further, a?'

@'

:=

sf, i =(),I.

I(@,* 1

Then we define for a given D-diagram

qx>* X* :=

(.f)*

:= allxp and

x:

X

the conjugate D-diagram

(.ox)*

x,.

.

,A(X)*

(a:')*

The fact that t h e object obtained is a 2)-diagram follows from formulas (9) and( 12). The definition of a morphism conjugate t o

T

:=

(To,TI) E L ( X ,Y )is

natural as well. Namely, by putting

T* := (T,,T;)

,

we can easily verify t h a t t h e following proposition is valid. ProDosition 6.

(b) (T*)c= (TA)*,(T*)A= (Tc)*.

We shall now describe the duality relation between the concepts of subdiagram and factor-diagram. Proposition 7.

If y is a subdiagram o f

X ,the following

B1-isomorphism takes place:

Comments and additional remarks ( X / Y ) * N Y'.

(30) Here

277

' Y

is a subdiagram o f

X*,defined

by the annulators

'x

c X:,

i=O,l. 0

In this case, the B1-isomorphism is realized by a couple of B1-isomorphisms (isometries) (Xi/x)*=

x',

i = 0,1.

T h e relation between the concept of dual couple

X

introduced in Defi-

nition 2.4.2 and the concept o f conjugate D-diagram is described in Proposition 8.

If X is a D-diagram generated by the couple I? for the embedding

& c B1,

then +

X' = Z(X*)

.

It should be recalled that the reflector a' was defined by formula (20). 0

In the category

B, we

can also introduce the concept of (projective)

tensor product. Definition 9. The t e w o r product of D-diagrams

X

and

F

(denoted by

X

B F) is the

pushout of the maps

8 :6 E L l ( A ( 8 ) 8 A@), X ; 8

6: Here X

x),

8 Y stands for the (projective) tensor product of the Banach

spaces X and

Y

(see, for example, Diestel and Uhl [l],Chap. Vlll).

0

Let us also introduce the B-valued tensor product

(31)

i =O,1 .

x 6Y

:=

q x 63 Y ) .

278

Interpolation spaces and interpolation functors

In other words,

x 6 Y coincides with the pushout of the D-diagram X @ Y.

We shall use the notation z 6 y for the element of the space (31), formed by the couple (z, y)

Here z = o,"(zo)

E C ( X )x A(Y). This element is defined by the identity

+ n;'(zl),

and the symbol of tensor product on the right-

hand side refers t o the category of Banach spaces

Xi

B (so that z; @ Sy(y) E

8 y t ) . The independence of this definition of the choice of the decom-

position

20, 2 1

for z follows from the definition of

X 6 Y.

A D-diagram can also be formed from operator spaces. Namely, let X , Y be objects of We shall consider the map cp; : T + o T, o 6f acting from L ( X , Y >into L ( A ( X ) , E ( Y ) ) ,i = 0,1.

a.

c~F

Definition 10.

The operator 2)-diagram of objects 8 ,Y [denoted by E ( X , Y)]is the pullback of the maps (po, (pl. 0

It can be easily seen that an operator D-diagram is generated by a couple only if R is regular (X= RO). As in the case of Banach spaces, tensor and operator D-diagrams are related through duality. Namely, the following proposition holds. Proposition 11.

The following B1-isomorphism takes place:

(33)

(X @ Y)*N E ( X , Y * )

1(

E(Y,X*).

0

This relation immediately lead t o B1-isomorphisms (isometries) (34)

(X 6 Y)* N

L ( X , Y * ) N L ( Y , X * ).

279

Comments and additional remarks

This concludes our review of the interpolation theory of D-diagrams. Some applications o f the concepts and results considered above will be discussed in the following sections.

2.7.3. Density of the Set of Dual Operators for Finite-Dimensional Couples Using the methods of the theory o f D-diagram, we shall prove here Lemma 2.4.26. Let us formulate it once again.

Let

2 be a finite dimensional regular couple, ? be an arbitrary couple,

and

B := { T ’ ;T E Ll(d,?)}. Proposition 1. The set

B

is dense in the operator ball L l ( Y ’ , X ’ )in the 7-topology gene-

rated by the system of seminorms

Pz,yt(T) := I(Ty’,z)l ,

Proof. We shall

2

E A(.&,

y’ E (?)*

.

require two auxiliary statements.

Lemma 1.

Let us suppose that

Y

E

&

and l e t

= (Zo,Z1) be a couple such that

2, Z Z1. We shall consider t h e diagram

Then C ( Y ) @I C(z) is a pushout of this diagram i n the category

Proof. Let X

B.

Ll(X 8 Z i , X ) , i = 0,1,be operators such that So o ( 6 r @ 6f) = Sl o ( 6 p 8 6f). We shall define the operator T E L(,Z(Y) €3 C ( Z ) , X ) by the formula be an arbitrary Banach space and let S; E

280

Interpolation spaces and interpolation functors

T [ ( o F ( ~ o ) + r l Y ( y8~ )61) := So(yo 8 6 )

+ Si(yi

8 6)

.

The consistency of this definition can be easily verified. Since for the operator T we have

S i = T o (u: 8 u2; ) ,

i=O,1,

C ( Y ) 8 C ( z ) is, according t o the definition, a pushout in the category

Bl!).

(but not in

B

+

Let us now suppose that

X E B1 and .f E F D ,i.e.

dimensional couple. We define the map ' p i :

it is a regular finite

Xf 8 2, + .C(z,,Xi)*by

the formula

8 z),T) = (x*,Tz)

(Qi(Z*

where

Z* E

X,t, z E Zi and T E Lc(Zi,Xi), i = 0 , l .

In this notation, the following lemma is true. Lemma 2.

The map cp

X*

@

:=

is a B1-isomorphic map from the 2)-diagram

(cp0,cpl)

2 into the 2)-diagram L(z,X)*.

Proof. The fact

that 9, :=

Xf 8 Zi

3:

L(Z;, Xi)* for a finite dimensional

Zi, i = 0,1, is well known from Banach analysis (see, for example, Diestel and Uhl [l],Chap. VIII). It can be easily seen that in view of the definition of

Q;,

the map

2)-diagram

Q

is a B-morphism from the D-diagram

x* 63

L ( z , X ) * .In order to prove that cp is a Bl-morphism,

t o the

it remains

t o show, in view of Proposition 1, that cpc is an injective map.

We shall use Lemma 1, putting Y := X* in it. Since in the case under consideration Zo S 2, because .f EFD, it follows from the lemma that --.).

cpc :

A(X)*

@ C(z)

4

L ( z , X ) * acts according t o the formula

(cpc(.* 8 z ) , T)= (Z*, TAZ) .

L(C(z'),A(X)) + L ( 2 , X ) and 1c, A(X)* 8 C(z) + L(,X(z),A (a ))* by the formulas Further, we define two maps 17 :

v ( T ) := (Sf o T o of,@ o T o ul), 2

:

Comments and additional remarks

281

($(z* 8 z ) , T ) := (z*,Tz). Then we obtain the identity (35)

77* o c p c = $ .

Since C ( z ) is a finite dimensional Banach space, i n view of the fact following from Banach analysis mentioned earlier, $ is an isometry. This and (35) imply that ' p ~ is injective. 0

Let us now prove t h e proposition. We shall use isometries (34).

If for

T E L ( X ,P*)and T E L ( P ,X*)correspond f E (X @ F), then

these isometries the operators t o the functional (36)

(f,z @ Y) = (Tc(z),y)= (m4,4,

where x E C ( X ) , y E A(Y), and z

8 y is defined by formula (32).

Further, in view o f t h e same relation (34). we have (37)

L(?*,Z*)

N

(?* @ Z)* .

Suppose that for such an isometry, the operator sociated with t h e functional

F E

(?* @ ?)*.

T

y* E A(?)* and x E A ( 2 ) . In view of (36), we have (38)

(F,Y*

6 4 = (TC(Y'),4 .

Lemma 2 gives the isometry (39)

'pE :

? * g Z N L(z,?)*

It can be easily verified that in this case

6 .),S) for any s E ~ ( x ' , ? ) . (40)

(cpC(Y*

From (37) and (41)

= (Y*,SZ)

(39), we have the isometry

L(?*,i*)

N

L(i,?)** .

E

L,(?*,i*)

is as-

We specify t h e elements

282

Interpolation spaces and interpolation functors

If with this isometry the functional F corresponds t o the operator T , then according t o (38) and (39) we have (F,YdY*

(42)

63 .I)

= (TC(Y*),.)

*

In view of t h e isometry (41) and the Goldstein theorem (see Dunford Theorem V.4.5), the image of the set and Schwartz [l],

Ll(d,?)for this

isometry is *-weakly dense in the ball Cl(P*,d*).Consequently, for a given E

> 0 there exists an operator T, E L l ( z l ? ) such that I(RYE:(Y*

6 .I)

63 .),T€)I < E .

- (cpC(Y*

In view of (40) and (42), this leads t o (43)

I(TC(Y*), ). - (Y*, T€(.))I

If now the couple

<E .

? is regular, then ?*

and TC is identified with

T.Therefore,

is identified with the couple

5,

in this situation we have

and the proposition is proved for the case under consideration. T o complete the proof, it remains t o note that the required r-density takes place for

? iff it takes place for Po.

0

Remark 1. The statement proved above is contained in an implicit form in the work of Janson [2]. In the review by Ovchinnikov [7] it is formulated explicitly. The proof proposed by this author is based on the Eberlein-Shmuljan theorem. Unfortunately, some essential elements are missing in the presentation, which makes the proof not very convincing.

It should be noted in conclusion that the condition of the finite dimend in Proposition 1cannot be weakened considerably even in the case of the category o f Banach spaces B . This is confirmed by the following result kindly communicated t o the authors by 0.1. Reinov.

sionality of

Comments and additional remarks

283

Proposition 2 (Reinov). There exist separable Banach spaces X and

Y , the first o f which is reflexive

while the second has the property o f metric a p p r ~ x i m a t i o n ~ and ~ , a compact operator

K E L(Y*,X*)such that K

does not belong t o the .r-closure of

any set

(44)

B,

:=

{T'; T E L,(X,Y)} ,

r

>0 .

Proof.We shall require 1 (Lindenstraw [I]). For any Banach space X , there exists a separable Banach space I" with the property of metric approximation and linear surjections Q E ,C2(Y,X)and P E LZ(Y*,X*)such that PQ* = &.. Lemma

0

It follows from this lemma that the following isomorphism holds

Henceforth, we shall choose a reflexive and separable space X

. Since it is well

known that the separability of the conjugate space implies the separability of the initial space, for such a choice of

X the space Y has an additional

property that all spaces conjugate t o it are separable. Further, we use the following classical result. Lemma 2 (Enflo [2]). There exists a reflexive separable Banach space X and a compact operator

k E L ( X ) which

possess the following properties.

> 0, there exists a number E > 0 and finite sets M c X and N c X * such that if for a certain operator T E L ( X ) of finite rank the For any r

ineq ua lity

23See,for example, Diestel and Uhl [l], Chap. VIII, for the definition.

Interpolation spaces and interpolation functors

284 holds, for the norm o f

IlTllx > r

T

we have

*

U

Let now the spaces X,

Y

and the operators P , Q and

I?

be chosen in

accordance with the lemmas formulated above. We put

v

(45)

:=

P*K E L ( X , Y * * )

.

Then V is a compact operator. Further, we put

K := V*Iy*E L ( Y * , X * ) .

(46)

Since an operator conjugate t o a compact operator is compact, it remains

K does not belong t o the .r-closure of any set B, [see (44)]. Suppose that the opposite is true so that K E B, for some r > 0. Since Y has the property of metric approximation, K must also lie in the .r-closure t o prove that

of the set of finite rank operators whose norms do not exceed a certain fixed constant y. Let us show that this is not so. let

T E .C,(X,Y)

be a finite rank operator. We take r

:= 27 in

> 0 and the finite Q * ( N )c Y * .

Lemma 2 and according t o this lemma find the number

E

N c X * . Further, we put N* := If (z,z') E M x N * , where Z' := Q* and Z* E N , according t o (45)

sets

M cX

and

and (46) we have

(T'z'

- Kx', X ) = (Q*x*,( T - V)Z)

=

= (Q*z*,(T- P*I?T)z)= (z*,(Q**T)z - ( Q * * P * ) ( k z ). ) However, in view of Lemma 1, @ * P I X

=

1 ,

and Q**(Y = Q (by the

canonical identification of a Banach space with the subspace of i t s second conjugate). Therefore, it follows that (47)

(T'x' - Kz', X ) = (z*, S X - I?z)

,

where we put S := QT. Since S is a finite rank operator, it follows from the inequality

Comments and additional remarks I(T*x’- K d , x)I

285

< E which holds for all (2,3:’)

E M x N*

in view of (47) and Lemma 2, that

IlQTll = llsll > 27 By Lemma 1, it follows that choice of

*

IlTll > 7,which is in contradiction with

the

T (E L , ( X , Y ) ) .

0

Let us show in conclusion that the majorization condition (2.4.24) i s considerably weaker than the condition of ?--density of t h e set

B := {T’; TE

Ll(Z,?)} in the ball L1(ff,g‘). We shall limit our analysis t o the category B of Banach spaces, where this circumstance is pronounced most clearly. It was established in Proposition 2 t h a t in the category B t h e T- density mentioned above does not take place even for “good” Banach spaces. We

shall not establish that an analog of the majorization condition (2.4.24) in the category B is satisfied for any Banach space. Namely, the following proposition is valid. Proposition 3 (Reinov). Let X , Y be Banach spaces and

T E L 1 ( Y * , X * ) Then . for

any x E X and

y* E Y* the following inequality holds: I(TY*,4I

(48)

Proof.

5 sup{(y*,Sx:); IISllX,Y 511 .

Consider t h e element y** :=

T’x E Y**(we

assume that

X is

canonically embedded in X**). Without loss of generality it can be assumed

llzll

= lly*11 = 1. Then IIT*xlI 5 1 and Ily*ll = 1 so that there is an element y E Y such that one has for fixed E > 0

that

IlYll 6 (1+ €1 11~x115 1+ c (Y*,Y) = (Y**,Y*>

7

(= (T*Z,Y*))

(a consequence of the so-called Helly’s lemma; see, for example, Pietsch

[l]).Consider the lines Lo c X , L1 c X spanned by the elements 3: and y respectively. One defined the operator SOE L(L0,L1) c L(L0,Y ) by the formula So(Xz) := Xy, X E R.Then one has from the preceding inequality

Interpolation spaces and interpolation functors

286

According t o the Hahn-Banach theorem one extends the one dimensional operator So from the subspace Lo t o the whole space of norm. Let

L E L1+,(X,L1)be the

X

with preservation

operator obtained by the extension.

Then

(Y*,SZ) = (Y*,SOZ) = (Y*,Y)

= (T*Z,Y*) .

It follows from this that

In view of the arbitrariness o f

E

> 0 the inequality

(48) is proved.

0

Remark. In view of Theorem 2.4.17, the majorization inequality (2.4.24) is not always fulfilled in the category

8 of Banach couples.

2.7.4. Some Unsolved Problems

Let us recall here some unsolved problems mentioned in the t e x t and formulate a few new ones. Most of them refer t o the material discussed in Secs. 2.4 and 2.5. (a) Does the set

Int(2) o f interpolation spaces define the couple generating

it (accurate t o transposition)? See Conjecture 2.2.32 for details. (b) Characterize the couples possessing the Hahn-Banach property. Here we speak of couples for whose arbitrary subcouples an analog of the Hahn-Banach theorem is valid. See Example 2.1.22 for details. (c) Do there exist unbounded interpolation functors? It is clear from Theorem 2.3.30 that this problem refers rather t o the subject matter of axiomatic set theory. (It should be recalled that for any model of set

Comments and additional remarks

287

theory, there exists a model containing it, i n which the classes of the narrower model become the sets o f the wider model.) (d) Formulate the criterion o f complete interpolation property for a given interpolation space X E Int(d). We recall that we speak of such X's for which

X' E I n t ( 2 ) . Describe complete interpolation couples (see

Definition 2.4.16).

In connection with these problems, see Theorems

2.4.17 and 2.4.24. (e) Formulate a duality criterion for the space X E

I(?'). It should be noted

t h a t the problem has the following not very satisfactory solution. We pro-

vide t h e space A ( 2 ) with the norm

llzll

:= sup { ( z ' , ~ ); 11z'11~5 1)

Y is the (abstract) completion of this space. Then a necessary and sufficient condition for X t o be dual is that Y be isometric

and suppose that

t o a certain intermediate space o f the couple

2.Recall that this condi-

tion for Y is equivalent t o the condition o f matching (A") in Proposition 2.2.27.

(f) Characterize complete interpolation functors. Here we speak of functors

F for which the map F' is also a functor (or, which is the same, F' =

DF).See in this connection (2.4.28)

and Definition 2.4.9.

(g) Characterize reflezive functors, viz. functors F such that

DDF = F".

(h) Is the analog of Theorem 2.4.34 of the form

valid provided that K A ( A )= B'? See Theorem 2.4.39 for details.

(i) Do there exist minimal functors which are not computable? See in this connection Theorem 2.5.12 which makes an affirmative answer t o this question quite probable.

(j) Describe the set o f all computable interpolation spaces of the couple x' (i.e. the spaces o f the form F ( 2 ) for a certain computable functor F ) .

Interpolation spaces and interpolation functors

288

(k) Is the intersection of computable functors a computable functor? An affirmative answer t o this question would make Theorem 2.5.26 much more stringent.

(I) The same question for minimal functors. (m) For which couples

A' is the intersection formula

valid?

(n) For which couples

A' is the following intersection formula valid?

For any elements a , b E C(A'), there exists an element c E

E ( i ) such

that

It should be noted that the previous formula follows from this one for the case when each orbit on the left-hand side is generated by a single element. (0) The

same question for the sum of c ~ o r b i t s . ' ~

(p) Prove that an analog of Theorem 2.5.23 on the interpolation of the Lipschitz operators is not valid for quasilinear operators acting in couples of Banach lattices. See Definition 1.10.2 as well as Supplement 1.11.4.

241t will be shown in Chap. 3 that the affirmative answers to questions (m), (n) and for couples 21 and ;3 , see Theorems 3.3.15 and 3.4.9.

(0)exist

289

CHAPTER 3 THE REAL INTERPOLATION METHOD 3.1. The

K - and J-functionals

A. The modern idea of t h e real method is that it is formed by two closely related families of functors, viz. on t h e concept of the

{ K a } and { J a } . Their definition is based

K - and J-functionals, which sporadically appeared

even in the previous chapters o f the book. We recall that

Here

x E C(x') and t > 0.

Furthermore,

for

x E A(x') and t > 0. In some calculations we also need the E-functional mentioned above.

Recall that

Here we assume t h a t inf

0 = +co.

Henceforth, the E-functional will be used for constructing t h e E-method of interpolation, which is close t o the real interpolation method. In a moment we shall establish a relation between t h e K - and E-functionals, based on the Legendre-Young transformation.

To formulate the final result, we require

some concepts and facts from the calculus of convex functions. Recall that a function

f

:

C -+ R,defined on the convex cone C

is called convez if Jensen's inequality is satisfied:

of the linear space,

290

The red interpolation method

Here q , x 2 E C and X

E ( 0 , l ) are arbitrary.

The function f is called concave if -f is a conzlez function. Henceforth, we shall also deal with convex functions which assume the value of +cx, (for a natural interpretation o f inequality (3.1.4)). a convex function

f , domf denotes the

function f is called proper if domf

set

For such

{x E C ; f ( x ) < +m}. The

# 0 (i.e. f # +m).

Definition 3.1.1.

We denote by Conv the convex cone formed by all continuous concave functions

f

:

1R+ + nt+ u (0).

0

ProDosition 3.1.2. (a) If f E Conv, then

f is a nondecreasing function, while t 4 t - ' f ( t ) is a nonincreasing function. Thus, for any s,t E R+, we have

(b) Conv is closed relative t o pointwise infimum.

Proof. (a)

Let s,t E (1 - A)t

lR+ be

+AN,

given and let N

>

s 2 t be arbitrary. Then s =

where X := (s - t ) / ( N - t ) , and in view of Jensen's

inequality

As N tends t o +m, we obtain f ( s ) 2 f ( t ) . Let us now suppose that 0 < E < t 5 s. Then t = (1- A)& A := ( t - E ) / ( s - E ) , and Jensen's inequality yields

f(t) As

E

t-&

Lf(s) 5 - &

tends t o zero, we obtain f ( t ) / t 2 f(s)/s.

+ As, where

The K - and J-functionals (b) Suppose that S

c Conv

291 := infS is

and is not empty, and that g

defined by the formula

g ( t ) := inf { f ( t ) ; f E S}

.

It follows from the fact t h a t f E Conv and from the properties of infimum that g satisfies Jensen’s inequality for concave functions. Besides,

the function g i s upper semicontinuous as the infimum of continuous functions, and hence is measurable. This and the concavity of g obviously imply that it is continuous. Thus, g E Conv.

Let us define the least concave majorant

R

f o f t h e function f

:=

R+---t

by putting

(3.1.6)

:= inf {g E Conv; g

L

If]} .

Corollary 3.1.3.

If t h e function f := R+-+ JR satisfies the inequality (3.1.7)

If(t)l

5 c max(1,t) ,

where c is a certain constant, then

t E =+,

f^ E Conv.

Proof. Since in view of (3.1.7) I f I does not exceed a certain linear function, the set on the right-hand side o f (3.1.6) is not empty. 0

We shall call the continuous function

f

:=

R+

--f

HE+

U (0) quasi-

concave if it satisfies inequality (3.1.5). Corollarv 3.1.4.

A quasi-concave function f is equivalent t o a function from Conv. To be more precise, f^ E Conv, and (3.1.8)

f 5 f^ 5 2f .

The real interpolation method

292

Proof.It followsfrom inequality (3.1.5)t h a t condition (3.1.7)with c is satisfied. Therefore, inequality in

(3.1.8). Let us

put

c

f(t) := s u p { c A i f ( t ; ) ; for

t > 0.

Obviously,

:= f(1)

E Conv, and it remains t o establish the right-hand

f 5 f , and it

Consequently, in view of

A; = 1, A; 2 0,

c

can be easily seen that

(3.1.6) and (3.1.5), we

Ad; = t }

f is concave.

have

and the supremum on the right-hand side does not exceed

s u p { c A;

+ t-'

c

A;t;} = 2

.

0

Remark

3.1.5.

A similar statement is also valid for continuous functions f :

R+-+ R+

which satisfy t h e inequality

(3.1.5') f ( t )2

c

max(l,t/s)f(s)

for a certain constant c

> 0 and for

all t,s

E R+.

Definition 3.1.6.

The convex cone of all proper convex nonincreasing functions f R U (0, +m} will be denoted by M C .

:

R+-+

S

c MC

0

Proposition 3.1.7.

The pointwise supremum of functions from a nonemp either is equal t o +m or belongs t o

b f . We put g := s u p s . Thus,

MC.

subse

The K - and J-functionals

293

Then g is monotonic and satisfies Jensen's inequality (3.1.4) since all

f

E

MC. 0

L e t us define the greatest convex minorant o f the function f :

R U (-00, (3.1.9)

R++

+m} by putting

f'

:= s u p { g

5

Ifl;

.

g E MC}

CorolIary 3.1.7 .'

If

If1 # m, then fv E M C .

Proof.Since g = 0 belongs t o M C , the set on the right-hand side of (3.1.9) is nonempty. If, in addition, If(t)l

f#

<

00

a t least a t a single point, then

0O.

0

Proposition 3.1.8.

2)belongs t o Conv, while E ( . ; 2 ;2)E M C . The function ,f(.; z ; 2)belongs t o the set Ex(Conv) formed by extreme

(a) The function

(b)

K ( . ;z ;

rays o f t h e cone Conv; here

$ ( t ) :=

1

-

cp(1lt) .

Proof. (a) The K-functional is concave as t h e infimum o f linear functions, so it belongs t o Conv.

For the E-functional we immediately obtain from

formula (3.1.3) t h a t it is nonincreasing and differs from the improper function

(3.1.10)

+00.

Further, definition (3.1.3) leads t o the identity

E(Xt ; 2 ; x') = XE(t ; X-'z ; 2)

and the inequality

The real interpolation method

294

(3.1.11) Here t k

> 0 and

This fact and

c

tk

< 00,

and the series

xk converges in

c(2).

(3.1.10) imply that

.(3.1.12) where

(Yk

:=

Xk/(C A,).

Tk := t k / X k , we obtain from

Taking here two summands and putting

(3.1.12) Jensen's inequality (3.1.4). Con-

MC.

sequently, the E-functional belongs t o

(b) In view of (3.1.2) it is sufficient t o verify t h a t every ray of Conv of the form

B2+ma, 0 < a < 00, where

(3.1.13)

ma(t) := min(1, t / a ) ,

t E B2

is an extreme ray. Assume this is not the case. Then for certain y o and y1 in Conv, which do not belong t o this ray and some

In view of inequality

(3.1.15)

cp(t)

X E ( 0 , l ) we have

(3.1.5), for any function cp E Conv we have

2 min(l,t/s)cp(s)

In the case under consideration, for s := a we obtain

and there exist values o f t for which this inequality is strict. In view of

(3.1.14) for such a value o f t we have

so that

The K - and J-functionds

295

We have arrived a t a contradiction.

Remark 3.1.9.

It will be shown below t h a t Ex(Conv) = {rn,R+; 0 5 a 5 m}. Here rno := 1 and moo := t. Let us now estalbish the relation between the K - and E-functionals. For this we define two operations on functions f

:

R+ -P R+U {+m} by

assuming t h a t

fv(t) := inf {f(s)

,

+st}

s>O

(3.1.16) f"(t) := sup { f ( s ) - s t } s>O

Since both operators are obviously related t o the operation o f transition t o a conjugate function in the calculus of convex functions, it follows from the corresponding duality theorem (see, for example, Rockafeller [l],Theorem

12.2) that (3.1.17) where

f

f = (fv)" , f^ =

,

is defined by formula (3.1.6) and

f

by (3.1.9). This leads t o

Proposition 3.1.10. The following formulas are valid:

K ( . ; 2 ; 2)= E ( . ; 2 ; 2)" , (3.1.18) E ( - ;2 ; 2)= I < ( . ;

Proof. In view of (3.1.1) ~ (; z t;

2)=

2;

d)V .

we have

inf 8>0

(

inf lblllX, 5 s

1 1 5

- zClllx,+ s t ) .

The red interpolation method

296

Combined with (3.1.3), this leads to the first identity (3.1.18). The second identity (3.1.18) follows from the first identities of (3.1.18) and (3.1.17) if we take into consideration that E ( . ; x ; 2)E M C (see Proposition 3.1.8). 0

The formulas (3.1.18) will be used below for calculating the K-functional of some couples. Here we point out as a corollary limiting relations for the

K-functional, which will be useful for the further analysis. Thus, in view of Lemma 2.2.21 and Propositions 3.1.2 and 3.1.8, we have

Remark 3.1.12. Since for a transposed couple X T := (X1,Xo) we obviously have (3.1.19)

K(t-';

+

2;

X T ) = t - ' K ( t ; x ; 2),

the second limiting relation is equivalent to the first one. Corollary 3.1.13. lim K ( t ; x ; 2)= id { l ~ x- yllxo ; y E X I ; x - y E x O )

t-+O

;

(3.1.20) lim t - ' K ( t ; x ; 2)= inf

t-+O

{llz - yllxl ; y E X O

Proof. In view of the first identity (3.1.18), lim K ( t ; x ; 2)= >lye inf t-+O

;x -y EXI>

.

we have +

( ~ ( ;sx ; X

I +ts)

=

S>O

the E-functional decreases). The last limit, however, is obviously equal to the right-hand side of the first identity (3.1.20). The second identity is obtained from the first one and relation (3.1.19). (since

0

The K - and J-functionals

297

Corollary 3.1.14. The element zbelongs t o C(r?)O iff +

lim ~ ( tz ;; 2)= lim t-'K(t; z ; X I = 0 .

t++O

t++m

Proof. If z E Z(r?)O, then for any E > 0 there exists an element z, E A ( 2 ) such that llz - z e l l c ( ~<) E. Therefore, z - z, can be represented in the = z o + s l with llzollxo+IIzl(I~l 5 E. Hence, in view of (3.1.20),

form z--1,

+

lim K ( t ; 2 ; X ) I t++O and since E

112 - 5,

- ~illx= o 11zollx0 I t:

9

> 0 is arbitrary, we obtain the first o f the required relations.

The

second relation is proved in a similar way. 0

B. Let us now calculate K-functionals of elements for some couples important for the further analysis. We shall start with the following remarks.

B; E BL(O), i = 0,1, i.e. they are Banach lattices on a measurable space ( 0 , p ) (see Definition 2.6.3). Since a Banach lattice Consider spaces

is known t o be continuously embedded into the corresponding linear metric space of measurable functions nach couple

M ( R , d p ) [see (1.2.5)], (Bo,B,)form a Ba-

I?.

This identity and the fact that a norm is monotonic on a Banach lattice lead t o the required statement. 0

The real interpolation method

298 Proposition 3.1.16.

> 0

If B E Bt(C2) and w

is an arbitrary measurable weight, then for

L,(w-l)

:=

LW(w-')(fi, d p ) [see (1.3.16)] the following identity is valid:

(3.1.22)

E ( t ; f ;B,LW(w-'))

=

Il(IfI

- t w ) + l l ~.

Here x+ := max(z,O).

Proof. For a function g in t h e closed ball Dt of the space L,(w-l) t , we have ~ ~ g 5 ~t so ~that~ 191 ~5 tw.~ Consequently, u ~ for ~

of radius such g we

have

If

- 91 L

(If1 - tw)+

1

whence it follows that t h e left-hand side of (3.1.22) is not less than its right-hand side. On t h e other hand, the function

f(.)

:=

{ f(x)

tw(.)sgnf(x)

obviously belongs t o

for lf(.)I

I t4.1

f

for If(.)l

> t4.1

7

D t , and hence the left-hand side of (3.1.22) does not

exceed

Ilf - f l l B

= ll(lfl - tw)+lIB

.

0

Let us now suppose that, as before, Lp" := L , ( t - S ) ( R + , d t / t )and

(3.1.23)

Em

4

:= (LO,,LL),L, := ( L : , L i ) .

Proposition 3.1.17. The following identities are valid:

K ( t ; f ; El) =

J

mql,tls)lf(s)lds ;

mt

(3.1.24)

K ( t ; f ; 3,) = f ( t ) . (For the definition of

Proof. The first

f

see (3.1.6).)

formula of (3.1.24) follows from the fact that, in view of

(3.1.21), we have

The K - and J-functionals

=

299

ds

J

I ~ ( S ) min(l,t/s) I

y

.

p1+

In order t o prove the second relation, we make use of (3.1.22) with

and

W(S)

:= s, s E

E ( t ;f ;

B

:=

Lo,

R+. Then we have

L)=

(Ifl(.) - t s ) +

SUP

=

S>O

SUP

{IfKs)- tsl .

s>o

Thus, in notation (3.1.16) we can write

E(. ; f ;

L)= IflA

.

It remains to apply the first identity (3.1.18) and then the second identity of (3.1.17). Thus,

E ( . ; f ; Z,)V

= (1flA)V =

p.

Let us now suppose that L, := L , ( R , d p ) , 1 5 p 5

00.

ProDosition 3.1.18. The following identity is valid: t

K ( t ; f ; Ll,L,) =

1

f’(s)ds.

0

For the definition of the decreasing rearrangement

Proof. In view of relation (3.1.23),

we have

r,see (1.9.3).

The red interpolation method

300

Proceeding in the same way as in the proof of (1.9.9), we see that the right-hand side is equal to J (f*(s)- t)+ds. Thus,

a

where we put

.(t)

R+; f*(S) 2 t ) .

:= sup{s E

In accordance with (3.1.18) for any s > 0 we then have 4s)

K ( t ; f ; L1,L,) 5

J

f * ( z ) d z- s u ( s )

+ts .

0

+

Let s := s ( t ) be such that ~ ( s 0)

5 t 5 u(s - 0). Substituting s ( t )

into

the previous inequality, we obtain t

(3.1.25)

~ ( tf ;; L~,L,)5

J

f*(x)da: .

0

Conversely, if have

If1

= fo+fl, f; 2 0,then in view of (1.9.18) and (1.9.19) we

-pt+

According

to

s>o

Proposition 1.9.6, the right-hand side is equal

IlfOllLl

+ t IlflllLcm .

Thus, we have

j 0

f*(s)dsI

id Ifl=f+O+h

f a 20

( I I ~ +~ ~llflllLm) I ~ ~,

to

The K - and J-functionds

301

which being combined with (3.1.21) leads to an inequality inverse t o (3.1.25).

Further, let

M be

a metric space with metric

the space o f functions bounded on

Next, for the function

T.

We denote by B ( M )

M and having the norm

f E B ( M ) we

define the modulus of continuity

w ( f ; .) by putting

(3.1.27) ~ (; tf) := sup {f(z) - f ( y ) ; for t

>

~ ( zY) ,

It )

0. Finally, we define the space o f Lapschitz functions L i p ( M ) by

assuming that (3.1.28)

IflLip(M)

:=

SUP

f ( X I - f (Y) .

r(x, Y)

It can be easily verified that B ( M ) and Lip(M) are complete (although (3.1.28) is just a seminorm, since it vanishes a t constants). The calculation of t h e K-functional of the couple

(B(M),Lip(M)) involves

Proposition 3.1.19.

If f E B ( M ) (3.1.29)

+ Lip(M), then

K ( t ; f ; B(M),Lip(M)) =

Proof.Suppose that f = fo 1

+

f1;

1

w(f ; 2t) I 5 2

N o ;

1

&(f; 2t) .

then

2t)

+ 51

w(f1;

2t) L

302

The red interpolation method

3 L j ( f ; 2t) also does not

Since the right-hand side is a function from Conv, exceed the right-hand side.

In order t o prove the inverse inequality, for fixed

f E B ( M ) and

s

>0

we define a function d := d ( f , s ) by the formula

d :=

1

-

SUP

{ 4 f ;t ) -

St)

t>O

in such a way that d =

(3.1.30)

u*(f; s) [see (3.1.16)].Let us show t h a t

E ( s ; B(M),Lip(M)) 5 d

For this we consider the function

Since the function (of

x) under the supremum belongs t o the space Lip(M)

and has a norm which does not exceed s, I f s l ~ i p ( ~ )

5 s as well.

Hence we have

1l.f

E ( s ; B(M),Lip(M)) 5

-fsllB(M)

>

and it remains for us t o estimate the right-hand side. y E

However, for any

M we have

whence for z = y we obtain

(3.1.31)

fa(.)

- f ( ~ 2) -d ,

On the other hand, for a fixed fs(z)

I f (

E

2

EM .

> 0 there exists a

~ c) ST(Z, ye)

-d +E

This inequality and the definition o f d leads t o

point yE E M for which

The K - and J-functionals Since E

> 0 is arbitrary, - f(.)

).(sf

Together with

303

we thus obtain

Id .

(3.1.31), this inequality leads t o the estimate Ilf-fslle(~,5

d, which proves (3.1.30). Using

(3.1.18) and (3.1.30), we now obtain

~ (; f t; B ( M ) ,Lip(M)) 5 inf { w A ( f ; s) + 2 t s )

=

s>o

1 = - (U*)V(f

; 2t) . 2 It remains t o note that in view of (3.1.17), the right-hand side is equal t o &(f; 2 t ) .

;

As a corollary, let us calculate the K-functional o f the couple (C,C'), where

C

consists lflcl

C[O, 11 with the norm of the maximum, while C' := C'[O,11 of functions f continuously differentiable on [0,1] and such t h a t :=

:= max

CorolIary

If'l.

3.1.20.

K ( t ; f ; C,C')=; &(f; 2 t ) .

Proof.We require t h e equality (3.1.32)

K ( . ; Z;2)= K ( . ; Z ;2') ,

which follows from the obvious equality

(3.1.33)

K ( t ; z ; 2)= IIZIIZ(X,,,~X~)

(2.2.12) according t o which C ( f c ) = C(?). Therefore, our statement will follow from Proposition 3.1.19 and equality (3.1.32) if we show that (C')' = Lip[O,l]. For this we take a function f E Lip[O,l] and relation

and extend it continuously as constants on

f" E

Lip(R+), and IfILipcm+)=

(fnLCm

c C'

by putting

R+.The

I ~ ~ L ~ ~ [ o , ~ Further, o.

obtained function

we define a sequence

The red interpolation method

304

Then

fn +

f

in

C

and, Ifn(c1 = max

.(fI

+ l / n ) - f(z)l L

IfI~ip[~,l].

X

Consequently, (3.1.34)

f

E

Then for

5

Ifl(C1).

Conversely, i f f E

h

(C')"and IflLip[O,ll

*

(C')',then for a certain sequence ( f n ) n E ~ c C'

> 0 we

we have

have

which leads t o the inverse inequality t o (3.1.34). 0

Finally, let us derive a formula for calculating the K-functional for the elements of a conjugate couple. Proposition 3.1.21. (a) If

2'

E C ( x " ) , then +

K ( t ; .'; X ' ) = sup{(z',z); J ( t - 1 ; (b) If x' E A(X'), then

2;

2)5 1) .

The K - and J-functionals

305

Proof. (a) In view of (3.1.32) and Proposition 2.4.6, we have

It remains t o note that

(b) The proof is similar. 0

A precise calculation o f the K-functional can be carried out only in some rare cases. In applications, however, it is sufficient t o carry out a calculation

up t o equivalence. For this purpose, we sometimes calculate instead of the K-functional a certain quantity similar t o it. The following two modifications of this kind will be useful for t h e further analysis. Definition 3.1.22. The Lp-functional o f the elements

2

E C ( x ‘ ) is a function defined by the

formula

PO, pl Here P’ := PO,^^), where 1I

< 00,

and t E

B+.

0

Definition 3.1.23. The K,-functional formula

o f an element

2

E C ( 2 ) is a function defined by the

The red interpolation method

306 Here

p=

t E R+and 1 5 p 5

00,

the ordinary modifcation corresponding t o

00.

0

Let us demonstrate the usefulness of the concepts introduced above a t the

hand of t h e following example. Let us consider a couple (L,(wo),

Lpl(ul))

:=

LAG), for which the following proposition is valid. Proposition 3.1.24.

The following equality holds:

where the function 1,- is defined by the equality

1A.s;t ) :=

inf

+

{szpo tyP1} .

2+y=l Z,Y>O

Proof.Arguing in the same way as when deriving the first identity in (3.1.24), we obtain

where we put

The K - and J-functionds

IAs,t)

M

min(s,t)

307

.

where we put

Proof. Let us make use of t h e following obvious equality: (3.1.37)

K p ( t )= (Lp,p(tP))l’p .

Then the proof is reduced t o calculating the function

Zp,p.

Since

lPIpis p

homogeneous here, it suffices t o show with the help of differential calculus that

Since we obviously have

(3.1.38)

Kp(.; z ; 2)x K ( -; z ; 2),

The real interpolation method

308

There is no simple relation similar t o (3.1.38) between the &-functional and the K-functional. Nevertheless, they can be expressed in terms o f each other, which follows from the useful proposition. Proposition 3.1.27. If z E C ( 2 ) and y E C(?), then t h e inequality

LA.; z ;

2)5 LA.; y ; ?)

is equivalent t o the inequality

Proof. For

t h e sake o f brevity, we put K ( t , z ) = K ( . ; x ; E ( . ; z ; x’),and so on. As in Proposition 3.1.10, we have

Ldt ; z)

=

inf ( E ( s ,z)”

z),E ( t , z ) =

+ tsP1) =

S>O

where we have put B ( s ; z) := E(s’/P1; 2)”.

Since the function s + s l / p l

is concave while the function s + s” is convex (since pi 2 I), the function E belongs t o the same cone M C as t h e E-functiona. Therefore, in view of (3.1.17), we have

B(t ; z) = ( B V ) A ( t ;

z) = sup

{ L d s ; z) - s t } .

s>o

5 L d t ; y) that k(t; z) 5 5 E ( t ; y), and applying (3.1.18), we

Hence it follows from the inequality L d t ; z)

k ( t ;y).

This means that E ( t ; z)

arrive a t

K ( t ; z) = inf { E ( s , z ) + t s } 5 inf { E ( s ;y ) + t s } + t s } S>O

S>O

=

The K - and J-functionals

309

Thus we have proved that the inequality for the K-functionals follows from the inequality for the Lrfunctionals. The inverse statement is proved in a similar way. 0

Remark 3.1.28.

Let w; be a convex function which bijects R3+ on itself and is equal t o zero

i = 0 , l . We put

a t zero,

(3.1.39)'

L;(t;

z;

2) :=

inf

+

{ W O ( ~ ~ Z O ~ tul(llzllJXl)} ~ X ~

.

z=zo+z1

We leave it t o the reader t o show t h a t t h e following fact o f a more general nature can be established from the above arguments: (3.1.40)

L;(.; ; x') 5 LG(-; y ; 9 )H K ( *; z ; 2)5 K ( *; y ; ?) .

Corollary 3.1.29.

If z E C ( 2 ) and y E C(?), then

K p ( - ;z ;

Proof.

For 1 5 p

x') 5 ITp(.; y ; f ) H K ( - ;z ; x') 5 K ( - ;y ; ?) .

< 00,

it is sufficient t o make use o f the equality

K p ( t )=

Lp,p(tP)l/Pand the previous statement. Let us consider the case p :=

03.

Then

and hence the inequality Km(t; z) _< K,(t

; y) leads t o the following state-

ment. For each

E

decomposition

>

0 and each decomposition y = yo

z = zo + z1 such that

Thus, we immediately obtain

+ yl,

there exists a

The real interpolation method

310

Taking here the lower bound over all decompositions y = yo+yl and making E

tend t o zero, we obtain

The converse statement follows from the case p limit as p

---f

< 00

by a passage t o the

00.

0

C . Let us indicate some generalizations o f the above analysis. These generalizations are connected with an extension of the category

B’ of Banach

couples. We begin with an analysis o f the widest category among those considered below, viz. the category

2 o f couples of nomzed

Abelian groups.

Here we shall list some properties o f this category. For details, see t h e monograph by Bergh and Lofstrom

[l], Sec. 3.10.

3.1.30. A function v : A + El+ specified on the Abelian group A is called a Definition

norm

if it satisfies the following conditions: (a) .(a) (b) .(-a)

=0

a =0;

= .(a);

(c) for a certain constant 7

2 1 and all a,b E A ,

The couple ( A ,v) is called a normed Abelian group. 0

In analogy with the case of metric spaces, the concept of open ball and the related concepts o f convergence, completeness, etc. are defined in ( A ,v). Let now ( A ,v) and ( B , p ) be two normed Abelian groups.

The K - and J-functionds

311

Definition 3.1.31.

We denote by L ( A , B ) the Abelian group of bounded homomorphism T : A --f B. Thus T E L ( A ,B ) if T is a homomorphism of the groups and

It can be easily verified that formula (3.1.41) defines a norm on the Abelian group L ( A , B ) and that L ( A , B ) is a complete normed Abelian group if B is such a group. 0

Having two complete normed Abelian groups A0 and A l , we say that they form an a-couple if A; are subgroups o f a certain Abelian group A , and the compatibility condition for the norms in Proposition 2.1.7 is satisfied.

L(A’,l?) of bounded + the a-couple B.

In analogy with the Banach case, we define the space

homomorphisms acting from the a-couple A ’ t o Definition 3.1.32. The category

A o f normed Abelian couples has a-couples as its objects and

bounded homomorphisms acting from one a couple t o another as its morphisms. 0

Proceeding in this way on the basis o f the analogy with Banach couples, we can obviously define the sum and the intersection, intermediate and the interpolation spaces, interpolation functors, and so on. We shall require these concepts very seldom. It should also be noted that the K-, J - , and E-functionals for an a-couple (3.1.1-3).

x’ are also defined

by the formulas similar t o

The properties of these functionals for a-couples will be described

somewhat later. Here, we consider two complete subcategories of the cate4

gory

A. The first

of them consists o f complete quasi-normed linear spaces

and bounded linear maps of such couples. We denote this category by

z.

For its description, it is sufficient t o explain what we mean by a quasi-normed linear space.

The real interpolation method

312 Definition

3.1.33.

A linear space V is called quasi-normed if it is supplied with a function I/ : V -+ R+satisfying the following conditions: (a) .(a)

= 0 ++a = 0 ;

(b) for a certain constant 8 E [0,1] and all X E

R,v

E

V,

v ( X a ) = IX1%(a) ;

(c) for a certain constant 7

2 1 and all v , w E V ,

+

Let us also introduce a subcategory Q o f couples of complete quasi-

normed spaces (quasi-Banach couples). Observe that a quasi-Banach space differs from a Banach space in t h e respect that the triangle inequality gets replaced by the less stringent inequality (c) in Definition 3.1.33.

T h e objects of the category f$ will be henceforth called q-couples, and i,Z-couples. Thus, we obtain the following chain of

those of the category

complete subcategories of the category (3.1.42)

A:

c i j c L' c A .

Further, let

x'

E

A and l e t v; be the norm in X i . The standard proper-

ties of the K-, E-, and J-functional are described in this case by the following

ProDosition 3.1.34. (a) If 7i are the constants for v; in inequality (c) in Definition 3.1.30, then

(3.1.43)

K ( t ;z

+ y ; x') 5

70

{K("

t ; z ;2)+ K ( Z t ; y ; x ' ) }

70

and a similar inequality is valid for the J-functionals.

70

The K - and J-functionals

313

(b) Under the same assumptions, we have

We leave the proof o f this proposition to the reader. 0

It should be noted that in some cases similar inequalities are required for an infinite number o f terms. They can be obtained with the help of the

Aoki-Rolevich theorem (see Bergh and Lofstrom 111, Lemma 3.10.2) from which it follows, for example, t h a t

2 1 and p E (0,1] which depend only on 2. Finally, it should be noted that, as in Proposition 3.1.8, the K - functional belongs t o the cone Conv, and j t o the set of i t s extreme rays. However, E generally belongs not t o the cone M C but t o the wider cone M consisting of proper nonincreasing functions f : R++ I3+u (0, +m}. For this reason, with certain constants 71,-yz

only the first of formulas (3.1.18) from Proposition 3.1.10 holds. The second formula is replaced by the equality (3.1.46)

8(.; z ; 2)= K ( . ; z ; 2)A .

Concluding the section, l e t us consider some examples of a- and I- couples (examples of q-couples were given in Chap. 1). Example 3.1.35.

V be

B

V be a quasi-Banach space. Suppose that a family A := {A, ; n E Z}is specified in V , which Let

a separated topological space and

satisfies the following conditions: (a)

fl An = (0);

(b) -An = A,;

L)

The red interpolation method

314

It can be easily verified that (3.1.47)

.(a)

U A,

:= inf (2"; a E A,}

Let us suppose that

U A,

is a normed Abelian group if we put

,

aE

U A, .

is complete relative t o the norm v (this is satisfied,

for example, in the case where A , = (0) for n

5 -no).

In this case we call

A an approsimationfamily. For an approximation family, the set (B, U A,) is obviously an a-couple.

If the stronger condition

is satisfied instead o f (b), the group

U A,

is obviously a linear space, and

norm (3.1.47) is 0-homogeneous, i.e. v ( X a ) = .(a)

,

Consequently, in this case

X

#0.

(B, IJ A,)

is an 2-couple.

Later we shall consider a number o f concrete realizations of this scheme. For the time being, we note that

where

n ( t )is the largest integer satisfying the inequality 2" 5 t .

Remark 3.1.36.

U A, as well. For example, 2" i n (3.1.47) can be replaced by q" with any q > 1. When A, = (0) for n < 0, we can also put v(u) := inf { n + 1 ; u E An}. It is possible (and useful for applications) to define other norms on

315

K - div is ibilit y

3.2. K-divisibility

A. One of the most fundamental properties of the K-functional is described in Theorem 3.2.7 on K-divisibility. Unfortunately, the proofs of this theorem known t o us are not very simple. In this subsection we shall consider some preliminary results which will be used in the proof presented in

this book. Some o f them are of interest themselves and are singled out as propositions.

Let us start with certain properties of the cone Conv. We put

(3.2.1) This is clearly a subcone of the cone Conv. Further, let us define an operator (3.2.2)

I given on the cone Conv by the formula

, t E R2+ .

I(cp; t ) := tcp(l/t)

Proposition 3.2.1.

The operator I i s an evolution on the cone Conv, and i t s restriction I1convo is an involution on the cone Convo. Here I is a monotone operator.

Proof. In view (3.2.3)

of Proposition 3.1.17, we have

cp = K ( . ; cp;

em)

for a function cp E Conv. Indeed, t o prove the validity of (3.2.3) sufficient t o verify that cp belongs t o C(Z,).

it i s

However, since cp(t) and t-lcp(t)

are monotonic (see Proposition 3.1.2). we have

= 241)

.

To complete the proof, it remains t o take into account identity (3.1.19) which implies that

Icp = K ( . ; $0; L'T ,)

.

316

The real interpolation method

Thus, Ip E Conv. The remaining statements are obvious. Theorem 3.2.2 (on descent). Let the inequality

be satisfied for an element f E Conv and for a sequence (p,), assume that

C p n ( l )< 00.

c Conv, and

c

Then there exists such a sequence (fn)nEN

Conv that

Proof.

It should be noted that the convergence of the series

point 1 implies, in view of inequality

t E [0, +m).

(3.1.5), its

C

p,, at

convergence a t any point

Thus, this series converges pointwise. Further we require

Lemma 3.2.3.

If { x a := ( I : ) ~ ~aNE ;A} is a linearly ordered subset o f the cone nonnegative sequences of space

Proof. Since infz*

=

( inf

Zlf

of

11, then

z:)nEN, the right-hand side obviously does

a

not exceed the left-hand side. To prove the inverse inequality, we take and choose for n E

lV

satisfied for an:

z!

&

inf xz+2”+’

for ,f3 5 a ,

.

a

We also fix a . and choose

5 N+1

x:<-

&

2

E

>0

an index a, so that the following inequalities are

N i n such a way that

K -divisibili ty

317

x: is a linearly ordered subset, this inequality is also valid for a 5 ao. We put ii := min an. Then for p 5 ii we have Since

OsnsN

0

Let us now prove the theorem. For this we consider a partially ordered set S-2 of sequences ( $ n ) n E ~ (3.2.4)

$n

Lpn,

c Conv, such that C &(l) < 00

f IC

+n

,

nE

and

N.

The order is introduced through the relation ($h) 5 (+:) @ 5 n E A T . Let {($:); cu E A} be a linearly ordered set in a.We put $, := inf $;

(n E

+:,

N).

a

Obviously,

($n) 5 ($:),

cu E A, and if (&) E

Zorn's lemma are satisfied for that

$n

a,then the conditions of

a.In order t o verify this, we note first of all

E Conv as they are lower bounds of concave nonnegative functions.

Further, the first inequality of (3.2.4) is obviously satisfied as well. Finally, in view o f Lemma 3.2.3, we have

so that t h e second inequality in

(3.2.4) also holds. L e t us now apply Zorn's

lemma, according t o which there exists in Let us show that

R a minimal element

( f n ) n E ~ .

f = C fn, which will complete the proof o f the theorem.

Otherwise, the open set

The real interpolation method

318 is nonempty.

Let ( a , b ) be one of the intervals constituting E , and l e t at least one of the functions f,, say, fk, be nonlinear on (a,b ) . Then “cutting” the graph of fk by a sufficiently small chord and replacing the function fk in (fn)

the sequence

by the obtained function fk, we obtain a new sequence from

52, which

This, however, is in contradiction t o the fact that (fn) If we have is minimal. Thus, the functions f, are linear on each (.,a). 0 < a < b < 00, it follows from the definition of E that for h := C fn we is less than

(fn).

have

f ( t ) = h ( t ) , for t

:= a , b

and f ( t ) < h ( t ) , for a


Since the function on the right-hand side is linear, this is i n contraction with the concavity o f f .

If such an interval (a,b) with 0 < a

< b < 00

does not exist, three cases

are possible: E contains one o f the intervals (0, b ) or (a,+w), where b and a

> 0, or E

< 00

:= (O,+oo).

If in the first case f(0) = h(O),everything is reduced t o the case considered above. If, however, f ( 0 ) < h(O), then for a certain point t o E (0, b ) we have

Let 1 be a supporting line t o the graph of f a t point to, so that

Further, suppose that f n ( t ) :=

Ant + B, for t E (0, b), where A,, B, 2 0.

Finally, l e t 6 be the root of the linear equation

5 I(b), the root 6 is unique, and 6 E ( t o , b ] . We put E := l(O)/h(O). Since h(0) > f(0) 2 0, this definition is consistent, and in view of (3.2.5) and (3.2.6), E < 1 . Finally, In view of (3.2.5), (3.2.6) and the fact that h(b) = f(b)

l e t us suppose that

K-divisibility

319 A,t+EB,

forO
fn(t)

for t

gn(t) :=

Here, t h e number

23,

is satisfied. Then

C gn(0) = Z(0)

,

>6

is chosen in such a way t h a t the relation gn(6) = fn(6) and

C gn(t) = l(t) 2 f(t) for t E (0,s).

C g,(b)

= h(6) = Z(6). Consequently,

Thus, we have constructed a sequence

(gn) E 52, differing from (fn) and such that (g,) diction with the minimality o f

The second case ( a ,+m) help of the transformation cp

5 (f,),

which is in contra-

(f,,).

c E,a > 0 is reduced t o the first case with the --f

Icp [see (3.2.2)]. We must only take into

account the fact that this transformation preserves inequalities and transforms linear functions into linear ones. Finally, the case

E

:= (0, +m) is

analyzed in the same way as the first case. 0

Corollary 3.2.4. The function cp E Conv lies on an extreme ray o f this cone iff for some constant 7 > 0 and some a E [0, +m] we have

cp=7ma.

< a < 00,

We recall that m a := min(l,t/a) for 0

Proof. The fact

mo= 1and

mm(t) := t.

that the function 7m, belongs to the set Ex(Conv) was

established i n the proof of Proposition 3.1.8 (for 0

<

a

<

00;

the case

a := 0,m is analyzed similarly). Let us prove that these functions exhaust all elements of the set of extreme rays. Let cp lie on an extreme ray of Conv. If cp does not have the form 7 m , with a := 0, +m, there exists a point (a,cp(a)) on the graph of this function at which the support function

Z,(t) := At

+ B has strictly positive coefficients.

Then cp

5 cpo + 9 1 , where

cpo(t) := At, 'pi := B . In view of Theorem 3.2.2, there exist functions f; E Conv, such that CP = f o + f i

Since cp(a) = Aa

,

+ B (:=

fo(t) I At

,

fi

I B.

Z,(a)), fo(a) = Aa and fi(a) =

B. As f; is

a concave and nondecreasing function, it follows hence that fo(t) = At for

320 t

The red interpolation method

5a

and

fi(t) = B

for

t >_ a .

On t h e other hand, 'p lies on an extreme

ray of Conv so that f i = y,'p for certain constants 7i > 0. Thus, 'p = 7 , y 1 f i , whence 'p = 7 m i n ( l , t / a ) for some y

> 0.

Henceforth, we shall need a special method of constructing from a given function in Conv and a number q

< 1 an

equivalent function in this cone,

which is the sum of elements in Ex(Conv). This construction starts with the following inductive process of constructing a sequence

( t i ) (which

can be

> 1. We put t o := 1. If a point ti, i 2 0 , has already been constructed, then assuming that i := 2n is even, we define ti 1 = tzn 1 as the root of the

finite) of points in (0, +m) for given 'p E Conv and q

+

+

equation

If i := 2n - 1 is odd, we define ti

+ 1 = tzn as an (arbitrary)

root of the

equation

(3.2.8)

~ ( t=)qcp(ti) .

If equation (3.2.7) (or (3.2.8)) has no solution, the process of constructing points with positive indices is terminated.

i's, we have from (3.2.8). in view of the fact that cp(t) is nondecreasing and cp(t)/t i s nonincreasing, For odd

ti+l q'p(ti) = ti+^) I -p ( t i ) .

ti

Thus,

Similarly, from (3.2.7) it follows t h a t the same inequality is valid for even i's as well.

Let us find out when and for which index the process of constructing of points ti with

if either

i 2 0 terminates. We shall show that this occurs if and only

K -divisi bili ty

321

p’(m) := lim

(3.2.9)

t++m

Q(t) > t

and then the process is discontinued for an even (3.2.10)

p(+oo) :=

i, or

/i. p ( t ) < +oo

and then the process terminates for an odd

i.

Indeed, if a point t2n+l from equation (3.2.7) cannot be found, then

and (3.2.9) is proved. Conversely, if (3.2.9) is satisfied, then p(+m) = +m. Therefore, equation (3.2.8) always has a solution, i.e. the process of constructing

t;

does

not terminate a t a point with an odd index. If, however, the process of construction continues unlimitedly, in view of the inequalities

< -4t2n+1) - 1 -

(~(t2n+2)

~

hn+2

hn+1

9

dt2n) t2n

we obtain, in contradiction t o (3.2.9),

If (3.2.10) is satisfied, the line of reasoning is the same, but instead of (3.2.11) we obtain the following inequality:

Since the point t z n + l is a root o f equation (3.2.7), it also follows that

Let us now construct the points ti with i

< 0.

For this purpose, we make

use of the preceding process of constructing sequences for the function Ip [see (3.2.2)], and then, having obtained this sequence ( & ) ; l o , we put t-;

:=

1

-,

ii

i20.

The real interpolation method

322

Considering then I ( p ; t ) := t ' p ( l / t ) ,we obtain from the properties proved above the similar properties for the sequence

t , with i 5 0.

Thus, t h e following proposition holds true. Proposition 3.2.5.

E Conv and t h a t q > 1 is given. Then there exists sequence (ti)i=-m,...,n of points lying on (0, +00), such that Suppose that 'p

to = 1, t;+l/t,2 q for -m

a

6 i
The relations

are satisfied.

05n5

(3.2.14)

00,

and n = +00 iff

cp'(+00)

:= lim t++-

cp(+oo)

:= lim t++m

cp(t)- 0 t

and

cp(t)= + 0 0 .

The nonfulfillment o f the first condition in (3.2.14) is equivalent t o the statement that n is even. In this case,

The nonfulfillment of the second condition in (3.2.14) is equivalent t o the statement that n is odd. In this case,

(3.2.16)

cp(tn) min(l,t/t,) 2 q-lcp(t)

for t

2 t,-l .

(d) m = +00 iff

(3.2.17)

cp(+O) = 0 and

cp'(+O)

:= lim t--+O

t

= +m

K-divisibili ty

323

Here, the nonfulfillment of the first condition in (3.2.17) is equivalent t o the statement that m is even. Then

for t 5 t ,

cp(t) 2 q-'cp(L,)

(3.2.18)

.

The nonfulfillment of the second condition in (3.2.17) is equivalent t o the statement that m is odd. In this case

Let us further consider for given cp E

Conv and q > 1 the function @

defined by the formula (3.2.20)

@ ( t ) :=

cp(t2;+l)min ( 1 , -m62i+l
t -) + h+l

Here Ek := 1 if k is even and Ek := 0 if k is odd or if k := +co. The sequence ( t i ) is constructed for cp and q according t o the processes described earlier. Proposition 3.2.6. The function @ is equivalent t o the function cp. To be more precise, the following inequality holds:

Moreover, for

Proof. If t

E

t E [tZ,,tZ;+2],we also have

[t2;,t2i+l], the fact that t-'cp(t) is nonincreasing and (3.2.13),

(3.2.16) and (3.2.19) imply that

The real interpolation method

324 Similarly, for t E [t2i+l,tZi+Z]we have

~ ( tI) ‘ P ( ~ z ~ +I z )~ ( t z i + l ) Taken together, these inequalities prove (3.2.22). Since the right-hand side

$ ( t ) for all t E R+, this proves the left inequality in (3.2.21) as well. It is sufficient to prove the right inequality in (3.2.21) only for t E [tzi+l; -m 5 2i 1 5 n],t := 0 and t := +m. Indeed, $ is a piecewise-linear function with vertices (tzi+l,@(t2;+l)),and the two functions (P and $ belong to Conv. When t := 0 or := +m, in view of Proposition 3.2.5 we have

of (3.2.22) in this case does not exceed

+

Let us now suppose that t := t2,+l. We note that

In view of (3.2.13). (3.2.15) and (3.2.18). and the fact that cp is monotone, have

we

it follows that ~1

5 (1+ 4-l

4 + q-’ + ...)V(tzi+l)I v ( t z i + l ). q-1

Similarly, in order to estimate S 2 ,we use (3.2.13), (3.2.16), (3.2.19) and the monotonicity of t-’cp(t). This gives

S2 i (q-’

+ Q - ~+ ...)cp(tzi+l)I~ ( t z i + l.) 4-1

Combining these estimates, we obtain

1

I(-divisi bility

325

C . We now have everything t o prove the main result, viz. t h e theorem on K-divisibility. To formulate this theorem, let us consider an arbitrary couple

2 and any element z E C ( 2 ) . We assume that for a certain sequence

((P,,)~,=w c Conv, such t h a t C cp,(l) < 00, (3.2.23)

K ( . ;z ;

2)I C cp,

we have

.

Then the following theorem is valid. Theorem 3.2.7

(Brudnyl- Krugljak).

There exists a sequence (Z,),,~Wc C ( 2 ) such that (3.2.24)

z =

,

z,

(convergence in

~(2))

and such t h a t its elements z, satisfy the inequalities

(3.2.25)

If we set (3.2.26)

+

K ( - ;2,; X ) I 7cpn ,

S(2) :=

12

E RV .

infy,' then

S(2) 5 8 .

Proof.The required sequence (z,,)

can be easily constructed when all sum-

mands cp,, belong t o Ex(Conv). The arguments given below aim at reducing the general case t o this particular case. In order t o avoid the obscuring of the main idea o f the proof by technical details, let us begin with t h e basic case. Namely, we assume t h a t conditions (3.2.14) and (3.2.17) are satisfied for the function cp :=

K ( - ;z ;

2).For a given q

( t , ) i c z(in view o f restrictions imposed on

+ [see (3.2.20)]. 'The quantity couple 2.

(P,

> 1, we

construct a sequence

m, n = +m) and the function

In view of the inequalities (3.2.21) and (3.2.22), we have

6(2)will be henceforth called the constant

of K-divisibility of the

326

The red interpolation method

Then according t o Theorem 3.2.2, there exist functions

+,,

E Conv, such

that

(3.2.27)

n E N .

and n

The definition of the function 8 implies that it belongs t o the subcone C9 c Conv, where C9 consists of all functions which are linear on each interval into which the semiaxis (0,+m) is divided by the points t z i + l . Consequently, all the functions also belong t o this subcone. In the further

+,,

analysis, we need Lemma 3.2.8. Every function

f

E

C9 can

be represented uniquely in the form of the con-

verging series

with non-negative a, b and c,.

Proof. It can be easily seen that i f f and

Ci

:=

tzi+l(f;(tz;+l)

E C’,taking a := f(+O),

- f , ! ( t z ; + l ) ) , where

b := f’(+co)

f/ and f,!are the

left-hand

and right-hand derivatives, we obtain t h e required representation with nonnegative coefficients. Let us prove that this representation is unique. It can be easily verified that for a function f represented in the form indicated in the lemma, we have

lim f(t) -b

t-a,

t

lim f ( t ) = a

and

Further, assuming g i ( t ) :=

ci

t-0

.

min( 1,t / t Z i + l ) , we have Ci

f((tzi+l)

= f:(tzi+l) = (gi):(tzi+l) - (gi):(tzi+l) = hi+l

Thus, the coefficients a, b and

ci

are uniquely determined by f

0

Therefore, each function $, in (3.2.27) can be uniquely represented in the form

K -divisibili t y (3.2.28)

327

$,,(t) = C

aniv(tzi+t)

min(1, t / t 2 i + 1 )

i

with non-negative a,,i. The first equality in (3.2.27) then implies that (3.2.29)

a,,i

= 1 for i E

Z.

n

We now have to specify the elements of the sequence (2,) c C ( 2 ) in the statement of the theorem. For this purpose, we note that in view of the definition of the K-functional, for a chosen e > 0 and any t > 0 there exist elements z , ( t ) E X i , i = 0,1, such that zo(t)

+ .l(t)

=2

,

(3.2.30) 11~0(t)llxo

+t 1 I 4 ) l l X l

I (1 + E ) ( P ( t ) .

Recall that here and below, cp := K ( . ; z ;

-+

x).

Lemma 3.2.9. There exists a set of elements {u; E C ( 2 ); i E 23) such that z = and (3.2.31) for all

t K ( t ; u i ; 2)5 (1 ~ ) ( 1q+ ) c p ( t z i + l ) min (1, -)

+

t2i+1

C

u;

, t E R+

i.

Proof. Using the sequence ( t i ) constructed for cp, we put (3.2.32)

ui

:=

zo(tzi+z)

- ~ o ( t 2 i ),

iEZ.

In view of (3.2.30), u, i s also equal to q ( t 2 i ) - z 1 ( t 2 i + 2 ) . It follows from this and from the definition of the K-functional that

328

The real interpolation method

From inequality (3.2.30) we also have (3.2.33)

llzo(ti)llxo

5 ( 1 +E)P(ti)

d t i )

I ( 1+ E ) -.

Ilxl(ti)llxl

7

ti

Combining this result with the preceding inequality and (3.2.13), we obtain

K ( t ; u i ; 2) 5

I(1 +€)(I The desired relation

2

+

q)v(tzi+l) min(l>t/tzi+l)

.

= C u , follows from the next lemma.

0

Lemma 3.2.10. The series u i converges absolutely in C ( i ) and its sum is equal to x.

xi

Proof.The inequality (3.2.21)

C

IIuill-qa,

=

C~

leads to ( 1 u i; ; 2)I ( 1

+ &)(1+q ) ( S 1 +

~ z .)

Here we put

As in the proof of Proposition 3.2.6, we have

Thus, the absolute convergence of the series is established. Furthermore, in view of the identity k Z-

Ui

=2

- ZO(t2k)

+

ZO(t-21)

= xI(tZk)

+

xO(t-'21)

-1

[see (3.2.32)] and the inequalities (3.2.33), we have k

5

(11xl(tZk)llX,

+

11ZO(t-ZI>llXo)

I

7

I<-divisibili ty

329

Since we have assumed that the relations (3.2.14) and (3.2.17) are satisfied for the function p, the right-hand side tends to zero as k,I -+ 00.

Finally, let us define the sequence (5,) c (3.2.34)

z,

:=

a,;ui

,

nE

E(d)by putting

N .

i

We recall that the coefficients a,; 2 0 are defined by the relations (3.2.28). Then it follows from (3.2.29) and the previous lemma that the series (3.2.34) converges absolutely in C ( x ' ) , while the series C z, converges absolutely in the same space and has the sum z. It remains to estimate the K-functional of the element 2., In view of (3.2.34), (3.2.28) and estimate (3.2.31) we have

.

Ci aniP(tZi+l)min(l,t/tzi+l)= (1+€)(I + q)$n(t)

Applying now estimate (3.2.27), we finally obtain

For q := 3, the right-hand side assumes the minimum value, and we also obtain inequalities (3.2.26). Let us now suppose that one of the conditions (3.2.14) or (3.2.17) is not satisfied for cp := K ( . ; z ; 2).We consider first the case when the first condition from (3.2.14) is not satisfied. Then the indices of the sequence ( t i ) vary from -00 t o a certain even index n > 0. In view of (3.2.20), the function d in this case has the form

+(t) :=

C

-00<2i+l
Therefore, the function

$k(t)=

t)k

+

Q(t ) v(tZi+l)min(l,t/tzi+l) 2t . qt n

from (3.2.27) can then be represented in the form

C -oo
dtn> akip(tZi+l)min(1, t/tzi+l)+ akm -. qt n

The real interpolation method

330 In analogy with (3.2.29), we then have

For -co

< 2i

+ 1 < n we now define the elements ui by the same formula

(3.2.32) and add t o this set one more element u+, :=

2

- 20(tn)

(= z l ( t n ) ).

Taking into account (3.2.33), we then have

As in Lemma 3.2.10, we verify t h a t

the series begin absolutely convergent in C(x'). Furthermore, we put

(3.2.36)

21;

c

:=

ski%

-m<2i+l
+ akmu+m

*

Using the estimates (3.2.31) and (3.2.35) and taking into account the definition of

$k,

we obtain

K ( t ; 26; 2) 5 (1 +€)(I

+q)

'

It remains t o apply (3.2.27) and put q := 3 in the inequality obtained. Let us now consider the case when the second inequality in (3.2.14) is not satisfied. Then n is odd, and @ has the form [see (3.2.20)]

+ ( t )=

C

-co<2i+l5n

In this case, we put

.

p(t2i+1) min(l,t/tzi+l)

K-divisibility According t o

331

(3.2.33) and (3.2.16), for t 5 tn we then have

K ( t ; u+ce; X) I t J l z l ( t n - l ) I I X 1 I (1 4

For

t > t,,

+

E)

dtn-1)

-t I tn-1

the same inequalities give

K ( t ; u+, ; 2)5 K ( t ; zo(tn-l) ; 2)+ K ( t ; 2 ; 2) 5

Combining these estimates, we have

(3.2.37)

K ( t ; u+, ; 2)L (1

+ e)(l +

q)v(tn)

min(1,t/tn) .

We define xk by the same formula (3.2.36). Then (3.2.31) and (3.2.37) lead t o

and it remains t o apply

(3.2.27). The case when (3.2.17) is not fulfilled is considered in a similar way.

Then we must put u-,

:= zo(t-,)

(=

z-z1(t-,))

if the first o f the conditions (3.2.17) is not satisfied (rn is even), and substitute into this formula t-m+l for

t - , if the second condition in (3.2.17) is

not satisfied (rn is odd). The reader can easily independently restore missing details.

D. Let us consider the validity o f the theorem on K-divisibility for the A o f normed Abelian couple (see Definition 3.1.32). The following

category

The red interpolation method

332

example shows t h a t in general this theorem does not hold. Example 3.2.11. --*

Z

Let

consist of two identical spaces

1 , 0 , Clearly, with this norm,

Z

22 supplied

with the norm

n#O n=O. becomes a complete normed Abelian group, and

ZEA.Then for any n E Z,we have

-

K ( t ; n ; Z)= min(1, t ) . := 2-N min(l,t), 1 5

Putting (P&)

k 5 2N and (PI;

:= 0 for

k > 2N,

we

have

I<(.; n ; r7)=

c

(Pk.

kizm If Theorem 3.2.7 were valid in this case, it would follow from this identity that n =

xi=,n k , (nk)c Z,and N

min( 1, t ) = ~

+

(; n;t ; Z )5 y2wN min( 1,t )

with a certain constant y. But then

y 2 2N for all N , and we arrive

at a

contradiction ,

Let us show that a less stringent version o f the theorem on K-divisibility is nevertheless valid. Namely, suppose that

functions (P,

(3.2.38)

x'

E

2,z E C ( x ' ) , and t h a t the

1 5 n 5 N in the cone Conv are such that

K ( . ; 2 ; 2)5

N (P,

.

n= 1

Theorem 3.2.12. There exist elements

x,, 1 5 n 5 N , C ( 2 ) such that

N

(3.2.39;

I

=

X, 1

and

K -divisibili ty

333

Here y depends only on

Proof.

N and on 2.

Let us limit ourselves t o the analysis of the basic case when the

bilateral sequence

(ti)

is constructed for the function 'p :=

K ( . ; x ; 2).

The remaining cases are analyzed similarly (see the end of the proof of Theorem 3.2.7). In view of (3.2.38) for each element

i E Z there exists a

5 n(i) I N , such that

number n = n(i), 1

Furthermore, we put

{

:=

&;

1

for n = n(i) ,

O

for n # n(i) .

This definition implies t h a t

c el N

(3.2.42)

=1

for all i

.

n=l

Let us now define the elements u; by the same formula (3.2.32).Using then the weakened triangle inequality 1 1 2

where c;

+ ~ l lI~c i (il l x I I X , + I I Y ~ ~ x ; )

i = 0, 1

7

2 1 depends only on X i , and following the

as in the proof of Lemma

same line of reasoning

3.2.9,we obtain the inequality

(3.2.43) K ( t ; u i ; 2)5 (1 +&)(I Here we put c :=

7

+q)c'p(tzi+l)min(l,t/tzi+l).

max(q,,c1).

It follows from this estimate that Lemma 3.2.10is valid in this case also. Thus, the series

C IIu;llzca,converges,

and x =

C

u; in C ( 2 ) . We now

Put

(3.2.44)

2,

:=

c

,

E ~ U ;

15 TI 5 N

.

i

In view of (3.2.42)and the absolute convergence of the series C u ; , we have

c xn=c N

u i = x ,

1

I

The red interpolation method

334

and hence the identity (3.2.39) is satsified. It remains t o verify inequality (3.2.40). However, according t o (3.2.43) and (3.2.44). we have

K ( t ; 2, ; 2)I (1

(3.2.45)

+ &)(I+ q)c

Ef&i+l)

min( 1, t / t 2 i + l )

.

I

Suppose now that, for a given t the indices

i

(3.2.46)

E:

= 1 and

Further, let s E (3.2.47)

E:

> 0,

the number

T

E Z is the maximum of

which satisfy the relation

tz;+l 5 t .

23 be the minimum o f the indices i for which

=1

and t 5 tzi+l .

Then the sum on the right-hand side of (3.2.45) obviously does not exceed

The sums Sl and Sz were estimated in t h e proof of Proposition 3.2.6, where

it was shown t h a t

Moreover, from (3.2.46), (3.2.47) and inequality (3.2.41) we have

Finally, from the inequalities (3.2.46) and (3.2.47) and the monotonicity of

cp,(t) and cp,(t)/t we also have

Combining the last three estimates, we obtain

Thus, the following inequality is obtained from the left-hand side of (3.2.45):

K ( t ; s,;

2)I 2 c ( l + E ) ( q 1)2 N v n ( t ) , +

q-1

K-divisibifity

335

which gives estimate (3.2.40) with y := 16cN(1

+ E ) for q

:= 3.

0

E. Let us now discuss the question o f the K-divisibility constant 6(@ in the statement of Theorem 3.2.7, as well as the absolute K-divisibility constant

6

(3.2.48)

:=

SUP

S(2).

2 Some informations about these constants is contained in Proposition 3.2.13. 4

4

= S(L1) = 6(L1, L,) = 1;

(a) S(L,)

(b) S(c,c') 2

= 1.609 ....

Proof.We shall only consider the couples 3,

and

i1and refer the reader t o

the references at t h e end o f this chapter as regards the remaining statements. Thus, let us suppose that

f E C(3,)

and that ( ( P ~ )c~Conv ~ Nwhere

C ( ~ ~ (<100) and that

In view of Proposition 3.1.17, the left-hand side is equal t o the concave majorant

f. For the same reason,

If1 I C

(3.2.49)

pn

we have

.

We put fn := f . (pn/C pn), IZ E N .Then f = C Since according t o (3.2.49) _< pn, we have

fn.

If n [

+

K ( . ; j n ; L,) = j n _< 4

Thus, 6(L,)

Gn

= pn

.

= 1.

In analogy t o the previous case, for the couple (3.2.50)

K(.; f ;

I C

pn

4

L1we

have

The real interpolation method

336

f E C(Z1) and ( p n ) , , , =c~Conv. Without any loss of generality, we can assume here that f 2 0. Then the left-hand side is equal t o Sf,where

for a

(see Proposition 3.1.17). For any given q

> 1,we replace the norm in Li

by

an equivalent norm, assuming t h a t

Obviously, this norm does not exceed q denoting the couple

(Ly, L:) by i l ( q ) ,

J 1g(s)1 2 = q 11g11L;. Therefore, 1R+ where the norm i n

L:

is replaced by

the one indicated above, we obtain

Applying Theorem 3.2.2 on descent, we obtain from this inquality and from

(3.2.50) that

(3.2.51) K ( - ;f ; XIQ)) = where (&)

c Conv and

C +,,

+,, 5 qvn, n E RV.

On the other hand, t h e I(-functional on the left-hand side can be calculated, according to Proposition 3.1.17, as follows

Thus, this function is a polygon with vertices lying above the points q' on the abscissa axis, i

25. Then, according t o Lemma 3.2.8, equality (3.2.51)

involves the representation

where all a;,,2 0 and

K -divisibility

337

W e now put

Since q

> 1 is arbitrary, it follows that 6 ( i l ) = 1.

Remark 3.2.14. Thus the absolute K-divisibility constant 6 [see (3.2.48)] satisfies the inequa Iit ies

1.52... 5 6 5 9

.

The second of these inequalities follows from (3.2.26). The exact calculation of this constant is obviously a difficult problem.

The r e d interpolation method

338

3.3.The K-Method A. Recall that t h e K-method forms t h e family {I<*}, where

(3.3.1) Here

llI<(.;

IIzIIKo(y, :=

2

; 2llo

.

runs through the family of Banach lattices over

( B + , d t / t ) , which

satisfy t h e condition

min(1,t) E @ .

(3.3.2)

The role of condition (3.3.2)is clarified in Proposition 3.3.1. K* is a functor iff condition (3.3.2)is satisfied.

Proof.Since the I<-functional belongs t o the cone Conv, in view of inequality (3.1.5)we have min(1,t)K(l; 5 ; 2)5 K ( t ; z ; 2).

(3.3.3)

Here K(1; 2 ; of

d ) = llzllc(~,.Therefore, applying the @-norm t o both sides

(3.3.3)and taking into account i t s monotonicity and condition (3.3.2),

we obtain the embedding

&(2) r;:

(3.3.4)

where y :=

C(X'),

(1 min(l,.)ll~.

Similarly, for z E A ( 2 ) we obviously have the inequality

K ( t ; 2 ; 2)I min(Lt) Il~lla(w,, which leads t o the embedding

(3.3.5)

A(2)

-.

Il*(z).

-

Consequently, in order to prove that the normed space h'a(X) is intermediate for the couple

X , it remains t o verify

i t s completeness. For this purpose, we

use

Lemma

3.3.2.

L e t the sequence of elements ( f n ) , , E ~ of the Banach lattice

be such that

The K-method

c llfnllo < and let

f

-

339

,

be the sum of the series

C f,

in @. Then this series converges

f

absolutely a t almost each point, and its sum coincides with

almost every-

where.

k f . We use the following well-known fact (see Theorem 1.3.2 for the case of

Lp): There exists a continuous embedding of @ into t h e corresponding space

M(R, d p ) of

measurable functions

(3.3.6)

L+

@

Let now s,

M(R,dp) .

be the n-th partial sum of the series

C

If,l.

In view of the

hypothesis o f the lemma, this series converges in @ t o a certain function s. On the other hand, the sequence (s,) does not decrease and hence converges a t each point t o a certain measurable function S. Let us show t h a t S E @.

For this purpose, we first prove that

(3.3.7)

s - s,

20

almost everywhere, n E N

.

Indeed, if condition (3.3.7) is not satisfied, there exists a set of positive measure E and a number

s(t)

y > 0 such t h a t

- s,(t) < -y

for t E

E

and for some n := no. In view of the monotonicity o f s, for n

2 0.

this is also true

But then 11s - snlle

L

II(s - s,)xcllo

L y llx.llo

( n 2 no)

,

which contradicts t h e fact that the left-hand side tends t o zero ,as n

4 00.

Thus, condition (3.3.7) is satisfied. Passing in this inequality t o the pointwise limit, we obtain s - S

2 0.

5 s-s,

On the other hand, s-s"

E @; by the de-

finition of Banach lattices, the element s-S, and hence 9, also belongs t o @. This means t h a t the pointwise converging series t o @. Then the series

C

fn,

C

lf,

has a sum belonging

which is majorized by the previous series, also

converges pointwise and has the sum

f,for which If1

5C

If,[

E @. But in

340

The real interpolation method

this case,

f" E +; in view o f embedding (3.3.6),

measure t o its sum

f

in t h e space

the series

0.Therefore,

f =f

C fn

converges in

almost everywhere.

0

Let us return t o the proof of the completeness of

K a ( 2 ) . Suppose that

c K a ( 2 ) is such that the sequence (zn)nEm (3.3.8)

C IIzntIK,ca,

< 00 .

We must show that the series

C x,

converges in

K a ( 2 ) . In view o f the

embedding (3.3.4), this series converges in C ( 2 ) . Suppose that its sum in this Banach space is x. Then for s, (3.3.9)

K(.;x-s,;2)<

:=

Em<,x,

we have

c K(.;x,;2).

m>n

In view of (3.3.8) and the previous lemma, the series on the right-hand side converges almost everywhere t o a certain function from @. Then the lefthand side also belongs to @, and therefore I - s, €

K a ( 2 ) . T h 'IS means

that x also belongs t o K a ( 2 ) . Applying the @-norm t o inequality (3.3.9) and letting n tend t o +m, we obtain that the series in the space

C

z,

converges t o x

Ka(2).

Thus, K a ( 2 ) E I ( 2 ) . Let us suppose not that T E

L(x',?). Then for

x E C ( 2 ) we have K ( t ; T I ; f)I

inf

+t

5

{ I I T ~ ~ I I Y ~~ ~ ~ x l l l y , >

Tz=Tzo+Tz~

Thus. we have established that

Applying the +-norm to both sides o f the inequality, we get

341

The IC-method This proves that if condition

(3.3.2) is satisfied, then Ka is an exact inter-

polation functor.

Let us establish the necessity of condition (3.3.2). Let Ka be a functor. Then for t h e couple

k

:=

(R, R),embedding (3.3.5) is valid

with some

y > 0. Here for x E A(*) we have

~ (; z t; k)= min(1,t) 1x1 . Consequently,

Remark

(3.3.5) leads t o the following min(1, t ) E 0 .

3.3.3.

Henceforth, Banach lattices satisfying condition (3.3.2) will be called the parameters of the IC-method. It will be shown below t h a t it is sufficient t o +

limit ourselves t o exact interpolation spaces of the couple such spaces are Banach lattices over of Example

L,.

The fact that

(R+, d t / t ) follows from the statement

2.6.12.

Let us now establish t h a t the functor Ka is maximal for the couple (see Theorem

z,

2.3.17). In order t o formulate the corresponding result, we

Put

(3.3.11)

8

4

:=

Ka(L,).

According t o t h e previous remark, sition

3.1.17, t h e norm in

(3.3.12)

ll.fll.5

=

8 is a

Banach lattice. In view o f Propo-

8 has the form

Ilf^lla .

3.3.4 (Brudnyi-Krugljak). + = Corb&(.;L,).

Theorem

Proof.We begin with the following statement. Lemma 3.3.5. + + Orb,(X ; L,) = L k , where w := K ( . ; z ; 2).

342

The real interpolation method

Proof. According

t o the definition o f the orbit functor [see (2.3.10) and (2.3.11)], we must show that

Let us first prove that t h e right-hand side does not exceed the left-hand side. Since the numerator does not exceed f(t) = K ( t ; f

; im), in view

of Proposition 3.1.7 and (3.3.10). the right-hand side does not exceed the quantity

In order t o prove t h e inverse inequality, we denote by nuous linear functional on

(3.3.13)

ft (t > 0) t h e conti-

C(d)for which for fixed 2 E C(d)we have

ft(z) = K ( t ; z ;

2)

and also

(3.3.13') Since

Ift(y)I

5 K(t ; y ; d )

K ( t ; .; 2)is a norm on

Hahn-Banch theorem.

for y E C ( 2 )

.

C(d),the existence o f ft follows from the

We fix an arbitrary q

> 1 and

define the integer

n ( t ) by the inequality q" 5 t < qn+l. Further, we shall define the operator T : C ( 2 ) -+ C(i,), assuming t h a t ( T Y ) ( ~ ):= .fp(t)(y)

7

YE

~ ( 2. )

If in this case y E X i , in view of inequality (3.3.13') we have

Consequently, T E

L ( i f , i m ) and , t h e norm of T does not exceed unity.

Further, t h e fact t h a t the K-functional is concave and equality (3.3.13) imply that

K ( t ; x ; 2)5 qK(q"('); z ; 2)= q f q n ( t , ( x )= q T ( x ) .

The K-method

343

+ +

Since Orb,(X,L,)

is an exact interpolation space of the couple

+

L,,

and

hence a Banach lattice, the norm is monotone in this space. Applying this norm t o both sides of the previous inequality, we obtain

Letting q + 1 we obtain the inverse inequality. Let us now prove the theorem. Since

+

& E Int(L,),

in view of Theorem

2.3.17 we have 4

(3.3.14)

Corb&(L,,i,)

= & = Ka(i,)

.

Since the co-orbit is maximal (see the cited theorem), the following embed-

-

ding is established:

Ka

1

+

Corb&(.;L,)

.

In order t o prove the inverse embedding, we take an element

x E Corbg(-f ;

z,)

for a certain operator

llfll&

=

such that its norm does not exceed unity. If

T E L(x',f,),

IITxIICorb+(t,,Z,)

Hence it follows that

f = Tx

then i n view of (3.3.14) we have

5

IITII,f,Z,

II"IICorb,(,f,Z,)

'

The red interpolation method

344

Since according t o Proposition 3.1.17 we have

the left-hand side of this inequality is equal to

Consequent Iy, we obt a in

which proves the inverse embedding. 0

Corollary 3.3.6.

Ka = K6.

Proof.Since the two functors

under consideration are maximal on

z,, it

is

sufficient t o prove their coincidence on this couple. Therefore, the problem boils down t o the proof of the following result. Lemma 3.3.7.

If @ E Int(~,), then

Proof. In view of Proposition 3.1.17, we have

llfllK,(zm)

=

llillo 2 llfllo .

Conversely, in view of Lemma 3.3.5, the function t o the space Orb,(Z,,&,)

space. Consequently, for any E for which

f

= K ( .; f ;

z,)

belongs

and has a norm not exceeding unity in this +

> 0 there exists an operator T E LI+~(L,)

f^ = T f . Since @ i s an interpolation space, we have

345

The K-method Making E

4

0 we obtain the required statement.

0

3.3.8.

Remark

-4

It follows from the proof of the lemma that

ip

= Int(L,)

iff

l l f l l ~ = Ilflla.

Remark 3.3.9.

h 0 '

is the maximal exact interpolation space embedded in ip. LS

ip and ipl

E Int(i,),

in view o f Lemma 3.3.7 we have

-4

ipl = K@l(L,)

Indeed, if

-4

L--)

K*(L,) = h .

Corollary 3.3.10.

The mapping @ -+ Ka bijects the set Int(3,)

onto the set of functors of

the K-method.

roof. If

ip

+

ip1

of a space from Int(ioo>,in view of Lemma 3.3.7

K a ( t o o )# K@I(~,).Conversely, the functor Ka equals Kb (see Lemma 3.3.6), where 4 E Int(i,). 0

B.

Let us now prove t h a t the family o f functors

{Kcp} is indeed an

interpolation method, i.e. is stable under superpositions. For this purpose, we consider three K-functors Kao, Kal and Ka. It should be noted that in view of embedding (3.3.6), the spaces

ho and 41 form a Banach couple.

We put (3.3.15)

S' := Ka(&o,&l) .

In view of the statements o f Example 2.6.12,

(R, ,d t / t ) . Theorem 3.3.11 (Brudnyi-Krugljak). The following relation takes place:

us a Banach lattice over

The red interpolation method

346 To be more precise, for a given couple

where we put

?

x'

we have

:= ( K a o ( f ) , K a l ( f ) ) .

S ( 2 ) 5 6. Proof. In view of Corollary 3.3.6, KQ = K6, where 6 E Int(Z,). We recall that the K-divisibility constant

Therefore,

according t o Lemma 3.3.7, we get

Since K I is ~ maximal on the couple

Ka(Kao,Kal)

1

+

L,, it follows

that

Kg .

In order t o prove the inverse embedding, we estimate the K-functional of

the couple

? i n the formulation of the theorem.

In view of (3.3.12),we have

K ( t ; K ( . ; z ; 2); Q o , Q , ) =

By virtue o f Theorem 3.2.7 on K-divisibility, the inequality

K ( . ; 2 ; x') 5 fo

+

fl

leads t o the existence o f zi E C ( x ' ) such that z = 10

K ( .; z i ; x') 5 ( 6 ( f ) Here e

> 0 is an arbitrary fixed

+ &)L,

K ( t ;K ( . ;z ;

i =0,l .

number and 6 ( x ' )

and the previous inequality that

2);i o , Q 1 )

1

+ z1,and

5 6. It follows

from this

The K-method

Thus, for

E

347

+0

K ( - ;z ;

P) 5 6 ( 2 ) K ( . ;K ( * ;2 ; 2);i0,il) .

Applying the @-norm t o both sides of this inequality and taking into account (3.3.15), we obtain 4

ll4lK*(P)

I 6 ( 2 ) I F ( - ; z ; x)llK*(&o,&l) = =

m I I ~ l l K " ( a ).

This proves the second embedding o f the theorem. 0

The result proved above is known as the reiteration theorem and has numerous applications. At the moment, we shall limit ourselves only t o two corollaries of this theorem. According to Definition 2.6.1, we immediately obtain Corollary 3.3.12.

The family of functors

K:

:=

{Ka ; @ E Int(i,)}

is an interpolation me-

thod. 0

In order t o formulate the second corollary, we consider three functions w, wo and w1 in the cone Conv and put

(3.3.17)

77 :=

WOU(W~/WO).

Corollary 3.3.13. -s

KLW,(KLW,O, KLw,')(X) not exceed 2S(J?). Recall that

K L ~ ( - ? )where , the isomorphism constant does

The red interpolation method

348

Proof.W e shall use a general statement situations t o replace the calculation of

which allows us in many practical

Ka

on the couple ( & o , & l ) by an

6 := (ao,Q1).To formulate the result, 6 there exists an operator Q : C(6) -+Conv

analogous calculation on the couple we assume that for the couple

which has the following properties: (a)

Q ( f + 9 ) 5 Qf + Qg, f , g E C(6);

(b) f

5 Qf

for

f

E Conv;

with a constant independent of

f.

Under these assumption, the following

lemma is valid. Lemma 3.3.14.

If 9 := K a ( 6 ) ,then

where the isomorphism constant does not exceed

Proof. Let

Mb(2).

f be a function in Conv. In view of Theorem 3.2.2 and the

monotonicity of the Qi-norms, we have

K ( t ; f ; & o , & l )=

If now

f

operator

= fo

+

fi,

where

fi

E @, we have, in view of the properties of the

Q,

Therefore, the right-hand side o f the preceding equality does not exceed

349

The K-method

Thus, for f E Conv the following inequality is established:

K ( - ;f ; & , & I ) 5 M I ( ( . ; f ; g) . Since the inverse inequality with

M

:= 1 is obvious, we hence obtain for

2)

f := K ( . ; z;

+

IlW. ; z ; 2>11\u Ri I F ( . ; 2 ; x)llK+(Oo,~l) . According t o Theorem 3.3.11 on reiteration, the right-hand side is equivalent t o t h e norm z in the space Ko(K~o,K~l)(i?), and the equivalence constant does not exceed S ( 2 ) . Thus, we have proved that

and the isomorphism constant does not exceed

M 6 ( 2 )I 8 M .

Let us return t o the situation under consideration. We have Oi := L z

L z . By t h e definition o f concave majorant [see (3.1.6)],for p E Conv the inequality I f 1 5 M q is equivalent t o the inequality f^ 5 Mp.

and O

:=

Consequently, (3.3.18)

Lg

= LL

.

Taking for Q the operator

f

-+

f , we see that it possesses properties (a)-(c)

in the lemma, and in view o f the above equality

M = 1 here. Therefore, an

application of the lemma leads t o the isomorphism (3.3.19)

KLL(KLw,o,KLw,') 2 K\u

where 9 := K , p ( L z ,L z ) and the isomorphism constant on the couple

2 does not exceed S ( 2 ) . It remains t o calculate the parameter

(3.3.20)

Z

L&

9.Let us prove that

,

where 77 is defined by formula (3.3.17), and the isomorphism constant = 2.

For f E

C(L$),where w'

:= ( w o , w I ) ,we have the following inequality:

The red interpolation method

350 K(t;

f ;LC)

:=

f belongs t o the space appearing on the left-hand side of (3.3.20) and has in this space a norm which does not exceed unity, then Therefore, if

Taking here t :=

we get

wo(s)'

i.e. ] l j l l L5~ 1. Thus, the left-hand side of (3.3.20) is embedded into the right-hand side, the embedding constant being

5

1.

In order t o prove the inverse embedding, we must verify that the following inequality is valid:

v ( s ) 5 max(wo(s),

w(t>

,

Indeed, since w is nondecreasing, for wo(s) 2 q ( s ) = wo(s)w(

s,t E pt+ .

F,we have

W l ( S ) ) 5 wo(s>w(t> = max(wo(s), -

+

WO(S)

while, since w ( t ) / t is nonincreasing, for

WO(S)

<

a, t

we have

w(t>

,

351

The K-method

Let

us now suppose that

for a fixed

t > 0.

IlfllLz

1. Then

we obtain the estimate

= w(t) +t $ ) = &(t) Thus,

.

f belongs t o the left-hand side of (3.3.20)and has a norm which does

not exceed two. The relations

(3.3.19) and (3.3.20)prove the

corollary.

0

C. Concluding this section,

let us establish some additional properties o f

functors of the K-method. W e shall first show t h a t the family

K: contains

infinite sums and intersections o f its elements. Theorem

3.3.15.

Let Qb :=

c Int(i,)

be a Banach family, and

are its intersetion and sum. Then

A(@) and C(@)

The real interpolation method

352 = KA(@)7

A(KOo)aEA

C(KQo)aEA

and the isomorphism constant on the couple

K,Z(@)

7

2 does not exceed 6(2).

For the definition o f sum and intersection, see Definitions 2.1.35 and 2.3.7.

In order t o prove the second equality, we note that in view of Lemma 3.3.7,

Kao(x,) = 9,. Therefore, Definition 2.3.7 and the statement of Example 2.2.5(b) lead t o

c (Ka,)(L) c =

= C ( @ )= KX(*)(L)

@(I

*

(I

(I

This equality and the fact that

KO is maximum (see Theorem 3.3.4) give

the embedding

c

24

(KO,)

&(*)

'

(I

In order t o prove the inverse embedding, we take r in K c ( ~ ) ( x 'Then ) . the

x)

K(.;r ; E C(@), and therefore can be represented i n the form sum C fn, where fn belongs t o !Ban. Here the summands should be

function of a

taken so that for a given

is valid. Then

E

> 0 the inequality

K ( .; z ; X ) 5 C

f,,

and according t o Theorem 3.2.7 (on

K-divisibility), for the chosen E there exists a sequence

(2,)

c C(I?)

such

that x = C x, and

K ( . ; s,;

2)I 6 ( 2 ) ( 1 + ~ ) , f n~ E N .

By the definition o f the norm in the sum, and taking into account Corollary 3.3.6 and Lemma 3.3.7, we obtain

The K-method

As

E

353

+ 0, we obtain the inverse embedding

Let us now give the intrinsic characterizations of functors of the

K-

method. For this purpose, we use Definition 3.3.16.

A functor F is called K - m o n o t o n e on the subcategory C c 2 if for any couples E C the following c o n d i t i o n ( K ) is satisfied. If K ( . ; y ; ?) I K ( . ; z ; where 2 E F ( z ) , y E C(?), then y E

z,?

z),

F(?') and IlYllF(y I Il~IIF(2). For C := B , the functor F is called

K-monotone.

0

Remark 3.3.17. If the condition ( K )only requires that the element y belongs to F(?), then F is called a K - m o n o t o n e (on C ) f u n c t o r in t h e side s e m e .

It is expedient to give an equivalent definition of K-monotonicity. For this purpose, we shall use the concept of the functor o f the K-orbit KO,, where the element z E C ( 2 ) ; namely, we put

It can easily be seen that

The real interpolation method

354

,

K ( . ; Z ;x') ,

(3.3.22)

KO, = KLG

so that for

z # 0, the I<-orbit is actually a functor.

w :=

A comparison of Definition 3.3.16 with the definition of the K-orbit immediately gives Proposition 3.3.18.

A functor F is K-monotone on the subcategory C iff (3.3.23)

~ ~ z ~ ~ & ~ ,F ( ??)E C ,

KO,(?)

for all z E F(x')\{O} and

x' E C .

U

Remark 3.3.19. In the case of the K-monotonicity in the wide sense, condition (3.3.23) is replaced by

F(?)

(3.3.24)

KO,(?)

for all z E

F ( 2 ) and x' E C .

cf

, ?E C ,

Indeed, it follows from the definition o f the K-monotonicity in the wide sense that there exists a set-theoretic embedding

KO,(?)

c F(?).

Since

the spaces under consideration are continuously embedded into C(?), the embedding operator is closed, and hence continuous. This proves the necessity of condition (3.3.24). The sufficiency of this condition is obvious.

It should also be noted that in view of Proposition 2.3.3, there exists a constant y(t) independent o f (3.3.25)

KO,(?)

p,for which

*)F(?)

, ?E C .

We can now give the intrinsic characterizations of functors of the method.

K-

This is t h e K-monotonicity, which follows from the statement

given below.

The K-method

355

Theorem 3.3.2Q (Brudnyr'-Krug&ak). If F is a K monotonic functor, then '(3.3.26)

F

,

S KQ

where

:=

F(i,) ,

and the isomorphism constant on the couple

r? does not exceed 6(2).

Suppose that w := ( x , Z ) where , x E C ( x ' ) is a couple with a singled out point and (pw := K ( . ; X ; 2). Let us consider the class of all 1 couples w := ( ~ ~ such 2that ) KO, ~f F . Since KO, = KL,(P~[see

-.Proof.

(3.3.22)] and all cpw belong t o the J e t Conv, this class canb e regarded as a set. We denote it by 52 and consider the family So := { K L; ~ w E R}. Since by the definition of the set R we have

KO,=KLe for all

(x,.??)

(3.3.27)

:= w E

1 ~f

F

7 ~f

C

R, and the functor

G := C ( K L C ) ~ ~ O

is defined consistently (see Definition 2.3.7). Let us verify that G = F . 1 1 Indeed, since K L z w = KO, c-, F , G ~f F. Conversely, if l l ~ l ( ~= ( 31) , view of the K-monotonicity of the functor F , the and w := ( ~ , r ? )in embedding KLe & F follows from (3.3.23). This means that w E R. Hence, in view of (3.3.27) we have 11X11G(a)

5 l141KLgd= 1 *

1

Thus, F c+ G, and the equality of these functors is proved. Let us now show that Lc E Int(z,). Indeed, it follows from (3.1.24) and (3.3.18) that

+

Thus, L z = KO,(&,), and it remains to take into account the fact that KO, is a functor. Now, in view of Theorem 3.3.15 and (3.3.27), we obtain G K Q , where 0 := C ( L c ) w Eand ~ the isomorphism constant does not exceed 6(r?) on the couple r?. According to the statement of Example 2.2.5(b), the space 0 E Int(z,), and then from Lemma 3.3.7 and the 'equality G = F and (3.3.27) we obtain

356

The real interpolation method

F ( L ) = C ( K L g 4 L ) )WER = C ( L z ) w E * := 0 .

Remark 3.3.21.

A similar statement, but without an estimate of the isomorphism constant

6(2),is also valid for the functor F which is K-monotone in the

in terms of

wide sense. In the proof, the embedding (3.3.23) must be replaced by the embedding (3.3.25), and then Proposition 2.3.3 should be used.

The following interesting property of K-monotonicity is also worth noting. Theorem 3.3.22 ( o n extension).

If F is a functor K-monotone on the subcategory C , there exists a parameter E

Int(3,)

such t h a t

F ( 2 ) 2 KQ)

, 2Ec,

where the isomorphism constant does not exceed

Proof.

Suppose that

fi

=

{(z,y)

E

R

:

S(2).

2

E C } , where s1 is the

set of couples with a singled out point in the previous proof.

Putting

x

G := C ( K , E ) ~ ~we A , prove similarly that

F ( 2 ) = G(2),

2Ec

G 2 K6 ,

d

and

where

:=

C ( L z ) w E f.i

As before, the isomorphism constant on the couple 2 does not exceed 6(2). O

D. Let us point out some generalizations of the K-method. Without altering the definition, we can extend this method t o the category 2 of the Abelian couples (see Definition 3.1.32). Here the parameters ip can be regarded as quasi-Banach spaces satisfying condition (3.3.2)

In analogy

with Proposition 3.3.1, the following proposition can be proved in this case.

‘TheIC-method

357

I’roDosition 3.3.23.

Ka

is a functor in the category

A.

[I In the proof, we can take into account the fact that embedding (3.3.6)

is also valid for quasi-Banach lattices and then make use of the following criterion of the completeness o f the normed Abelian group A (see BerghLofstrom [l],Lemma 3.10.2).

The group A is complete iff convergence in S of the series C z, follows from convergence of the numerical series C llznll; for some p E (0,1].

While proving Theorem 3.3.4 about the maximality of Ka on the couple we used the Hahn-Banach theorem. Therefore, for the category this theorem is apparently not true. For applications it is important, however, that t h e reiteration theorem similar t o Theorem 3.3.11 is valid in the cate-

,t,,

gory

A

2. Namely, the following theorem is valid.

:Theorem 3.3.24. There exists the isomorphism

where \k := Ka(&o,&l). -Proof.

The continuous embedding

is proved as in Theorem 3.3.11. Indeed, in this proof the K-divisibility was

used for two addends, and this is true for the category A as well (see Theorem

3.3.12). The inverse embedding for Theorem 3.3.11 was proved with the help of Theorem 3.3.4 mentioned above. In the case under consideration, this line of reasoning is inapplicable. However, t h e required embedding can be proved directly on the basis o f the inequality

The red interpolation method

358

where

c(2)

:=

max

.(Xi)and

where by

c(A) we

denote the constant

i=OJ

in the weak triangle inequality v(z

+ y) 5 c(v(x) + v(y))

for the normed

Abelian group ( A ,v ) (see Definition 3.1.30). indeed, applying the @-norm t o both sides of (3.3.28), we obtain the embedding inverse t o (3.3.28). In order t o prove inequality (3.3.29), we note that its left-hand side does not exceed

For

x = 20 + zl,the following inequality is also valid [cf. (3.1.43)]: K(.;z;~)~c(~){K(.;zo;~)+K(.;zl;~)}.

Therefore, taking g; :=

K ( . ; 2 ; ;2)in the previous inequality, we see t h a t

the left-hand side o f (3.3.29) does not exceed

c(2)

{~IK(. ; 20 ; 2>11o,+ t IW(* ; 2 1 ; 2 > 1 1 Io ~ .

inf x=xO+xl

However, the expression obtained coincides with the right-hand side of inequality (3.3.29). 0

In some problems the discrete version of a K-method is useful. For i t s definition, we fix a number q (3.3.30)

> 1 and put

K ( ' ) ( t ;2 ; 2) :=

C

K(q" ; Z ;-f)X[,n,,n+l)(t)

.

h € z

Then the discrete K-method consists o f the functors { K p ); QnCConv #

81,

where

IIzIIKp(f) Here t h e condition

:= l l K ( P ) ( - ; 5 ; 2)llo

.

n Conv # 0 is equivalent

t o condition (3.3.2).

The

properties of the discrete method follow from the corresponding properties of the K-method in view of the following fact.

The K-method

359

F’roposition 3.3.25. E K k ) , the isomorphism constant being

5

q.

-Proof. Since the K-functional is concave, we have

I&)(.; CI

a:;

2)I K ( * ;a:; 2)I qK(P)(*;a:; 2).

The r e d interpolation method

360 3.4. The 3-Method

A. The 3-method is a family of functors {Ja} considered in Example 2.6.5. We recall t h a t the definition of Ja is based on the concept of the

J-functional [see (3.1.2)] and has the following form. An element 5 E C ( 2 ) is said t o permit a canonical representation if there exists a strongly measurable vector valued function u

:=

R+

--$

b

A(x') for which the Bochner-Lebesgue integral J u ( t ) d t / t exists for any a a > 0 and b < 00 and b

J

z = lim

dt u(t) -

t

a

a-0

(convergence in

~ ( 2.) )

b-oo

Let us agree t o write this limit relation in t h e form (3.4.1)

z =

/

u(t)

dt

.

m+ Naturally, such a representation is not unique. Let us consider a Banach lattice @ over ( R + , d t / t ) , which satisfies the following conditions: (3.4.2)

dt

n L~w(R+,-) t

+ (01 ,

o

-

c(&) .

We define the space J a ( 2 ) with the help of the condition (3.4.3)

lI~lIJ*(B):=

inf I(J(t;4 t ) ;d)lla <

,

where the lower bound is taken over all canonic representations (3.4.1). Naturally, in this definition we can take lattices that do not satisfy conditions (3.4.2). These conditions, however, are due t o the following result. ProDosition 3.4.1.

Ja is a functor iff conditions (3.4.2) are satisfied.

Proof. Since every function

t

-+

J ( t ; u ( t ) ;x') is locally integrable, the

nonfulfillment of condition (3.4.2) would lead t o the equality J a ( d ) = (0).

The 3-method

361

Let us prove t h e necessity o f the second condition from (3.4.2). For this we assume that it is not satisfied. Then there exists a function

f’ E @nLlW, but IlfIlc(~,,

f > 0 such that

= 00. Since Ilfllc(t,,coincides with K(1; f ;

el),

according t o (3.1.24) we have (3.4.4)

C(z1) = Ly

Therefore,

Ilfllv

:=

there exists a subset

5 =

J

6

,

where w ( t ) := max(1,t)

f(t) dt = 00. J -

1R+

.

Consequently, for any

4t) t such that

N >0

c a,

dt

5 .

a(t)t .

R+

. a(t)

Since we also have for u ( t ) := z

J ( t ; u ( t >; 2) I m W , t ) a ( t >II”ll&(d)

7

we obtain in view of (3.4.3) (:3.4.5)

11211.J,(d)5 11 max(l,t>a(t>ll*11211A(a)

Here max(l,t)a(t)

.

5 & If(t)l < & If(t)l, so that from (3.4.5)

IIZII.J*(d)<

1 IIZllA(f)

we obtain

llfll* .

!Since N is arbitrary, it follows hence that the quantity (3.4.3) vanishes on

.A(x’) and is not a norm for A(x’)

#

(0). Let us now prove the sufficiency. From the norm axioms, only condition 1(~)(~,(?) = 0 j 2 = 0 has t o be verified. For this purpose, we shall use the inequality l(3.4.6)

K ( s ; a ;x’) 5 min(l,s/t)J(t; a ; r?)

,

a E A(r?)

,

which directly follows from the definitions of the K - and J-functionals [see (3.1.1) and (3.1.2)]. Applying this inequality t o the element u ( t ) from the canonical representation (3.4.1) and taking into account (3.4.4), we have

The red interpolation method

362

In view of (3.4.2), we hence obtain

where o is the norm of the embedding operator embedding

Jcp(2)

(3.4.7)

-

C(2)

-+

C(&).

Thus, the

,

= 0 only for z = 0, are established. Ja(2) is a normed space. Let us show that it is a Banach space. Let ( I , ) , ~ Nc J a ( X ) be a sequence such t h a t

as well as the statement that IIIIIJ*(X) Consequently,

In view o f (3.4.7), the series

C

I, converges in

E(d). If

I is i t s sum in

C ( x ' ) , it is sufficient t o prove that z E J a ( 2 ) . Let (3.4.8)

z, =

1

dt u n ( t )t

R t

be a canonical representation of (3.4.9) Then

IIJ(t ; .n(t>;

I,, such that

2113 . < 2 II%IIJ*(a,'

C I I J ( t ; u,(t); d?lla < co and

c J(t

; un(t) ;

by Lemma 3.3.2 the series

2)

converges almost everywhere and its sum belongs to function

J ( t ; C un(t) ; x'), which

belongs t o

a.

a.

Therefore, the

is obviously majorized by this series, also

If we also prove that

C un(t) E A ( 2 )

that z allows the canonical representation

for almost all

t and

The J-method

363

the required fact will be established. By the definition of the

J- functional,

we have

so that

C un(t)E A ( 2 ) for almost all t.

Further, from the local integrability

of the right-hand side o f this inequality it follows that

a

a

Let us show that here we can proceed t o the limit as a

4

0, b + 03. Then

,the required representation (3.4.10)is obtained from (3.4.8).T h e possibility

(of a passage t o the limit follows from the inequality

b

which ensures the uniform (in a, b ) convergence o f the series

CJ

un(t)

a

$

in C(x'). In view o f (3.4.6),the left-hand side o f (3.4.11)does not exceed

By virtue of (3.4.4) (3.4.2)and (3.4.9),the integral on the right-hand side does not exceed

I I J ( ~un(t>; ; -f)ll,(zl)

I

Q

I I J (i ~un(t>; -f118

5 20

IlZnll.r,(2)

5

.

Thus, (3.4.11)is proved and thus the fact that J @ ( - f )is a Banach space is established. In order t o prove that it is intermediate for the couple remains t o establish, taking into account (3.4.7),the embedding

(3.4.12) A ( d ) L) J a ( d ) .

3,it

The red interpolation method

364

For this purpose, we take a nonzero function

f

in CP and suppose that the

interval A := [a,b]is such t h a t the measure of the set {t E A ; f ( t )

# 0)

:= y l f l x A , where y is chosen from the dt a(t) - = 1. By analogy with the proof of (3.4.5), we obtain t

differs from zero. We put a condition j"

m+

llxllJ,(a)5 11 max(l,t)a(t)ll@ Here max(1,t)

'

IIZllA(a)

5 max(1,b) for t E A, so that

exceed y max(1, b ) llflla

'

t h e right-hand side does not

11zllAca).Thus, the embedding (3.4.12)

is establis-

hed. Finally, let us verify the interpolation property of Ja. It follows immediately from the obvious inequality

J(.; T x ; ?) 5 IITlla,yJ(-; Z ;2).

Remark 3.4.2.

A Banach lattice CP satisfying condition (3.4.2) will be henceforth called a parameter of the 3-method. It will be shown below that it is sufficient t o limit ourselves t o Q in Int(i1). An important property of the 3-method is described in Theorem 3.4.3 (Brudnyi-Krugbak).

Ja = Orba(i1 ; .).

Proof.We shall need Lemma 3.4.4. Qr

L &(El).

Proof. Ja(zl) is a

it is sufficient to establish the corresponding embedding for Q+. Thus, suppose that f E and

Banach lattice (see Example 2.6.12)

f 2 0. We take an arbitrary q > 1 and

(3.4.13)

R,,

:= {t

50

put

E [ q " ; q " + l ) ; f ( t ) E [q",q"">}

.

'The J-method

365

Z.The family (On,) obviously forms a partition of the set rrupp f , and hence f = C f xn,,. We define the function u : R++ A(&) Here, n,m E

by the formula

f=

(3.4.14)

c f xn,,

J

=

dt u ( t ) 7,

R t

and for t E On,

we have, in view of (3.4.13) and the choice of ,,c

nnm

From this inequality and from (3.4.14) it follows that

IlfllJ,(z,,

I IIJ(t;4 4 ; L d l l o I q2 llfll@ .

Making q tend t o unity, we obtain the required embedding. Lemma 3.4.5. Let

5

E C ( 2 ) have a canonical representation (3.4.1). Then there exists an

'operator T E (3.4.15)

Proof.

.&(el; x') such t h a t

T ( J ( t ; u ( t ); 2))= z

For g E C(Z1). we put

We assume that the fraction in the integrand is equal t o zero for those

t E R+which annihilate the denominator (and hence the numerator). In view of (3.4.1), the identity (3.4.15) is satisfied, and it remains t o show that

IITllt,,n 5 1. The required estimate is obtained as follows:

The red interpolation method

366

Let us pass t o the proof of the theorem. For this we first establish the embedding (3.4.16)

Ja

A

Suppose t h a t I E representation of

x

OrbQ(zl; .) .

Ja(x'). Then for a given

E

> 0 there

exists a canonical

[see (3.4.1)] such t h a t

IIJ(t; u(t>; m

a

I (1 +

llXIlJ*(2)

.

Let us take the operator T mentioned i n Lemma 3.4.5.

Then from the

definition of the orbit [see (2.3.17)] and the above inequality we obtain l1410rbo(tl; R )

5 IITllz,,n IIJ(t i 4 t ) ;@lie I(1 + &) ll"llJ*(d) .

Since E is arbitrary, this leads t o (3.4.16).

It remains t o establish the embedding inverse t o (3.4.16).

In view of

the minimality property of the orbit (see Theorem 2.4.15), it is sufficient t o :=

El. Thus, we have t o establish

.la(&).

Moreover, the space J a ( L 1 ) E

verify this embedding for the couple the validity of the embedding (3.4.17)

Orba(Z1,zl)

A

According t o Lemma 3.4.4,

Int(&).

Ja(il) .

But Orba(el,Z1) is the minimal (with respect t o embedding)

among all spaces from Int(&) which contain

(see statement (c) of The-

orem 2.3.15). This proves the validity of (3.4.17). Corollarv 3.4.6 If

9 is a parameter of the 3-method and

The 3-method (3.4.18)

&

367

:=

Ja(&) ,

the following identity holds: l(3.4.19) -.Proof.

Ja = J b

.

Ja + Orbo(Z1 ; .), the space @J coincides with 1 which is minimal among all @ E Int(L1) for which 0 -t \k

In view o f the equality

.the space

(see Proposition 2.2.6 and Theorem 2.3.15(c)). But according t o statement 3. -

(d) of this theorem, OrbaAn(&; -) = Orbo(L1; .). It remains t o make use of t h e coincidence o f Jb with OrbaAn(Z1; .). 0

Corollary 3.4.7.

If @ E Int(el), then (3.4.20)

Ja(Z1) =

@

Proof. In this case, 9-

. coincides with 0. Therefore, according t o Theorem

2.3.15(d), we have +

Ja(L1) = Orba(Z1,el) = ni,$

=0

.

0

Remark 3.4.8. Let us show that the map @ -+

Ja is a bijection o f the set Int(il) into a

set of functors of the J-method. Indeed, it follows from (3.4.19) t h a t each

functor Ja is generated by the space @J E Int(e1). If 0 and @' belong t o

Int(zl) are different here, in view of (3.4.20) Ja

#J~I.

B. Let us analyze some more important properties of the functors of t h e 3-method. The first of the results t o be considered below plays a significant role in the proof o f the corresponding reiteration theorem. Theorem 3.4.9. Let

( @ P a ) o E ~ be

a family of Banach spaces from Int(&). Then the following

statements hold:

The red interpolation method

368

(3.4.22)

.

A ( J @ , L ~ A J(AO,),~A

Proof. According

to statements (a) and (b) of Example 2.2.5, the spaces ( C @ , ) , e ~and A ( G a ) , e ~belong t o Int(L1). Therefore, it follows from Corollary 3.4.7 that relations (3.4.21) and (3.4.22) are satisfied on the couple Then in view of the minimal property of J (see Theorems 3.4.3 and 2.3.15), we obtain the embeddings -4

zl.

(3.4.23)

JC(@,)

A

1

Since here

~t

C(J@,)7

A

JA(@,)

A(J@,) . 1

have Ja, L) J q @ a ) a Efor A any a E JC(Q,),,, (see Definition 2.3.15). Combined

C ( @ a ) a E A ~we , also 1

A. Consequently, C(J@,) L)

with the first embedding in (3.4.23) this proves the equality (3.4.21). The proof of isomorphism (3.4.22) is based on the following fact. +

Let us suppose that canonical representations of the element z E C ( X ) are given:

(3.4.24)

J

z =

u,(t)

dt t ,

aEA

.

m+ Let us show that there exists a canonical representation (3.4.25)

z =

1

u(t)

dt

,

m+ which is not worse than the previous ones in the sense that for a certain absolute convergence y we have (3.4.26)

K ( . ; j ; 31) 5 y inf K ( . ; j,;

31).

0

Here we put (3.4.27)

j ( t ) := J ( t , u ( t ) ;2);

j,

:=

J ( t , u,(t) ; x').

In order to prove this statement, it should be noted first that according to

Lemma 3.4.5, z = Ta(j,)for some T, E ,Cl(zl,-?). Therefore,

The J-method

369

which leads t o t h e inequality (3.4.28)

K(.; z;

2)I

inf K ( . ; j,;

:= cp

z1)

.

d

In view of Proposition 3.1.2(b), the function cp belongs t o Conv. According

to Corollary 3.1.14, the regularity of the couple (3.4.29)

L',

implies t h a t

Q(t> lim cp(t) = t+w lim t-0 t --0 .

L e t a constant q

> 1 be given and l e t (ti)-msis,,be the sequence of

points

constructed for cp by the process indicated in Proposition 3.2.5. Then according t o statements (c) and (d) of this proposition and (3.4.29) the numbers 1%

and m are either odd of equal t o +m. Therefore, the function @ con-

structed from cp in Proposition 3.2.6 has in this case the form (3.4.30)

C

@(t)=

cp(tz;+i)

min(l,t/tzi+i)

,

-2k
!where Ic,l E

Z+U {+m}.

Further, from the four possible cases (depending on whether

k

and I are

.finite or infinite), we shall limit ourselves to a detailed analysis of only two. Then we shall briefly discuss the proof in t h e two remaining cases.

k = 1

Thus, let us first suppose that

= +m (this is equivalent t o

limt.++oocp(t) = limt-o c p ( t )= +m. As in Lemma 3.2.9, we put

t

(3.4.31)

vi := zo(tzi+Z) - Z O ( t Z i )

Here z o ( t )

7

2

E

+ zl(t) = z is chosen for a given

Then, according t o Lemma 3.2.10, the series

Z. E

>0

[see inequality (3.2.30)].

xiEzwi absolutely converges

in C ( x ' ) , and (3.4.32)

z =

~i

Estimating J ( t , vi ; x') in the same way as was done for t h e K-functional of the element vi in Lemma 3.2.9, we obtain t h e inequality (3.4.33)

~ (; vit ; 2)I (1

+

+ q)V(tZi+l) max(1, t/tzi+l) .

The red interpolation method

370 We now put

Since tZi+z/tZi+l

2q

(see Proposition 3.2.5(a)), we have

Here use is made o f the absolute convergence of the series (3.4.32). Thus, the function u gives the canonical representation (3.4.25) for the element 2.

Let us show that for this function inequality (3.4.26) is valid. In view of

(3.4.27) and (3.4.34) we have

1 j(t>I T;;;E

C ~(qt2i+1;vi ; 2)

~[tZ,+i,qtZi+i)

.

Here the K-functional of the characteristic function in the couple

ilon the

right-hand side does not exceed lnq min(1, t/tzi+l) [see (3.1.24)]. Taking into account (3.4.33) and (3.4.20), we obtain

+

~ (; j t; 31) 5 (1 € ) ( I +q)q

C

p(tzi+l) min(1, t/tzi+l) =

t

= (1

+ E X 1 + q)qd(t>.

Using inequality (3.2.21) t o estimate @, we arrive at the required inequality

Further, we consider the case k = +00, 1 < +00. In this situation, limt-,+o cp(t)/t = 00, but limt++m p(t) < 00. In the analysis of the previous case, condition c a r d A < 00 was not used, while in the remaining cases it plays a significant role. Here, it is sufficient t o consider A := (0, l}since from the validity of (3.4.22) for c a r d A = 2 follows its validity for any finite A. The required vector function u in the situation under consideration is obtained as the sum of three terms wi, 0

5iI 2, which have noninteresting

supports. Namely, in the notation o f (3.4.31), we put

The 3-method

371

c

1

wo := h q

ui X[tz,+l,Ptz,+l)

*

-oo
In analogy with the estimate obtained for the function (3.4.34), for

jo(t) =

J ( t ; wo(t); 2)we have the estimate K ( t ; j o ; E l ) L (1

(3.4.35)

Since limt,+,

cp(t) <

+

E)Q(I

+q) C

cp(tZi+l)min(1, t/tzi+l).

-k
in view of the inequality preceding (3.2.12) we

00,

have

mqcpo, cpl)(t) := cp(t>I qcp(t2r-1)

t 2 t21-1. Consequently, in view of the concavity of number a0 E (0,l) and a constant y(q) for which

for

cpi,

there exist a

v a o ( t )I~ ( q ) ~ ( t z r - i ) ( t 2 h).

(3.4.36)

We then put

where the vector functions u, are taken from the expansion (3.4.34). Further, we put

and finally u := wo

+ + Wl

w2

Then from the definition of the function wi we have, taking into account (3.4.31)

-

zo(t2,-2) =

J

u,,(t)

dt t .

1Re From this relation and from (3.4.25) we obtain the required representation

The red interpolation method

372

2

=

J

dt u(t>

t.

R+

(3.4.36), we have for t > tzi-l the following estimate for j z ( t ) := J ( t ; wz(t); X ) : Further, in view of

4

Therefore, since the K-functional belongs t o the cone Conv, for all

t > 0 we

have

(3.4.37)

K ( t ; jz ;

G)I r(a>cp(tzl-l>min(1, tltzl-1)

.

In order t o estimate the similar K-functional for j l ( t ) :=

J ( t ; wl(t) ; 2).

we shall use the representation

and the inequality

from which we obtain, in view of (3.1.24) and t h e definition o f (pa,

In view of t h e previous inequality, we have from the definition o f wl(t)

The 3-method

373

Taking into account (3.4.26), we hence obtain the final estimate:

K ( t ;j 1 ;

El) I 7(q)(P(tz1-1) min(1,tltzI-l) .

Combined with t h e estimates (3.4.35) and (3.4.37) and definition (3.4.30),

this gives the required inequality for the vector function:

It remains t o consider the two remaining cases. In both cases, k

< +m,

and hence limt-r+oo cp(t)/t < m. Therefore, inequality (3.4.36) is replaced by the inequality

E ( 0 , l ) . The function wo is defined as earlier, while the analogs of the functions w; (we denote them by w-;),i = 0,1, are given by where

a1

The r e d interpolation method

374 Further, for 1 = co we put u := w-2 +w-1 + w o

and for

,

I < co,

c w;. 2

u :=

;=-2

All t h e remaining reasonin is the same a

I

th previous cases.

Thus, statement (3.4.26) is established. To complete the proof of the theorem, we shall use the following important fact. Lemma 3.4.10 (Sedaev-Semenow).

If f,g E C(zl)

are such that

I((., f ; E l > Ih'(.,g ; 31) , then for any

E

> 0 there exists an operator T E L l + c ( ~ l such ) that

Tg= f .

The proof of a more general statement (see Theorem 4.4.12) will be given later in this book.

We now have everything needed in order t o prove the embedding inverse t o the second embedding in (3.4.23). Thus, l e t z belong t o the open unit ball of the space

A ( J @ , ( d ) )Then . for any

(Y

E A there exists a canonical

representation

such that the following inequality holds:

(3.4.38) IljaIlo, := IIJ(t,U a ( t > ; -f)l1am < 1 . Let us show that then the norm of z in the space J ~ o ~ , does ( @ not exceed ~ ( q ) which , corresponds t o the required statement. For this purpose, using

statement (3.4.26), we shall find a representation

The 3-method

i(3.4.39)

z =

375

1

u(t)

dt t ,

pL+

such that for the function j ( t ) := J ( t , u ( t ) ;x') the inequality

(3.4.40) K ( -; j ; El) 5 r(q)K ( . ; j , ; El) ,

(Y

E

A

,

is valid. We will use the statement of the previous lemma. Then it follows

(3.4.40)that for any a E A there exists an operator T, E L?(*)(Z1) such that j = Ta(ja). Since a, E Int(&), it follows from this and (3.4.38) from

that

113'11& I II~allt,lljallo, < =i.(q>(1+ E l 2 . Taking in this inequality the least upper bound for account

cy

E A and taking into

(3.4.39),we then obtain 11+A(*,(n,

<

SUP

.

Iljlleo <

P

Thus we obtain t h e required estimate of the norm. 0

Remark

3.4.11.

It should be noted that the condition cardA < oo and statement (3.4.22) k = 1 = 00. In other words, this formula is also valid for does not contain a infinite families only if the intersection @ :=

are inessential for

function 'p in Conv for which

lim cp(t>< oo or t-+m

:yo

t

< oo .

It can easily be verified (see the following section) that this condition is satisfied iff

@\(Ly u L:) # 0.

In the general case, however, formula families. This will be shown in Sec.

(3.4.22)is not valid for infinite

3.5.

Finally, it should be noted that statement

(3.4.26)combined with the

result obtained by Semenov and Sedaev can be used for the proof (in the case

A'

:=

El) o f the formula

any functions fo,

f1

o f intersection from 2.7.4(e).

E C(&), we have

Namely, for

The red interpolation method

376

where

f

can be defined by the condition

Sf x

rnin

Sf;.

i=O,1

Let us now find out which o f the functors of the J-method putable (see Definition 2.5.7).

are com-

A similar problem for the functors of the

K-method are considered in the following section. An exhaustive answer t o the above question is given in Theorem 3.4.12 (Janson).

A functor Ja, where 0 E Int(,fl), is computable iff the space 0 is regular in the couple

zl.

Proof. In view of Theorem 3.4.3, (3.4.41)

; .)

Ja = Orbe(&

we have

.

Therefore, in conformity t o statement (b) o f Theorem 2.5.12, the computability of

Ja is equivalent t o the fulfillment o f the following two conditions.

0 is regular in the couple

zl,and this couple satisfies the approximation

condition (see Definition 2.4.22). Condition (2.5.16) is satisfied, which in the situation under consideration states that for 2

E A ( 2 ) we have

where the lower bound is taken over all finite families of functions

(fm)me~c (3.4.43)

+

A(&)

z =

and operators ( T m ) mc EL ~ ( & , z ) such that

C

Tmfm.

mEM

In view of (a), the regularity condition for

0 is necessary for computability

of Ja. Let us now prove the sufficiency of this condition.

For this purpose,

we shall establish the validity o f the above conditions (a) and (b).

The

The J-method

377

first of them follows from the statement o f the theorem and from what was proved in Example 2.4.23. In order t o prove (b), we shall first note that the norm in orbit is defined as the greatest lower bound of (3.4.42), taken over

infinite families (T,) and

(fm),

where

(fm) c C(il).Therefore, in view of

(3.4.41). the left-hand side o f (3.4.42) does not exceed the right-hand side, and it remains for us t o establish the opposite inequality. We shall first prove a weaker inequality

(3.4.44)

inf{CllTmIle,,a Ilfmll}

where 7 does not depend on z E

< 711~11Jo(2) 7

A ( 2 ) . But first we show that the required

condition follows from it. For this, using (3.4.41). we find for a given

E

>0

the representation

C

z=

Tnfn

(convergence in

E(d))

nE N

such that

IITnIIel,f IIfnII4 L (1 + €1IIzII.ro(f) .

C n E N

Since 9 is regular, without any loss o f generality we can assume that all fn

E A(Z1). Further, let us choose N := N ,

50

that

2 := Tnfn. Then 2 = - Cn
and assume that fn

the previous inequality, here llZllJo(?)

< E.

the existence of finite families ( s , ) , ~ M

2=

Then inequality (3.4.44) leads t o

and ( g , ) , € M

c A(Z1). such that

C Smgm

and

c

IISmllZ1,d Ilsmll*

For the element z =

CnENTnfn

< 7 II2IlJ*(R) < 7&.

Cn
have the representation z =

+ C smgmo f the form required by (3.4.43)

infimum in (3.4.42) does not exceed

and such that the

The real interpolation method

378

C

+C

IITnIIE1,2 IIfnIIa

ndV

IIsmJItt,2 IIgmIIO

<

< (1 + E ) llzllJ4(2)+ YE . Since E

> 0 is arbitrary, it follows hence that the right-hand side of inequality

(3.4.42) does not exceed the left-hand side. Thus, it remains t o prove (3.4.44). For this we need Lemma 3.4.13. Let 9i, i = O , l , be Banach lattices given on t h e same space with a measure and having the following property

(F)(of Fatou):

( F )The unit ball of 9;is closed relative t o convergence in measure, a = 0 , l .

6

Then the couple

Proof. Suppose t h a t

:=

(ao,aPl) is relatively complete.

the sequence ( f n ) n E ~ converges t o f in

belongs t o the unit ball of

@i.

C(@)

and

It is sufficient t o prove that f also belongs t o

this unit ball. In view o f the statement of Example 2.6.12, the space C(@) is also a Banach lattice, and hence it is continuously embedded into the space of measurable functions

M

[see (3.3.6)]. Consequently, the convergence of

(fn) t o f in C(6)leads t o the convergence of this sequence t o f in measure. Then according t o Fatou’s property llfllai

I 1.

0

Let us now suppose that

2

E A ( 2 ) . Without loss of generality, we can

assume that (3.4.45)

IlzllJ,(a,

= 1.

Then according t o Lemma 3.4.5 and the definition of

JG

[see (3.4.3)], there

exist a function f and an operator T such that (3.4.46)

z = Tf

7

llTlIt,,z I 1

7

Let a , b E nt, satisfy the condition 0 for m (3.4.47)

z = 2+

+ + 20

2-

,

IIfIIa < 2 < a < b < 00. We represent z

in the

The 3-method

379

where we put 20

:= T ( f X ( a , b ) ) 7

z+ := T ( f X ( O , a ] )

7

z-

:= T ( f X [ b , o o ) ) .

If we can show that for a certain choice of a > 0 and b < have zrt = &(grt), where

llR*Ilz,,y < 3, g* E A(J%)

considering that X ( a , b ) f belongs t o A (:,),

00

and 11grtllo

we will

< 6,

so

we obtain the required statement.

Indeed, in this case z = R+(g+)+T(t ~ ( ~ , b+)R) - ( g - ) , so that the infimum in

(3.4.44)does not exceed, i n view of (3.4.46)and (3.4.45), IIRtllz,,R 119+11o + llTllz,,a

llfll@ + IIR-Ilt,,a lI9-llo <

< 3 . 6 + 1 2 + 3 . 6 = 3811~115,(2,. *

Thus, it remains t o obtain the required representation for the elements z*. We shall limit ourselves t o the analysis of z+ since for z- the situation

is similar. Let first fbe such that

3.1.11,f E (L:)",and in view of the previous lemma and Theorem 1.3.2,this space coincides with L i . Then f X ( O , ~ I E L:. Moreover, this function obviously belongs t o Ly. Consequently, f ~ ( 0 . ~E1 A(z,), and putting R+ := T and g+ := f ~ ( o , ~ i]n, view of (3.4.48)we Then according t o Corollary

obtain the required representation of z+. Let us now suppose that

K ( t ; f ; x i ) = +oo . (3.4.49) lim t-0 t Then in view of (3.4.46) (3.4.47)and the formula for the K-functional on [see (3.1.21)],we have

&

By the same reason,

380

The real interpolation method

Using (3.4.47), we choose for a given

E

>0

a number a := a,

>0

such

that the right-hand_side o f the last inequality becomes less than

+E )

(1

K ( a ; f ; "I. a

Then the previous inequalities give

We now put

Since a

> 0, g+ E A(&).

K-functional on

In view of formula (3.1.24), we also have for the

El

In view of (3.1.5), the right-hand side does not exceed e K ( t ; f ; 21). Thus,

Using Lemma 3.4.10, we hence establish the existence of an operator

RE

C ( 2 , ) such that g+ = Rf.Consequently, taking into account (3.4.46) and the interpolation property of Q, (see the statement of the theorem), we obtain /19+Ilo

I IlRllz,

llfllo < 6 .

Finally, l e t us consider the operator

R+

: C(21) +

C(x'). defined by the

formula

Then R+(g+)= I+. If in this case h E L f , then taking into account the definition of g+ and inequality (3.4.50). we have

The 3-method

381

Here i = 0 , l . Thus,

IIR+llz,,a < 3, and the required representation is obtained.

C. In the proof o f the reiteration theorem and in some other problems, it is useful t o consider a discrete version of the J-method. Its definition is based on the use o f the canonical representation of the form (3.4.51)

z=

C

Z ,

,

(2,)

c A(-?),

nE Z

where the series converges in Let (3.4.52)

C(2).

be a Banach lattice o f bilateral sequences satisfying the conditions (0)

Here II :=

# @ c C(<) .

(Zy,Ii)

and

Then in analogy with (3.4.3), we put (3.4.53)

ll~llJi(2):=

*

inf (I(J(2";xn ; X ) n E Z J I O

7

where the greatest lower bound is taken over all canonical representations of the element z in the form (3.4.51). Naturally, in definition (3.4.53) we can take @ which does not satisfy conditions (3.4.52). However, repeating the same line of reasoning as in the proof of Proposition 3.4.1, we arrive a t Proposition 3.4.14.

J d is a functor iff @ satisfies condition (3.4.52). The family of functors {Ji} forms the z d - m e t h o d (discrete 3 - m e t h o d ) . 0

If we repeat exactly the same arguments as used in the proof of Theorems 3.4.3, 3.4.9, 3.4.12 and all lemmas, propositions and corollaries accompanying them, the analogs of the relevant results will be valid for the functors

Ji as well. We leave it t o the reader t o formulate the corresponding results. Here, we shall only establish the relationship between the functors of the 3-method and its discretization. For this purpose, we put

The red interpolation method

382

ProDosition 3.4.15.

The following relation holds:

Ja 2 ( J a y

(3.4.54)

.

Proof. Let us show that the identity operator l a bijects the space J a ( 2 ) into the spaces ( J a ) d ( g ) .Since these spaces are continuously embedded in C ( x ' ) , according t o the theorem of the closed graph, they have equivalent norms. Then Proposition 2.3.3 allows us t o conclude that relation (3.4.54) is satisfied.

In order t o establish t h e bijection mentioned above, we first take an element

2

from J*(r?)and show t h a t it is contained in

t o the definition o f 2

=

J

JQ,there exists a canonical U(t)

( J a ) d ( g ) .According

representation

at t .

Rt 2nt1

We put z, = J

dt u ( t ) t,n E Z.Then

(5,)

c A(@,

I

= C x,, and

2" 2-t 1

J

~ ( 2 ;"5, ; 2)I

dt J ( t ; u(t>; 2)t .

2"

(r

Let us consider the operator

Tf

:=

T E L(&, <) defined by the formula

f(t)

$)

,

nc.ZG

It follows from the previous inequality that

T ( J ( ~ )u;( t > 3) ; L ( ~ ( 2 "; 5,; Since in this case the function J(t ; u(t); image formed by the mapping noted that we assume that

T

x)),~-#

.

2)belongs t o GJ = J*(Ll), i t s -+ ad. It should be

belongs t o J a ( l l ) :=

E Int(Zl), which in view of (3.4.19) does not +

restrict the generality. But then the sequence ( J ( 2 , ; 2, ; X ) ) , E z , which

The 3-method

383

is majorized in this way, also belongs t o the lattice @ d . Consequently, the

x equal t o C x, belongs t o (J@)'(Z). Conversely, suppose that x E We shall verify that this element belongs t o J a ( 2 ) . According t o the definition of for the element x

element

(J@)d(z).

.there exists a canonical representation (3.4.51), where

According t o Theorem 3.4.3, there consequently exists an operator

R

E

.C(xl, <) and a function g E @ such t h a t

Let us define the operator Sby the formula

'The prime on the symbol of sum indicates that the terms with x, = 0 are omitted. As in Lemma 3.4.5, it is verified that I l S l l ~< , ~00. According t o the definition of S, we also have

SRg = x and S R E 'to Orb@(Z1,2)= J@(Z). Consequently,

L(z,,Z),which

means that

x

belongs

0

Corollary 3.4.16. l(J@o,

J@,>(Z) 2 ((J@,)d,(J@Jd)(J?).

-.Proof.

According t o Theorem 3.4.9 and its discrete analog, J@o

+

J@l

= J@O+bl

9

(J@,)'

+ (Jdd = (J@o+Ol)d .

It follows from these relation and the previous proposition that the operator

112 maps the spaces C(Ja,, Ja,)(-f) isomorphically onto a similar space for the functors ( J Q ~ i) = ~ ,0,1. 1 7

384

The real interpolation method

We now have everything t o formulate the reiteration theorem of the method.

3-

Theorem 3.4.17 (Brudnyi-Krugljak). The following relation holds: (3.4.55)

Ja(Ja,, J a , )

= J& ,

where we have put (3.4.56)

6

:= J a ( & ~ , & i ).

We recall that

&

Proof. According

:= J a ( e 1 ) .

t o the definition of

6,the

left- and right-hand sides of

-#

(3.4.55) coincide on the couple

L1.Since the functor

J6 is minimal on this

couple (see Theorem 3.4.3 and 2.3.15) this leads t o the embedding

J&

A

Ja(Ja,,Ja,> .

In view of Proposition 3.4.15 and Corollary 3.4.16, it is sufficient to prove Thus, we put the inverse embedding for the functors of the form

p

:=

((Jadd, (.la1Id)(ri)

and assume that z E C(?) is such that (3.4.57)

lIzll(J+)d(p)

=

.

Then there exists a canonical representation (3.4.58)

z=

2,

(convergence in C( p ) )

such that (3.4.59)

l ( ( ~ ( 2 ;"z n ; p))nEzIJad <2 +

Further, if we make use of the fact that according to the definition of the J-functional and the couple f ,we have from Theorem 3.4.9

The 3-method

385

Therefore, the previous inequality leads t o the existence of a sequence (tdnk)&z

(3.4.60)

C

A(-f), such that

rn =

C unk

(convergence in

~(2))

k

and t o the fulfillment o f the inequality

llfn ll~(ot,2nof)< 7 ~ ( 2 ;'zn n ; P) +

for the sequence

fn

independent of z and

:=

(,7(2k;t d n k ; X ) ) k E z (here n E

z and 7

is

71).

Let us consider the sequence

f:=

C

fn.

n&7

Since in view of the previous inequality

according t o (3.4.59) we have

Consequently, f E C ( l d ) and

Considering the definition o f

fn,

we have

Therefore, in view of (3.4.61), the left-hand side of this inequality belongs to

( J a ) d ( 6 d and ) its norm in this space does not exceed 2 7 .

t o the definition of

@ [see (3.4.56)],

we have

But according

386

The red interpolation method

Thus we have established that the element 2 :=

x k

(En Unk)

belongs

t o (J&)"(X'), and its norm does not exceed 27. If we manage to prove that 2 = 2 , in view of (3.4.57) we shall obtain the inequality

i 27 II"II(Je)d(P)

lI"11(J4)d(d)

7

which proves the required embedding. Thus, considering (3.4.60) and (3.4.58), it remains for us to show that

c (c

(3.4.62)

Unk)

k

Since f E

n

=

(c

Unk)

n

.

k

C ( 6 d )c C(6). we have

c

f(k)+

k
c 2k < f(k)

+oo

.

k>O

Consequently, in view of the equality f(k) =

En

J(2k;

;

x)(see the definitions of fn

and f ) , we have

Considering the definition of the J-functional, we hence obtain

Thus, the series Ck En U,k converges absolutely in C ( 2 ) . Thus. the change in the order of summation in (3.4.62) is justified 0

Corollary 3.4.18. The family of functors { J a } forms an interpolation method.

Proof.The statement follos from Theorem 3.4.17 and Definition 2.6.1. 0

Equivalence theorems

387

3.5. Eauivalence Theorems

A. We shall establish the relation "etween functors o f the K- and J methods. A similar result was obtained early for a narrow family o f parameters which will be referred t o as quasi-power parameters. This family is described by Definition 3.5.1.

A parameter 0 o f the K-method is called a quasi-power parameter if the opera tor

(3.5.1)

(sf)(t> :=

J

du m i n ( l , t / u > j ( u )U

R+ is bounded i n the space 0. U

Example 3.5.2. (a) In view of Hardy's inequality, the parameter

:= L:, 0

< 19 < 1, is a

quasi-power parameter (naturally, such parameters will be called power

parameters). It should be recalled that by definition,

L:

:=

L,(t-'

; dt/t).

(b) Let us consider a more general situation L; := L , ( l / w ( t ) ; d t / t ) , where w

:

lR+ -+nt, belongs

to the cone Conv. We call the function w a

quasi-power function if (3.5.2)

w M Sw

It will be shown later that (see Sec. 4.5) the operator S is bounded in

for quasi-power functions w. Consequently, L; is a quasi-power

parameter i n this case. Here L, can be replaced by an arbitrary space

0 E Int(L1, L W ) . Theorem

3.5.3(Lions-Peetre).

If

The red interpolation method

388 (3.5.3)

h-'$= J* .

0

Below we shall establish a general theorem on the relation between the I(-and J-functors, which will lead to the equivalence theorem (3.5.3). Here we shall only show t h a t the condition imposed on the parameter @ is also a necessary condition. Indeed, let condition (3.5.3) be satisfied. Then from

(3.5.3) and Lemma 3.4.4 we can write t h e following relation for the couple

t,: @

L

,2 K & ) .

J&)

Since in view of Proposition 3.1.17

K ( - ;f ; XI) = S(lfl),we obtain from

the above embedding

Thus, S E L(@),and @ i s a quasi-power parameter. For a parameter of t h e general form, the equivalence theorem splits into two: t h e first theorem contains a formula reducing a J-functor t o a I
@\(Lo, u L A ) #

0.

The parameter @ of the 3-method is called nondegenerute if (3.5.5)

@\(L? u L:) # 0 .

In the theorem formulated below, we assume that -+

(3.5.6)

9 := Ja(L,).

Eq uivdence theorems Theorem (a) If

389

3.5.5 (Brudnyi- Krugljak).

@ is a nondegenerate parameter o f t h e 3-method, then

J e Z K* ,

(3.5.7)

where t h e isomorphism constant does not exceed 18. (b) If @

c-,

Lf for

a certain

J4 Z

i

E {O,l}, but @

# A(Z1), then

Kq n Pr;

It should be recalled that P r ; ( z ) :=

Xi.

(c) Finally, if ip = A(Zl), then

Proof. We

begin with the analysis of case (c). Since A

-

J a , we need

only establish the inverse embedding. But in view of Corollary 3.4.7 applied t o ip := A(Zl), we have A(Zl) = JA(zl)(tl).Since @ is minimal on the couple

z,,

this leads t o the embedding Ja

A.

Let us now consider the main case (a). Since the parameter ip is nondegenerate, in view o f (3.5.5) there exists a function

(3.5.8)

jE @

and

f

LYUL:

f 2 0 such that

.

Let us prove that this leads t o the nondegeneracy of the parameter Q in

f belongs t o J a ( i 1 ) . Since the operator S [see (3.5.1)] belongs t o L l ( z l , i m ) , Sf belongs to Je(Z,) = 9.But in view of Proposition 3.1.7, Sf = IT.(.;f ; 2,). Hence

(3.5.6). Indeed, in view o f Lemma 3.4.4, the function

g :=

I{(.; f ; Z,)E @ .

It follows from t h e second condition o f (3.5.8) that g =

Sf # Lo, U L k .

Thus, the function g has the following properties: (3.5.9)

g(+m) :=

t21im g ( t ) = +m ,

g’(0) := lim t++O

g(t) = +m . t

390

The r e d interpolation method

It can be easily verified that the presence of such a function in the lattice B is equivalent to the fulfillment o f condition

(3.5.4).

Thus , \I, is a nondegenerate parameter of the K-method. 0

Proposition

3.5.6

If B is a nondegenerate parameter of the K-method, the functor Kq is mi-

zl.

nimal on thecouple

Proof.

We put

6

K ~ ( z 1 ) .According

:=

t o Theorem

3.4.3, J5 =

-+

Orbe(L1; .), and hence from the minimal property of the orbit and from

6 we have

the choice of

(3.5.10) Je

1

Kq .

Let us establish the inverse embedding. For this, i n view o f the same theorem

3.4.3 and the definition of orbit, it is sufficient t o establish the following statement.

E K q ( 2 ) and E > 0, there exists a function f E K*(Z,) and an operator T E LC,(zl; x') such that For any element I

(3.5.11)

5

llflle L ;u Il"llK*(~).

= Tf and

Here y := Q(1+

;u

q)(l+4, In9

:=

lnq

q+l and q > 1 are arbitrary. q-1

It follows fro m this statement that for small

This proves the embedding inverse t o

E

and q = 2

(3.5.10).

It remains t o construct for a given element I the function and the operator indicated i n

(3.5.11).We choose an arbitrary

cp :=

A'(.;

-+

z;

X)

E

> 0 and

put

+ Eg ,

E Conv satisfies conditions (3.5.9)and is arbitrary in all other respects. Then cp E II, n Conv. Let { t i } be the sequence in

where the function g

391

Equivalence theorems Proposition 3.2.5, constructed for the function cp and for a given q

>

1.

(3.5.9),this sequence is infinite on both sides, and corresponding function $, constructed for cp in accordance with

In view of conditions hence the

Proposition 3.2.6, has the form

$ ( t )=

C

~ ( t 2 i + 1 )min(1, t / t 2 i + l )

.

ieZ Since ~ t 2 i +5~ t2;+25 t 2 , + 3 , the characteristic functions

i

E

Z

xi

:= X [ t Z , + , , 4 t 2 , + 1 ) ,

have pairwise nonintersecting supports.

We now put

f

:=

C

~ ( t 2 i + l ) ~ i a

&Z Since

and, moreover,

(sxi)(t)I Inq . m;n(l,t/t2i+l) < In q),

(considering the inequality 1- q-l

we have

K ( . ; j ; Z,> < (1nq)g .

E

5

5 cp E 9.Therefore, f E > 0, we have

In view of Proposition 3.2.6, $ for a sufficiently small

h'q(&),

and

This established the second inequality from (3.5.11). In the same way as in Lemma 3.2.9, we now put

ui :=

zo(t*i+2)

- zo(t2i) ,

where the vector functions z o ( t ) and ditions (3.2.30).

2

E

z,

z l ( t )are chosen according t o t h e con-

Then, according t o Lemma 3.2.10, the series

r j cUi~

absolutely converges t o z in the space C(x'). It follows from what has been established in Lemma 3.2.9 t h a t

392

The real interpolation method

Then

Tf = C

u, = x since the supports o f

do not intersect. Moreover, in view o f

the functions in the family

(3.5.12),we

{xi}

have

and further,

Thus,

T E L7(,?1,z),where y is the constant in (3.5.11).

Remark

3.5.7.

Thus, for a nondegenerate Q we have established the isomorphism

where

6

+

:=

K a ( L 1 )and the isomorphism constant does not exceed 18.

Let us return t o the analysis o f case (a) of the theorem. proposition proved above it follows that

Ka

L

Recall that Q :=

Orb&(,fl;.)

.

J@(,f,)

6

(3.5.13) Orbi(Z1;

0

)

and

JQ ,

which will lead t o the embedding

(3.5.14)

Kq

~f

J@

.

:=

K,p(z,). Let us show that

From the

Equivalence theorems

393

2 0

For this purpose, we take g

&

:=

in the open unit ball of the space +

K a ( i l ) . Then the function j := K ( . ; 9 ; L,) belongs t o 9 :=

+

Ja(L,)

and lljlla

< 1. Therefore,

for which IIJ(t; u ( t ) ;

there exists a canonical representation

~?,)ll~ < 1. Using Lemma 3.4.5 and putting h(t)

:=

J ( t ; u ( t ) ; t,),we find an operator T E L 1 ( & , i m )such t h a t

Further, in view o f Lemma 3.4.4, we also have IlhllJ,(Z,,

I llhllo < 1 .

Consequently, in view of the second equality from (3.1.24) and the choice of

g, we obtain

Thus, we

This inequality and Lemma 3.4.10 imply that there exists an operator S E

&+,(&),

E

> 0, such that

g=Sh.

+

+

I 1. Consequently, Il9llJ,(Z1, 5 (1 E ) llhllJ*(~,,< 1 E , whence Il9llJ,(Z,, Thus we have established that if g belongs t o the unit ball of the space

6,it also belongs t o the

unit ball of the space J,(Z,).

embedding (3.5.13) on the couple

El.

This leads t o the

Using the minimal property of t h e

orbit, we obtain statement (3.5.13) from this embedding. Thus, the embedding (3.5.14) has been established. The inverse embedding can be easily obtained. Indeed, from the definition of 9 [see (3.5.6)] it follows that +

J*(L,)

+

:=

9 = K*(L,).

394

The real interpolation method

Using the maximal property of (3.5.15)

Ke (see Theorem 3.3.4), we hence obtain

KQ .

JQ

It remains t o consider case (b) of the theorem. Thus, let @ C L i , but @ # A(fl). Then it follows from t h e second condition that (3.5.16)

GJ c L:

,

@

g

,

i E (0,l)

Let us consider a new parameter

Obviously, it is nondegenerate, and hence in view of item (a) of the theorem under consideration and Theorem 3.4.9 [see (3.4.21)], for

@

:=

J&(Em)

we obtain the following equality:

Let us now show t h a t (3.5.18)

1

JL;

~f

Pr; . -+

Indeed, in view of the minimal property of the J-functors on the couple L 1 ,

it is sufficient t o verify the embedding only on this couple. Let

2

E

JL;(fI).

Then from t h e inequality

where u :

R+--t A(&)

it follows that

is the function in the canonical representation

Eq uivdence theorems

395

which proves (3.5.18). From (3.5.17) and (3.5.18) we now obtain

Thus, the embedding (3.5.19)

Kq n Pr; c+ Ja

-

has been established. Conversely, from (3.5.16) and (3.5.18) we have

Ja

-+

JL;

Pr; .

Moreover, embedding (3.5.15) which has been proved without using the nondegeneracy of CP is valid. Consequently,

which together with (3.5.19) proves case (b) of the theorem. 0

Corollary 3.5.8. For the functor Ja to coincide with a certain functor of the IC-method, it i s necessary and sufficient that the parameter CP be nondegenerate.

Proof. The

sufficiency follows from item (a) of the above theorem. Let CP be a nondegenerate parameter and let, say, L:. Then in view of (3.5.18).

us prove the necessity. Let

@

c

Ja

-+

Prl .

On the other hand, if Ja coincides with a functor of the IC-method ( J a = I<*), in view of Corollary 3.1.11 and Theorem 3.3.15 we have

The red interpolation method

396 (3.5.20)

A' = KACz,,

Kq

Ja .

Thus, we obtain

A'

~ - JG f

L)

Prl .

However, the embedding obtained is incorrect. For example, from what has been established in the proof of Corollary 3.1.20, we have

Ac(c,c') = (c')' = Lip [0,1] , while

Prl(c, c') = c' $ Lip [0,1].

0

B. Let us consider the second equivalence theorem which expresses functors of the K-method in terms of functors of the j - m e t h o d . For its formulation, we consider a certain parameter 9 of the IC-method and put (3.5.21)

Qj

:= Kq(z1).

In addition we are restricted by the case

QJ L)

qzm, .

The general situation can be reduced t o t h a t one, see Lemma 3.5.10. T heorern 3.5.9 (Brudnyl- Krugljak). (a) If

9

is a nondegenerate parameter, then

where the isomorphism constant does not exceed 18.

(b) If 9 L-) L'b, for a certain i E {O,l}, but 9 # A(Z,), then

(c) Finally, if

9 = A(E,), then

Eq uivdence theorems

Proof. Case (c)

397

follows from equality (3.5.20). Case (a) of the theorem has

already been proved (see Remark 3.5.7). Hence, it remains to consider only case (b). Thus, suppose that

(3.5.22)

9 -+ L',

and

9 .ft L',-'

,

i E (0,l) .

It follows from the definition of @ [see (3.5.21)] and Corollary 3.4.7 that 4

J&)

= @ := K q ( L 1 ) .

Therefore, in view of the minimality property of Ja on the couple

+

L1,

we

have (3.5.23)

JQ

1 L)

Kq . 4

On the other hand, from the relative completeness of the couple L,,

we

have

whence the embedding

Pr; n A' v K q

-

the maximal property of K q on the couple L,. embedding (3.5.22) this gives

follows from

(3.5.24)

Together with

JQ + ( P r , n A=)-+ K q

Let us now prove the inverse embedding. First of all we note that i n view of Corollary 3.3.6, we can assume, without any loss of generality, that

9 E Int(e,). Then from the first relation in (3.5.22) and Theorem 2.2.28 we conclude that only the following cases are possible: (3.5.25)

9 c-) Co(i,) ;

(3.5.26)

9

Lk .

The real interpolation method

398

Let us consider them separately. We shall require Lemma 3.5.10. The following isomorphisms are valid:

KB z (KBn Pro)+ (Kqn Prl) ;

where 9; :=

XP n L i .

Proof.The nontrivial part is the proof of t h e embedding (3.5.27) I&

L)

(1cq0n Pro) + ( K 9 , n P r l ) .

For this purpose we consider, for a given E

> 0, the functions s;(.): lR+ +

X ; , i = 0,1, such that

Thus, for the element ii := ~ ~ ( we 1 )obtain the inequality

K ( . ; s o ;2)L: 2(1+E)K(.;

s;

2).

Equivalence theorems

399

A similar line of reasoning is carried out for the element

21

:= zI(1) for +

which an identical inequality is valid. Here 2 = 5o +il. If now z E K * ( X ) ,

5; E X i , and in view of the inequalities proved above for functional, 2; E K q ( 2 ) . Thus, then

K-

the

and since in view of Corollary 3.1.11 and Theorem 3.3.15 we have

K* n Pr; = (K* n Pr;) n Pr; L+ (K* n (Pr;)')n Pr; =

the embedding (3.5.27) is proved. 0

Let us establish now t h e inverse embedding t o (3.5.24).

For this pur-

pose we use the first embedding in (3.5.22) from which it follows that Ql-; :=

\k

n L:'

-4

L)

A(L,).

Hence, in view of Lemma 3.5.10, we

obtain

It now remains t o verify that (3.5.29)

Pr; n Kq

L)

Ja .

However, the application of the previous embedding t o the couple sidering the definition of @ and the relative completeness of

@ :=

@

e,, gives

~ ~ (-,2 (L:~ n a) )+ (L;-; n ~ ( i-,,) 9n )L;-;

Hence it follows that @ (3.5.30)

&,

~t

con-

.

L;. If, besides,

# A(zl) ,

we can apply statement (b) of Theorem 3.5.5 t o 9. According t o this theorem,

400

The real interpolation method

n Ka

Pr1-i where

\k

E' Ja

,

:= Ja(Ew).

If we manage to establish the validity of the equality

(3.5.31)

Ja(3,)

=9

,

we have thus proved the required embedding

(3.5.29). Indeed, statement

(3.5.30)also follows from equality (3.5.31). Namely, if @ coincided w i t h A(El), it would follow from (3.5.31)and statement (c) o f Theorem 3.5.5 that 9 Z JA(zl)(Lw)= A(2,) in spite o f the hypothesis of the theorem [see (3.5.22)]. Formula (3.5.31),which has still t o be proved, follows from 4

the statement, which has a certain interest o f its own.

3.5.11. If the space 9 E Int(Z,), and if 9 c)Co(Z,), formula (3.5.31)is valid. Proposition

while

CP

:= K q ( L l ) ,then

Proof.We shall require two auxiliary results. Lemma

3.5.12.

If @ belongs t o the cone Conv and E > 0 is given, then there exists a function @ E Conv which is twice continuously differentiable such t h a t

+

(3.5.32) cp I I ( 1+ E ) c p .

Proof. For 6 > 0, we put

Since cp is monotone and concave, we can write

Further,

$06

is obviously non-negative, nondecreasing, and since

Equivalence theorems

401

cps E Conv. From the representation

1

t(l+6)

(P6(t)=

J

Y(U)~U

t

it follows t h a t y~ is continuously differentiable on ( O , + o o ) . Let us repeat this technique and choose 6 by t h e equality (1+6)2 = 1+ E . Then we obtain the function

+

:=

$966

with t h e required properties.

0

Lemma 3.5.13.

If the function

'p

E Conv is twice continuously differentiable on R+and is

such that

lim p(t) t -0

then for the function +(s)

:= - s 2 y n ( s ) 2 0 we have

Proof. Integrating twice by parts and taking into account the concavity of y , we obtain the identity 00

cp =

J

min(1, t / s ) {--s2cp''(s))

ds

.

0

It remains t o make use of Proposition 3.1.17 and the definition of t h e operator S [see (3.5.1)]. 0

Let us now prove equality (3.5.31). By definition, the space

E Int(Ll).

J a ( z l ) = a. Consequently, Ja(zl) = K*(&) (:= a), i.e. the functors .Ivand K,,j coincide on the couple L1. The fact that J , is minimal on this couple leads t o t h e embedding

Then, according t o Corollary 3.4.7,

402

The real interpolation method

The last equality follows from the fact that 9 belongs t o Int(,f,)

and from

Lemma 3.3.7. In order t o prove the inverse embedding, we take a function

h'w(z,)

(= 9).In view

of Corollary 3.3.6, the concave majorant

llfllv = Ilflle. Further, from the condition 9 3.1.14 and t h e equality K ( - ;f ; 3 ), = f , we have and

Therefore, using Lemma 3.5.12, for a given

E

c

> 0 we

Co(z,),

f

from

f

E 9

Corollary

can find a function

g E Conv n Cz such that

(3.5.33)

f I g I (1 + E ) f .

Further, we find with t h e help of Lemma 3.5.13 a function h

Let us estimate the norm of h in

a. We have

IIhlle = IlhllK,(t,) = Ilsllw

= (1

2 0 such t h a t

I (1+ &) llflllv

=

+ €1llflllv .

Finally, using the fact t h a t S belongs t o

C l ( e l , Em)and the left

(3.5.33), we obtain

Together with the previous inequality, this gives

l.fllJ*(z,)

I (1+ & > llflllv

which in view of t h e arbitrariness of

9

7

E

> 0 proves the embedding

JQ(~,).

This completes the proof of statement (b) of the theorem. 0

inequality

Eq uivdence theorems

403

Remark 3.5.14.

It is useful to note that the correspondence f + h, constructed in the proof of Proposition 3.5.11, gives “almost” the inversion o f the operator S. To be more precise, if f

E Conv (:= Conv n Co(x,)) and

rf = h, then

C . Let us consider a few corollaries of the basic equivalence theorems. First of all, we shall establish the classical Lions-Peetre theorem [see (3.5.3)]. Corollary 3.5.15.

If the operator S E L(G),where G is a parameter of the 3 - m e t h o d , then

Proof.We shall first Ja

(3.5.34)

~f

prove that

KQ .

For this we use inequality (3.4.6). It follows from this inequality t h a t

if

an

Applying the @-norm t o both sides and using the boundedness o f S in

a,

element

x E C o ( X )has the

where u :

canonical representation

nt+ + A(x‘), then

5

J

m i n ( l , s / t ) J ( t ;u ( t > ;X’)

dt t .

nt, Thus, the following inequality holds:

K ( . ; z ; 2)5

we obtain

s [J(t; u ( t ); x’)] .

The real interpolation method

404

This proves the embedding (3.5.34). In order to prove the inverse embedding, it is sufficient to note that the boundedness of S in @ leads t o the nondegeneracy of this parameter. Therefore, statement (a) of Theorem 3.5.5 is applicable, according to which (3.5.35)

Ja

Kq

,

where \k := Ja(Zm). Since for cp E Conv we have Sp 2 cp [see inequality (3.1.5)], we obtain (see Corollary 3.1.17, (3.5.15) and Lemma 3.4.4)

Ilfll& := ll.fllq I IlS.flllY Thus,

6

~t

f^

= llK(.; ; Zl)ll*

5

4, and in view of Corollary 3.3.6, from (3.5.35)

Ja

%

Kq = K4

t-' K6

= Ka

llf^lla := I l f l l 6 .

we obtain

.

Thus, the inverse embedding is established. 0

Let us now show that the equivalence theorems 3.5.5 and 3.5.9 are con-+ siderably simplified for relatively complete couples. Namely, if the couple X is relatively complete, the following statement is valid: Corollary 3.5.16. (a)

For an arbitrary parameter @ of the 3-method, the following isomorphism holds: (3.5.36)

J a ( 3 ) 2 K q ( 3 ), +

where 9 := J*(&). (b) For an arbitrary parameter 9 of the K-method, the following isomorphism holds:

Eq uivdence theorems where 9 :=

405

K*(i,).

Proof. (a) It is sufficient t o consider only the case

L+

Li since for a nondegenerate

parameter 9, the statement follows from item (a) of Theorem 3.5.5. In

this case, we have, by item (b) (or (c)) of this theorem (3.53)

~~( 2~ 2 ~) ( n xi 2, )

where 8

:=

Ja(z,).

Further, from the relative completeness of

x'

and Corollary 3.1.11, we have

xi =xi= K L 6 , ( 2 .) Therefore, t h e right-hand side of (3.5.38) is given by

~ ~ ( n ~2 ~) ~ = ~( ~ 2 )~. If 8

L-)

L k , the right-hand

~

(

side o f this equality is isomorphic t o

and the statement is proved. But since @ relative completeness of

-

~

z,

2

h'*(z),

L f , we have, in view of the

and relation (3.5.38),

(b) Taking into account Theorem 3.5.9, it is sufficient t o consider only the case when

9 L-) L&. In this case, the above-mentioned theorem yields

(3.5.39)

~ ~ ( E ~2 ~)( + ( p2T l -); n A C > ( ~, ) -4

where 9 :=

)

Kg(L1).

However, in view of Proposition 2.2.20, the relative completeness of

2

leads t o the equality Ac(x') = A(x'). Therefore, the right-hand side of (3.5.39) is equal t o

~~( + (xi 2 n)~ ( 2 = )~ )~( + ~2()2 ~) ~ (. 2 )

The red interpolation method

406

For the further analysis, the following statement o f technical nature about relatively complete couples will be useful. Corollary 3.5.17.

If the couple

2 is relatively complete,

then for each element

2

E

Co(z)

there exists a canonical representation

such that

K ( . ; J ( t ; u(t); Here y is independent of

2);i1)I y K ( . ; 2 ; 2).

x.

k f . We take for Q in relation (3.5.37) a lattice defined by t h e norm

Then @ :=

K a ( i l ) is defined by the norm

Ilfll@ = Ils(lfl>llv =

SUP O<S
U

du

J

=

du J min(Ls/u)lf(u)l - = R+

min(l,t/u> lf(u)l

= K(t;

f;

el>.

m+ If now x E C o ( z ) , relation (3.5.37) with

and QJ specified above gives,

taking into consideration the definition of .J*,

5

y

sup O<s
K ( s ; 2 ; 2)= y K ( t ; 2 ; 2),

407

Eq uivdence theorems

where t h e lower bound is taken over all functions in the canonical representation of 2. 0

Concluding the section, we shall consider an example t h a t shows t h a t t h e intersection formula

(3.5.40)

A(J@,)crEA

in Theorem

Example

JA(O,),,,

3.4.9is valid only for a finite

set of indices

A.

3.5.18.

We consider a sequence

('p,),€~

c

Conv such that qn $ LA for

i =0,l

and

inf 'p,(t) = m(t) := min(1,t) n

For yn we can take, for example, the function equal t o val ( 1 / n ,n) and coinciding on (l/n,1) and interpolating

outside t h e inter-

(1,n) with t h e linear functions

& a t t h e ends of these intervals.

Then according t o Theorem

3.3.15, A(KLE))~= ~ PI (vA ( L c ) , , ~ = K L ~ 7

where

L z = Lo, n L k , so that from Corollary 3.1.11and Proposition 2.2.20

it follows that t h e right-hand side is equal t o A'. Hence,

(3.5.41) A(KLL"),EN = A' . We now define a function w, by putting w,

:=

r ( y , ) , where

operator of t h e right inversion of S, which is defined in Remark Sec.

3.8,it will be established that if 'p E Conv and

'p

is the

3.5.14.In

$ L L , i = 0,1, then

the following relation holds:

K L z 2 JL$

,

where w :=

r'p,

and the isomorphism constant does not exceed

18.

Applying this isomorphism t o the functions 'pn and using the identity

(3.5.41),we obtain

The real interpolation method

408

It follows from this identity that the functor on t h e left-hand side cannot

JQ for any a. Indeed, otherwise we would have A' 2 Ja so that A(z,) = Ac(zl) 2 Ja(zl). Then the minimality of the -+ functors A = J A ( t , ) and Ja on the couple L1 implies that Ja 2 A, and be represented in the form

hence Ac 2

Ja

E A, which is not true.

Thus, we have established that the relation

A ( J L % ) ~ , N JQ does not hold for any

CardA = CO.

a.

Consequently, formula (3.5.40) is not valid for

Theorems on density and relative completeness

409

3.6. Theorems on Density and Relative Completeness

A. We shall start with an analysis o f the density o f the set A(x') in

J a ( 2 ) for

an arbitrary couple

r?. Actually, we shall establish the following

more general fact. Theorem 3.6.1.

(Ja)"

Jao .

Proof.We require Lemma 3.6.2.

If @ is a regular parameter o f the 3-method, then A(X) is dense in .la(-?) for any couple

2.

Proof.Since by Theorem for any element

2

3.4.3 the functor Ja coincides w i t h Orba(2, ; .),

E J+(r?) there exist a function f E @ and an operator

T E L(fl,r?) such that

x=Tf. Since

A(&) is dense i n 0 by hypothesis, for any given

function

such that

fE

112

Ilf

-

f,l @

< E.

> 0 there

exists a

- f€lla <

IITII E .

Then

- TfcllJ*(a, = IIT(f - fc)llJ*(n,

Since the element

E

5 IlTll Ilf

Tf,in this case belongs to the space A ( 2 ) . this space is

dense in Ja(r?). 0

Let us now prove the theorem. By the definition o f regularization, A(,<) is dense in

(Ja)'. On the other hand, this set is dense in J ~ ino view of

the

lemma proved above. Therefore, it is sufficient to establish the equivalence

@"

1 L,

J ~ on o

A(@. Here the embedding 1 0 obviously leads t o the embedding J ~ oc-, J a , and hence t o the

of the norms of

and

embedding

J ~ oA (Ja)" .

elements in

T h e red interpolation method

410

Therefore, we need only establish the inverse embedding. Let

xE

A(d)n

J G ( ~ )Then . by what has been proved earlier, [see (3.4.46) and (3.4.47)], there exist elements a 1 , a 2 , a 3E A(&) and operatorsTl,T2,T3 E Cl(zl,z), such that

and for a certain absolute constant y we have

Ilaillo

5 Y I141J4(d) 7

i = 1,273 .

From this inequality and from the coincidence of the norms of Je and

5:

on A ( z ) and of the norms o f @ and @' on A(&) we obtain

so that

( J g n A)(-?)

5

( Jaon A)(-?).

0

Corollary 3.6.3. (a) If 0 E Int(il), the regularity condition for 9 is necessary and sufficient for the functor Ja t o be regular. +

(b) If 9 E Int(L,),

the regularity and non-degeneracy of @ is necessary and

sufficient for the functor K e to be regular.

Proof. (a) Only the necessity has t o be proved. It follows from the interpolation

property of

that

J @ ( i l= ) 9 so that the regularity of 9 follows from

the regularity of Ja.

A(Z,) is dense i n 9,9 L) C o ( i , ) , and hence Co. Moreover, since Q is a nondegenerate parameter of the G m e t h o d , from item (a) of the equivalence theorem 3.5.9 we obtain

(b) Sufficiency. Since

Xu

L)

Theorems on density and relative completeness Therefore, the fact that

411

A ( 2 ) is dense in K q ( 2 ) follows from Lemma

3.6.2 and the following statement. Lemma 3.6.4. If \k E Int(i,)

is a regular space o f the couple

a regular space of the couple

Proof.

then @ :=

E 0 we shall construct a sequence

f

converging t o this function in 0. Then the lemma

will be proved. Without any loss of generality, we assume that

f $! A(&). (3.6.1)

-+

K*(L,) is

il.

For an arbitrary function

(fn)n,-m c A(&)

+

L,,

Let us first consider the case when f ( t ) = 0 for

f 2 0 and

t > 1. Thus,

f $!A(&), f 2 0 and f ( t ) = 0 fort > 1 .

We put +

g :=

A calculation of (3.6.2)

Sf for t 2 1 [taking into account ( t 2 1)

g ( t ) = g(1)

Further, since (3.6.3)

S f = K ( . ; f ;L,) . (3.6.1)] shows that

.

f E Ly and f @ A ( i , ) , we have

lim '('1 = lim t-0

t

t-0

(1

j

min (1,

t o

:-f(s)

e)

= +oo

.

We suppose now that Bn := {x : IIxIIa(t,) < n } and e,(g)

:=

id

119 - hllv

.

heB,

In view of the hypothesis of the lemma, (3.6.4)

lim en(g) = 0

n-+m

On t h e other hand,

where gn := n min(1,t). Indeed, if h E Bn,we have Ih(t)l

5 n min(1,t) = gn(t) and

hence

The real interpolation method

412

Taking into account (3.6.2), (3.6.3) and the concavity of g, we see that for n

> g(1) the equation

(3.6.6)

(= n min(1,t))

g(t) = g,(t)

has a single root b,, where 0

< b, < 1. Since g is differentiable and g ( 0 ) = 0,

applying the Rolle theorem t o the function g - g, on the interval (0, b,), we further see that there exists a point a, E (0, b,), such that (3.6.7)

g’(a,) = n

.

Finally, we put

and show that

[If

- fnll0

--+

0 as n + 00.

For this purpose, let us verify that (3.6.8)

(Sfn)(t) =

{

+ const

for t >_ a, for 0

5 t 5 a,

7

.

Indeed, the calculation of Sfn taking into account the fact that f,(t) = 0 for t < a, shows that the function Sf, is linear on (O,a,> and is equal to zero a t the origin. Therefore, in order to establish the second equality in (3.6.8), we have only to show that (Sfn>’(an) = g’(an> = n

[see (3.6.7)].

But this follows from the continuous differentiability of the functions Sf, and g and the first equality in (3.6.8). The first equality of (3.6.8) follows from the fact that g = Sf, and hence

Since f * ~(o.~,) = 0 for t > a,, S ( f x ( ~ , ~ ,=) )const for t > a,, and so (3.6.8) is established. Taking now into account the definition of i p , we have

Theorems on density and relative completeness (3.6.9)

IIf - fnlle

=

IIK(.; f - fn

;i1)~lQ=

=

llsf - S f n l l Q

= 119 - S f n l l Q

413

. -+

Since g - S f n = const for t > a, and g - S f n = K ( . ; f - fn; Ll), the concave majorant of the function (g - S ~ , , ) X ( ~coincides , ~ , , ) with g - f n . Moreover, since 11$110 = llpll~(see Remark 3.3.8), it follows from (3.6.9) and (3.6.8) that (3.6.10)

Ilf

- fnllo = 11(g - n t ) ~ ( o , a , ) l l *

.

But an lies in the interval (0, b,) in which, in view of (3.6.6) and the concavity 2 nt = g n ( t ) . Therefore, equation (3.6.10) can also be written in the form of g, we have g ( t )

It now follows from (3.6.4) and (3.6.5) that

Ilf - fnllo -, 0 as n -+

00.

Let now f 2 0 be such that (3.6.11)

f

6 A(&)

and f(t) = 0

for t

< 1.

We shall reduce this case t o the one analyzed above. Let us consider the map

( T f ) ( t ):=

1 u(,)

Since for the transposed couple

K ( . ;z ;

7

ZT

t E lR+ . :=

(Xl,Xo)we have

P)= T ( K ( . ;z ; 2)),

we get

T(@)= KT(U)(Ll). Here T maps A(Z,) onto itself so that A(Z,) is dense in T(O).Besides, Tf satisfies conditions (3.6.1). Therefore, according to what has been proved above, there exists a sequence of functions (hn)ncm C A ( z l ) , such that in view of (3.6.11)

The red interpolation method

414

It remains t o take into account that llTf - h , l J ~ p= ) Ilf - T-'(h,)lJo and that T-'(h,) E A(&). T h e general case is reduced to those considered above since any function

f from 0

can be represented in the form

f = fX[O,lI + fX[l,+,) . This proves the sufficiency of the conditions in Corollary 3.6.3(b). +

Let us now prove the necessity. Since

K&)

=9

Consequently, if A(Z,)

9 E Int(L,), we obtain

. is dense in

Ka(,f,), then 9 is a regular interme-

4

diate space of the couple

L,.

Thus, the necessity of the first condition is

established. Let us prove the necessity o f the non-degeneracy condition for the parameter

+

9 E Int(L,).

Suppose on the contrary t h a t Q is a degenerate

parameter. Suppose, for the sake o f definiteness, that

We shall indicate a couple

x' for which A(x') is not dense in K a ( x ' ) ,which 4

is in contradiction t o the regularity of Ka. Namely, we put X

:=

(C,C').

Then from what was proved i n Corollary 3.1.20, we have

(C')" = Ac(C,C') = Lip [0,1]

.

Further, in view o f item (c) of Theorem 3.5.9, we have

Ac(C,C') = K,(t,)(C, C')

L)

Ka(C,C')

L)

K L ~ ( CC') , .

Since the right-hand side is equal to (C')" = Lip [0,1], it follows from the last two relations that

Kq(C, C') E Lip [O,1] .

Theorems on density and relative completeness

415

At the same time, A(C,C') = C' and C' is known t o be not dense in Lip [0,1]. The contradiction obtained proved the necessity of the condition of nondegeneracy. 0

Remark 3.6.5 (Atzenstein). When 9 is an arbitrary parameter of the K-method, the criteria of the regularity of the functor

KO are more complicated. Namely, in this case the

necessary and sufficient conditions are as follows. (a)

9 n Conv c ~ ~ ( 3 , ) .

(b) Q

n Conv

L'b, n Conv, i = 0 , l .

(c) For any function

f E Q , we have

The proof can be obtained in the way described above. Remark 3.6.6. Lemma 3.6.2 can be made stronger by arguing in the following way.

If

ip,

6 E Int(&),

then for

J Q ( ~ t)o

2 it is necessary and sufficient that

be dense in J & ( i )for any couple

be dense in

6.

B. Let us now formulate the criterion of the relative completeness of the space K q ( 3 ) in the couple 2. For this purpose, we assume t h a t that (3.6.11')

9

L)

Co(3,) .

Then the following theorem is valid. Theorem 3.6.7.

K i E Ksc.

9 is a parameter of the K-method and

The real interpolation method

416

Proof.We shall need Lemma 3.6.8.

r? we have

If 9"= 9 ,for any couple

= KQ(2)

K&)"

Proof. By the definition of

I

relative completion, it is sufficient t o establish

t h a t t h e conditions

lead t o the inequality 11~11K,(n)

51 .

In order t o verify this, we note that in view of Proposition 3.1.7,

l l ~ ( ;. z ; 2)- K(*; zn ; y ) ~ l x ( L ~-) +

= ( ~ ( . ; 2 ; ~ ) - K ( . ; 2 n ; ~ ) ) ( l ) 5 K ( l ; 2= -xn;x) =

112

- xnllc(2)

.

Therefore, it follows from the first condition in (3.6.12) that +

( K ( . ;2,; X ) ) n E N converges in C(Zm) t o K ( . ; 2 ; 2).Moreover, it follows from t h e second condition in (3.6.12) t h a t

l l ~ (; *x n ; ~ > I =I *SUP II"nIIKu(2) < I

SUP

n

Since the lattice

9 is relatively complete by hypothesis, the last two

state-

ments imply t h a t 11~11K,(~, :=

1 1 1 " . ; z ; x')llly I 1 .

0

Let us continue t h e proof of the theorem. We shall use the fact that

A'

Z

K,(t,)

(see Theorem 3.5.9).

3.3.11, this gives

Together

with -the-reiteration

theorem

Theorems on density and relative completeness

where

\k

:=

KA(,r,)(@,C(2,))

417

@'

Thus, we arrive at the relation

Therefore, t o complete the proof of the theorem, it remains t o prove the relation (3.6.13)

&'

2 (9'7.

The validity of this relation is established in Lemma 3.6.9.

If 9 is a parameter of the K-method, and condition (3.6.11') is satisfied, the relation (3.6.13) holds.

Proof. Since 9 A

Q',

& + (9'7 as well.

Moreover,

(Q.7

= K,p(im)

(Corollary 3.3.6 and Lemma 3.3.7), and in view of the previous lemma the space ( 9'7 is relatively complete. Thus, (3.6.14)

4' 4 (9'7.

In order t o establish the inverse embedding, we take f in the unit ball of

(9'7.Obviously, without loss of generality we can assume that f E Conv. Under this assumption,

Ilfllpcr = I l f ^ l l ~ = ~ Ilfllq.,

and we thus have

Let us represent f as the sum of two terms (3.6.16)

f = f ~ ( o . 1 )+ f ~ ( i ,:=~ fo)

and make sure that fo

+ fi

E @' (the inclusion fi E

6' is proved similarly).

we have t o establish the embedding

(9"y-

6' ,

which is inverse t o (3.6.14). We can assume that

Thus

418

(3.6.17)

The r e d interpolation method

f o ( t ) - +0O lim t

t-0

Indeed, otherwise fo belongs t o A(Z,),

and hence t o

Thus, relation (3.6.17) is satisfied. We put induction a decreasing sequence (tn),ZO,

6' as well.

t o = 1 and construct by t , is equal to the

assuming t h a t

smallest root of the equation

f o ( t ) / t does not decrease for t 5 1, such a root exists and is strictly less than tn-l. In view o f (3.6.17), t, --+ 0 as n + 00.

Since

Let us now define the function yo on R+as a polygonal line with vertices

( t n , fo(tn)/2,),

12

= 0,1,

... . It is clear that cpo(t,) 5 cpo(t,-l)

and moreover

W t n ) >-%(tn-1) tn tn-1 since this is equivalent t o the inequality

which is valid in view o f (3.6.18). Thus, cpo is a quasi-concave function (i.e. it satisfies inequality (3.1.5)) and hence is equivalent t o a function from Conv. Let us also verify that (3.6.19)

lim i-0 t - +0O

.

Indeed, in view of (3.6.18), (3.6.17) and the relation limn.+oot , = 0, we have

Let us now prove that (3.6.20)

cpOX[O,l]

E

*.

Since in view of (3.6.15) we have

Ilfollvc I Ilfllw 5 1

7

there exists a sequence (g,),,=N such that

Theorems on density and relative completeness

419

For a given natural number N , the function (fo - g n ) x ( ~ - 1 , 1obviously ) belongs t o A(,f,).

Therefore, its norm in C(,f,)

in n) t o the norm in

A(Z,). This and the first relation of (3.6.21)give

Il(f0 - gn)X(N-l.l)IIL\(tm) x Il(f0 (n + m)

is equivalent (uniformly

- gn)X(N-1,1)IIqt,)

+

0

.

Combining this with the second relation from

II~OX(N-I,~III*

Illgnll*

+

Il(f0

-

(3.6.21),we obtain

s~)x[N-~,IIII*

I

Thus, we have established that

(3.6.22)

IIf0X(N-',1]II*

51

(N E

w.

Since according t o the definition of 'po we have

the first inequality from

(3.6.22)and the relation limn+00 tn = 0 imply that

Thus the series

c II 00

(P0X[tn ,*n-1 )

n=l

II*

converges. Since here cpOx(o,~]= C n E'poX[tn,tn-l) ~ (convergence everywhere), in view of Lemma 3.3.2we obtain 'p0x(0,1] E Q.

Let us now return t o the function fo in (3.6.16)and prove that Ilfox(N-f.l]lls

I T

9

where y does not depend on N . For this purpose, we take a natural number inequality

A4 > N and write the

420

The red interpolation method

(3.6.23)

(fOX(N-',l]r

I

fo(N-') N-' tX(0,M-q + f0X(M-',1] + fO(l)X(l,+rn)

which follows from the fact that fo belongs to Conv and from the definition

of concave majorant. Let us evaluate every summand of the sum on the right-hand side. If y is the norm o f the embedding operator IlfO(~)X(l,m)lllV 5

A(Z,)

L)

Q, then

fo(1)rl l X ( l , ~ ) l L ( L m I ) ?fo(l) .

Further, in view o f inequality (3.6.22) we have IlfOX(M-',llllry.

I 1

7

and it remains to evaluate the first term from (3.6.23). In view of the condition M > N and the fact that fo(t)/t does not decrease on ( O , l ) , we obtain

L e t us now choose M > N so that for a given E

> 0,

which is possible in view o f condition (3.6.19). Then the left-hand side of

(3.6.24) does not exceed

Here we use the quasi-concavity of cpo (so that cpo

5 Go 5 2cp0, see Corollary

3.1.4). Summarizing these results, we obtain from (3.6.23) Ilfox(N-',l]ll$ L 2 E Since ~OX(N-I,~I -+

+ 1 + ;ifo(l)

fo i n C(Z,),

completion we obtain fo E

:= 71

N ).

according t o the definition of the relative

9'.

0

Thus, the required relation (3.6.13) is proved. 0

(n E

Theorems on density and relative completeness

421

Corollary 3.6.10.

If 9 E Int(i,), for any couple in 0

~(i,).

then for the space

K a ( 2 )t o

be relatively complete in C(,f)

2,it is necessary and sufficient that 9 be relatively complete

The real interpolation method

422 3.7. Duality Theorem

A. L e t us consider another i m p o r t a n t relation between t h e functors of the

K -and 3-methods,

which is based on duality. I n order to formulate this

associated lattice. W e shall consider this concept only for t h e case of t h e measurable space (R+, d t / t ) . Therefore, relation, l e t us recall t h e concept of

t h e following definition will b e convenient. Definition 3.7.1. T h e Banach lattice

(3.7.1)

9+defined by t h e n o r m

Ilflle+ :=

SUP

dt f ( t > s ( l / tt > ; llgllo I 1

is called t h e lattice associated to t h e Banach lattice 9. 0

It is useful to n o t e t h a t if 9 is also an intermediate space of t h e couple

i,,t h e n there exists a simple relation between t h e dual space 9’ and t h e 9+.Namely, i n this case 9‘ L+ A(&)*. Since A(&) 2 L;”, where m(t) := min(l,t),we can assume t h a t all functionals in A(Z,)* have t h e dt form h + J h ( t ) g ( l / t )- . T h e n by definition t R+

lattice

Therefore, t h e f a c t t h a t

A(E1) is dense i n 9 leads to t h e equality 9’ = 9’.

In t h e general case, t a k i n g into account t h e equality

(3.7.2)

(9’)’ = 9’ ,

we o b t a i n t h e following relation:

(3.7.3)

9’= (a”+

Duality theorem

423

After these remarks, we shall give the main result.

For i t s formulation, it should be recalled that if F is a functor, then F’ denotes the map X’ + F ( 2 ) ’ specified on the subcategory 6’of dual couples. Theorem 3.7.2 (Brudnyi-Krugljak).

(Jay 2 KO+.

Proof. According to the definition of dual space and the identity Jg

Z Jp

(see Theorem 3.6.1) we have

(3.7.4)

(Ja)‘ = (Jg)’ Z (Jao)’ .

Further, according t o Theorem 3.3.4 and 3.4.3, and equality (3.7.2) we have

JOO= Orbao(il; .) ,

Kat = Corb(,q,(.; 3,) .

Here (9’ is regular on the couple 21, and therefore the functor Orb,p(& ; .) is computable (Theorem 3.4.12). In view of the general duality relation for computable functors (Theorem 2.5.18), we obtain -+

(3.7.5)

Orboo(L1; .)’ = Corbpo)t(. ; 3 ),

.

Together with the previous relation, this gives (3.7.6)

(Ja)‘ Z I{*, .

Since according t o (3.7.3) O+

~t

a’, we obtain the following embedding:

K*+ L-, (Jo)’. Let us now prove the inverse embedding (3.7.7)

(Ja)’

A

K*+ .

It is sufficient to carry out the proof for the case when O E Int(i1). Indeed, let the embedding (3.7.7) be established for this case and l e t O be an arbitrary

parameter of the 3-method. We put and in view of (3.7.7). (3.7.8)

(J5)’

1

K&+ I

6

:=

J*(Zl).

Then

6E

Int(&),

The real interpolation method

424

On the other hand,

3.4.4)so that &+

J* = .J& (see Corollary 3.4.6),and O A 4 (see Lemma A O+. Hence and from (3.7.8)we obtain (3.7.7)for an

O. Thus, O E Int(Z1). Let

arbitrary

(3.7.9)

~ ' n c o n vA

us prove that

a+ .

Kot = KBtnConv, the embedding (3.7.7)will follow from this equality O under consideration. To prove the embedding (3.7.9), + we shall use the duality relation in Proposition 3.1.21for a couple X := 21.

Since

and from (3.7.6)for This gives

qt-' ;f ;

Z,)

= sup {(f,9 ) ; J(t ; 9 ; J f l )

I 1)

7

where we put

(3.7.10) ( f 7 g ) :=

dt

J

f(t)s(llt)t .

lR+

f E Conv n O'. Then the left-hand side of (3.7.10)is equal to f ( t - ' ) (see Proposition 3.1.17). Consequently, for given q > 1 and i E Z, there exists a function g; E A(Zl), such t h a t

Suppose now that

I (f,g) I f(q-i)

(3.7.11) q-lf(q-i) and

(3.7.12) J ( q ' , g ; ; Further, we put

Z 1 )=

1.

A, := (q',q'+'] and define the operator T : C ( z l )

4

C(Z1) by the formula gi

ieZ

Let us verify that into account

/ h(t) t . dt

(3.7.13) T h :=

T

E

A,

L,(Jfl). Indeed, if h E LI,s E {O,l},

(3.7.12),we obtain

then taking

425

Duality theorem Let now h >_ 0 be a function in

k < 1.

a. We put hk,l

:=

hX(g1,9k]r

k,I E

z,

In view o f the concavity o f f and inequality (3.7.11), we then have

Taking into account the definition o f

1

hkJ,

we thus obtain

:

q'

(3.7.14)

T and

L Q2(f,Thk,l).

f(l/t)h(t)

Pk

On the other hand, the interpolation property of 0 implies that

qk

Since f , h

2 0, we can proceed t o the limit as k,1 --+ 00. Taking into account

the fact that q

> 1is arbitrary,

(f,h)

we hence obtain

I Ilfllw llhllo ' *

Taking here the least upper bound for h >_ 0 with Ilhlla

I 1 and taking into

account (3.7.1), we obtain

Ilfllo+ 5 Ilfllot

*

Thus, the embedding (3.7.7) is established. 0

Remark 3.7.3. Since for a regular @ we have minimal on the couple exact realtion

(JQ)' = J*

(in view of the fact that

El), in this case relation (3.7.5)

Ja is

is replaced by a more

426

The real interpolation method (J@)'= K@t.

The above line of reasoning leads in this situation o t t h e equation

Let now @ be a quasi-power parameter. Since in this case K@ S (Theorem 3.5.3), we can also use the notation

~ @ ( x ' ) , ~ @ (In+x particular, '). for x' use t h e notation

:=

L:,

x',

JQ

for any o f the spaces

o < 6 < 1, 1 5 p 5 m, we

XSp.

Corollary 3.7.4.

If 0 is a quasi-power parameter, then (3.7.16)

( 2 ~E) (J7')@+ ' .

Proof. Since for

( f , g ) :=

1

f(t)g(l/t)

dt t we have ( S f , g )

= (f,Sg),

R+

the boundedness of the operator S in @ leads t o boundedness in @+. Then the application of Theorem 3.7.2 immediately leads t o (3.7.16). 0

In particular, if we choose @ :=

L:, 0 < 29 < 1, we obtain t h e following

classical result. Corollary 3.7.5 (Lions). +

(XS,)'

+

XJ,,, where

+4 = 1.

0

B. Let us now consider the duality theorem for t h e functor K Q . Here (Kq)' Jq+, which seems quite probable, does not hold. This follows from the situation is more complicated. For example, the formula

T heorem 3.7.6 (Brudnyi- Krugbak). Suppose that

(3.7.17)

E Int(Z,).

(Kq)' E Jq+

Then t h e relation

427

Dudi t y theorem is satisfied iff Q is a nondegenerate parameter of the K-method.

Proof. Let us first establish the embedding (3.7.18)

J*+(Z')A

KQ(.~)'.

For this purpose, we take a functional y in there exists the representation (3.7.19)

y=

/

J*+(2'). Then for a given q > 1

4 t ) dt , -

,nz,

t

where u ( t ) E A(x') and (3.7.20)

IIJ(t ; u ( t ) ;*Ill*+

I q IIYIIJ,,~~~).

According t o the identity in Proposition 3.1.21, we have

Consequently, for z E A ( 2 ) and t

> 0 we obtain

J ( t ; u ( t ) ; 2')* K(t-1; z ;

2)2 I((u(t),z)(.

Together with (3.7.19) and (3.7.20) this inequality leads to I(y,z)I I

1 l(u(t)7z)lt

dt I( J ( t ; u ( t ) ;d ' ) , K ( t ;z ;

2))5

PL,

5 IIJ(t; 4 4 ; ml*+ IlK(t; 5 ; d)ll* I 4 II"IIK,(R) IlYllJ*+(a~)~ Thus, y E K Q ( ~ ) and ', IIYllKv(n,l

I Q llYIlJ,+(2~)

*

Passing to the limit as q + 1, we arrive a t (3.7.18). Let us now prove the sufficiency of the nondegeneracy condition for the validity of (3.7.17). In view of (3.7.18), we must only establish the embedding (3.7.21)

Kry(2)' L) JQ+(~??') .

The real interpolation method

428 For this we put

and show first t h a t

(3.7.22)

1

K&, L) J&

Indeed, in view of Proposition 3.5.11,

9 = Ja(z,),

and hence considering

t h e interpolation property of Q, we have

J@(i,) = K*(Z,)

(= 9). +

Consequently, since

I<* is maximal on the couple L,,

JQ(2)A

we obtain

K*(Z).

Passing t o the dual spaces, we obtain (3.7.22) Further, let us establish the embedding

(3.7.23)

l<~+(x") ~f J*t(d') .

For this we note that a dual couple is relatively complete (Theorem 2.4.3). In view of Corollary 3.5.16, the functor J*t coincides with a certain functor

8' of such couples. Therefore, J*t has -+ relative t o the couple iw E B on this subcategory.

of t h e K-method on the subcategory

the maximal property

But this means that it is sufficient t o verify the embedding (3.7.23) only on +

the couple L,.

Thus, it remains t o prove t h e following statement.

Lemma 3.7.7. +

If 9 E Int(L,),

:=

I<*(&)

and

9 is a

nondegenerate parameter, then

Ka+(Zw) A J&,).

Proof.Without (3.7.24)

loss of generality, we can assume that

9 c C-(i,)

.

Otherwise, we shall replace Q by the space choose t h e constant

9

y so that the embedding

+

:= Q

n y . Eo(L,).

Let us

Duality theorem

(3.7.25)

Q !c

429

y

. C(,f,)

is valid. Then

6

+

:=

=

qil> =~ ~ ( n i ~,K)~ , , ( ~ , ) =( L ~ ) 1 anyz(i1) ct a.

Here we took into account the fact that K ( . ; f ; El) E C o ( i , ) . Therefore, we have the embedding (3.7.26)

&+(em).

A

Kat(i,)

Let us verify that the set Q n Co(i,)

is dense in Q relative t o the topo-

logy generated by the pairing (f,g) [see (3.7.10)]. Indeed, if g E Q , then gn := ~ [ ~ - l ,E~Q] g n C o ( i , ) and for any h

( g , h ) = n+m lim

j

g(t)h(llt)

E 9+ we have

dt

t = ;il (gn,h) .

n-1

Therefore, considering the choice of y and the definition of $ [see (3.7.25)], we have

Hence it follows that (3.7.27)

Jqc(Z,)

= J$,(Z,)

.

Let the statement of the lemma be proved for %. Then we obtain from (3.7.26) and (3.7.27) a similar embedding for 9. Thus, the embedding (3.7.24) will be henceforth satisfied. Suppose now that (3.7.28)

f

E

Kat(i,)

.

Then the concave majorant f is in a+. Let us now verify that Indeed, otherwise, as will be shown below,

f

E C-(,f,).

The red interpolation method

430

@ n Lf E A(,f,) for some i E (0, l} .

(3.7.29)

On t h e other hand, it will be shown later that this equality is in contradiction with the nondegeneracy of t h e parameter '4'. Thus, let

f

E @+, but f

is valid for a certain

4 Co(e,).

We shall prove that equality (3.7.29)

i. In view of the second condition imposed on

f

and

Corollary 3.1.14, a t least one of the following inequalities holds:

lim f(t) t-0 Then

>0 ,

f(t>> o lim -

t-w

t

.

f , having been multiplied by an appropriate constant,

the function rnw(t) := 1 or the function rno(t) :=

majorizes either

t . This

means that

L j , cf @+ for some j E ( 0 , l ) . Since

L&, = (L{)+, it follows

consequently (3.7.29) holds for

hence t h a t 0 cf L i , and

i = 1- y.

Let us now show that (3.7.29) contradicts the nondegeneracy of Q. Precisely this property of '4' implies that there exists a concave function

h E (Q f l L&)\A(z,). In view of (3.7.24) the operator I' in Remark 3.5.14 is applicable t o h. For this operator,

and hence

Ilrhll@= l l ~ ~ l l K & = ) IISrhll* I P IlhIlQ< +m . This means t h a t I'h E @. Similarly, it can be verified that rh E L i . At the same time, it follows from the left inequality in (3.7.30) that S r h =

K ( . ; r h ; El) does not belong t o A(e,), i.e. r h $! K A ( ~ , ) ( &= ) A(Z1). This means that I'h E (@ n Li)\A(e,), which contradicts t o (3.7.29). Thus, f E Co(i,), and hence I'f does exist. In view of (3.7.30) and the embedding 0 c) J@(,fl)(see Lemma 3.4.4), we have

IlflIJy+(tm) 5 ~ ~ s ( r f ) ~ ~ J y5+ lIsllL(t1,t,) (~m)

. llrfll Jy+(tl) I IlI'fllQt= sup {(I'f,h, ; Ilhl(Q

'

=

'1

'

Duality theorem

431

Thus,

Let us verify that it is sufficient to take the least upper bound in (3.7.31) only over concave functions h. Indeed, ( r f , h ) 5 (rf,h)and at the same time (Ih(lo= llkll,p since Q = J4(io0)is an exact interpolation space of the couple

,fm(see

Proposition 3.5.11).

In view of (3.7.24), the operator r is applicable to the functions h specified above. Using inequalities (3.7.31) and (3.7.30), we thus obtain

Since JJSgJllv = IlgllK,(t,, = Ilgll0, it follows from the above inequality combined with (3.7.30) that

ll.fllJ*+(~m) L q2 SUP {(f,S> ; 119110 = 1) = q2 Ilfll0+ . But since Ilfll@+ = llfll K,+ (x-), making q tend t o 1we obtain the embedding K@+(Z,) A Jq+(Z,). 0

Let us finally show that the required embedding (3.7.21) follows from embeddings (3.7.22) and (3.7.23). For this we shall also use the relation [see (3.7.7)j

J4(2)’

A K@+(Z’).

-

By this embedding and from (3.7.22) and (3.7.23) we get

Kq(2)’L: J @ ( 2 ) ’L-, K @ + ( 2 )

&+(2) .

Thus, the sufficiency of the conditions of the theorem for the relation (3.7.32)

K& 2 JQ+

The real interpolation method

432 t o be satisfied is established. Let us verify that for \k

Int(z,)

the nondegeneracy of the parameter

Q is also a necessary condition for this relation to be fulfilled. Indeed, let

this condition not be satisfied and let, for the sake of definiteness, P -+

L;.

Then

since according t o Corollary 3.1.11, K L o , ( ~coincides ) with the relative completion of X o in C ( 2 ) . Therefore, from the validity of (3.7.32) it would follow that

(Xi)'~t K,p(x')' = J,p+(Z1) E o ( i ' ) .

-

It remains t o specify a regular couple x' for which the embedding

(XG)' C " 2 )

(3.7.33)

does not hold.

2

(II)" = lI (see Lemma 3.4.13), and hence the left-hand side of (3.7.33) is equal t o I,. On the other hand, x" coincides with Let

:= (Zl,c,,). Then

the couple

(Irn, 11),

and since Zl is not dense in

I,

we have Co(*)

L+

# .,I

0

Remark 3.7.8. Since for an arbitrary parameter of the IC-method K,p = K , and

\ir

E

4

Int(L,),

the following statement results from Theorem 3.7.6 in the situa-

tion under consideration. The nondegeneracy o f the parameter $ is a necessary and sufficient condition for the following equality t o be valid:

Let us consider one more duality relation in which the parameter \k is not any longer assumed t o belong t o Int(z,) we shall assume that

or t o be nondegenerate. Instead,

Duality theorem

(3.7.34)

\k

433

=a+ ,

where Q is a certain parameter of the 3-method, and that Q

~t

Co(z,).

Theorem 3.7.9. For the identity

K*(Z)’

Ja(2’)

it is necessary and sufficient that the closed unit ball of the space J a ( 2 ’ ) be closed in the *-weak topology of the space A(Z)*. to be fulfilled,

Proof. In view of Theorem 2.4.34, the identity (3.7.35)

Corbat(2; 2)’ = O r b A ( i ; 2)

is satisfied iff the closed unit ball of the space on the right-hand side is closed

in the *-weak topology of the space A(x’)*. It is assumed here that

(3.7.36)

A

L)

(Ao)“,

Choosing A :=

a, A’

:=

zl and using Theorems 3.3.4 and 3.4.3, we

obtain from (3.7.34) and the embedding \k (3.7.37)

K‘&’

L*

Co(t,)

the relation

= Ja(2’) .

But as was proved in Theorem 3.7.2 [see (3.7.9)],

Consequently, K ~EII{*+ = I{* and hence we obtain from (3.7.37)

K*(Z)’ E Ja(2’) . To complete the proof, it remains to verify the fulfilment of condition (3.7.36) for A’ := 21,A := @. For this purpose let us take f E 9;then

The red interpolation method

434

Remark 3.7.10 (Aizenstein). There exists another version of the duality relation, in which the assumption about the *-weak closure made in Theorem 3.7.9 is excluded. Namely, as in Theorem 3.7.9, we assume t h a t

9 = O+

(3.7.38)

and

@

~t

Moreover, we also assume that

Co(Z,)

8

.

is a nondegenerate parameter of the

Ic-

method. In these conditions, the following relation is valid:

K;

(3.7.39)

E JF

.

C . Concluding the section, we shall compute dual functors (see Definition 2.4.29) for the functors of the real method. Proposition 3.7.11. (a)

DJa 2 K a t .

(b) If

8 is a nondegenerate parameter o f the K- method, DK9

S

then

Kg ,

4

where $ = J e t ( & , ) .

Proof. (a) According t o Definition 2.4.29, the following statement is correct. If

F'

:

x" -+ F ( 2 ) ' is a functor

o f the subcategory of couples dual t o

regular couples, then D F is its maximal extension t o the entire category (concerning the maximal extension, see Theorem 2.3.24).

In view

o f Theorem3.7.2, the conditions of the statement formulated above are satisfied for the functor F

:= Ja. Since in this case the functor J& Z Ka+ is defined on the entire category B, we have

Duality theorem

435

K@++ DJ@ .

(3.7.40)

On the other hand, the functor K@+is maximal on the couple

z, (The-

orem 3.3.4), and this couple belongs t o the subcategory of dual couples. Therefore, in view o f the same Theorem 3.7.2,

which leads t o the embedding inverse t o (3.7.40). (b) In the case under consideration, Theorem 3.7.6 is applicable, which together with Corollary 3.5.16 yields

where

@

+

:= Jq+(L,).

While applying Corollary 3.5.16, we took into account the fact that the couple

2’ is relatively complete (Theorem 2.4.3).

It remains to apply the arguments of item (a). Remark 3.7.12.

It is useful t o note that for regular couples, D K q ( 2 ) can be computed G. Namely, in this case

without the assumption about the nondegeneracy of the formula (3.7.41)

(DKq)(r?) K e + ( z ),

is valid, where

:= I ( r y ( ~ l ) .

In order t o verify the validity o f this formula, we shall use Theorem 2.4.31 for the functor F := we have

K q . According t o this theorem, for regular couples

The real interpolation method

436 Since the couple

r?'

is relatively complete,

(r?')'

also possesses this pro-

perty (see Corollary 2.2.23). Therefore, we can make use of Corollary 3.5.16 according t o which

Kry ((x")O)

2

J* ((2)') .

Thus, from the last two relations and Theorem 3.7.2 we get

(DKry)(r?)g

K$'(K@+(r?"))

K>'(J@((r?')')')

The norm of the space on t h e right-hand side is equal, by definition, t o

IIK(.; K

i:")ll@+.Therefore,

~ X ;

t o complete the proof of the validity of

formula (3.7.41), it is sufficient t o verify t h a t (3.7.42)

K(.;K

2")= K ( . ; X ; x') .

~ X ;

+

+

Y ) = I I y l l c ( ~ ~ , ~and v ~ the ) , couple X is regular. Therefore, in order t o prove (3.7.42), we must only verify that for 2 E A(Z), Here K ( t ; y ;

(3.7.43)

Il"a~llc(a,,)= Il"llc(a, .

Since C(x'") = A(r?')' and

A(x") = C(13')',in view of Propositions 2.4.6

and 2.4.11(a), the left-hand side is equal t o {(K,?x, Y)A(BI) ; IlYllA(nj)

=

{(Y,z)A(f)

5 '1

; Il?/llc(m)t5

= '

Here (1yl(c(2), := sup {(y, ~ ) ~ ;( I a J z )l l c ( ~5) l},and hence the right-hand side of (3.7.43) is equal, according to the Hahn-Banach theorem, to I I X I I ~ ( ~ ) and relation (3.7.43) is proved.

It should be noted that relation (3.7.41) entails that (A')' is not a functor (see Theorem 2.4.17). Indeed, otherwise for the regular couples

r? we would

have +

= l'A(Em)(x')'

= (DKA(E,)>(X'>

.

The calculation of the right-hand side with the help of formula (3.7.41) gives

( D K A ( ; m ) ) ( 2z) I<*+(?) = C ( 2 )

,

,

Dudi ty theorem since

9

:=

437

KA(e_,(tl)= A(&)

and

=

A(Zl)' = C(z,).

Thus, if

(A')' is a functor, then (3.7.44)

(A'(2))' 2 C(-f') .

Since, on the other hand,

A(x')' is also equal to C(x"), and A ( 2 ) can A(@', even for regular couples, the equality

be considerably smaller than

(3.7.44) is generally incorrect.

The red interpolation method

438 3.8. Computations

A. In this section, we will obtain corollaries from the basic results for “concrete” parameters of the real method (for example, for parameters of

the form L z ) . We shall begin with the corollaries of the reiteration theorem

3.3.11. In order t o formulate the first result, we require the concept of modified

F . It should be recalled [see (2.3.9)] t h a t the fundamental function ( P F is defined by the equality

fundamental f u n c t i o n @F f o r the f u n c t o r

(3.8.1)

F ( ( s R , t R ) ) = ~ F ( s ,, ~ s), tR >0 .

According t o Proposition 2.3.10, v F differs from zero, is positively homogeneous and nondecreasing. Therefore, the function

is quasi-concave, i.e. $F does not decrease, while $F(t)/t does not increase with increasing

t.

Since

it follows from definition (3.8.2) that

Proposition 3.8.1. Let w be a quasi-concave function and l e t F coincide with K,, Then

Proof.According t o (3.8.3), for the K-functor we have

In view of the quasi-concavity o f w, the denominator is given by

or

JL;.

439

Computations

and hence the statement is proved in this case. For the J-functor, we consider the couple X := (apt,R)and establish

that for x E

Z(d)= R,

(3.8.1)and (3.8.2)the left-hand side is equal t o Q/@F((Y) {for F := J L ~ )+F , = w i n this case, too. By definition, the norm on the left-hand side of (3.8.4)is given by Since in view of

In view of quasi-concavity o f w , we further have

max(a,t)

4t)

L-

a

4 a >'

and hence the right-hand side o f the previous inequality is not smaller than

Thus, the left-hand side of Conversely, we put u

JU(t)J dt

condition

IL+

not exceed

t

:=

(3.8.4)is not smaller than the right-hand side. where ye

>

0 is chosen from the

= 1x1. Then the norm on the left-hand side does

The red interpolation method

440 In view of the arbitrariness of

E

> 0,

the left-hand side of (3.8.4) does not

exceed the right-hand side. 0

Further, we shall require Definition 3.8.2.

A nondecreasing function w (1 5 p

5 co) if for

a certain

R+ + R+ is called a pquasi-concave constant y > 0 and for all t > 0 we have :

Here m t ( s ) := min(l,s/t). 0

Remark 3.8.3. Since w is a monotone function, the integral on the right-hand side obviously does not exceed y ( p ) w ( t ) . Thus, for the pquasi-concave function w,

uniformly in t

>0

Example 3.8.4. (a) For p := co,the function w satisfies the inequality

i.e. is equivalent t o a quasi-concave function (see Remark 3.1.5). The

inverse statement is obviously valid as well.

(b) As can be easily verified, condition (3.8.5) is equivalent t o the boundedness in the lattice Ls of the operator S, defined by the formula

Computations

441

(c) If the function

‘p

is a pi-quasi-concave

(i = 0,1) and po < pl, it

is

pquasi-concave for any p satisfying the condition

1 P

- 1-9 Po

+-P19l

(OIS
This follows immediately from Holder’s inequality.

(d) If ‘p is pquasi-concave, then

Indeed, using (3.8.3) i n the case

II,

:=

Lr,we obtain

and it remains t o use relation (3.8.6). (e) If the increasing function w is equivalent in the vicinity of zero and

td J l n t ( 0 ,where 0 < 9 < 1 and /? is arbitrary, it is pquasi-concave for any p E [I,m]. infinity t o the function

We now have everything in order t o formulate the first result. Let Fo and Fl be interpolation functors and let their fundamental functions q, := + F ~ ,i = 0 , l satisfy the following conditions: (a) q; is 1-quasi-concave,

(b) the function cplo :=

i = 0, I;

‘ p l / q o maps horneomorphically

El+ onto itself.

For the sake of definiteness, we shall assume that cplo increases. We put (3.8.7)

w := (‘plo)-’

and consider the operator

(inverse function)

S,

defined

by the formula

The red interpolation method

442

Under these assumptions and in this notation, the following theorem is valid. Theorem

3.8.5.

If Q is a parameter o f the X-method and the operator any p E [I,m] we have

S, E L(9),then for

Proof.We require Proposition 3.8.6.

If cp :=

(3.8.9)

G F , then Jq

Proof. Let x

1 L-+

F

1 L-+

K L .~

belong t o a unit ball of the space F ( i ) . Let us verify that in

this case

which proves the first embedding in

(3.8.9).

Since for a given to > 0 the function z -+ K ( t o ;z ; x’) is a norm on Z ( d ) , according t o the Hahn-Banach theorem there exists a linear functional f E C(x’)* such that for all y E C ( X ) we have

and moreover

(3.8.11) f(x) = K(t0; x ; d ) . Then from the first inequality it follows that

443

Computations and hence the operator T : y

(&,t;'&)

4

f ( y ) acts from the couple

2 into the couple

and has a norm which does not exceed unity. Consequently,

considering (3.8.1),(3.8.2) and (3.8.11) we have

Thus, inequality (3.8.10) is established.

Let now z belong t o the unit ball of the space Jp(d).Let us verify t h a t

(3.8.12)

1141F(a,I 1

7

which will prove the left embedding of (3.8.9). In view of the assumption, for

E

>

0 given there exists a canonical

representation

such that

(3.8.13)

/ nt,

J(t ; u ( t ); 2) dt -<1+€. t dt)

We choose an arbitrary q

> 1 and put

qntl

u,

:=

dt u(t) 7

J a"

Then in view of (3.8.13) and the quasi-concavity of cp, we have

-dt < t

'

-

J ( t ; u ( t ); x') dt I '(1 t Q(t)

+E)

Let us consider the operator T : (R,q-"R) 4 x' defined by the formula

T X :=

U,

*

X .

Since for s E ( 0 , l ) we have

The red interpolation method

444

so the norm T does not exceed

Hence it follows that

From this inequality and (3.8.14) we obtain

Making E

-+

0 and q + 1, we establish inequality (3.8.12).

0

Let us now prove the theorem.

Since the functions

concave, the operator S is bounded on

pi

are l-quasi-

LF (see Example 3.8.4(b)). There-

fore, according to Theorem 3.5.1, we have

JL" z K LlPI , which together with the embedding (3.8.9) gives

From this embedding and from the relation (3.8.16)

K L L) ~ K.7

c+

I
that will be established below it follows that in order to prove the theorem, it is sufficient to obtain the embedding (3.8.17)

Kq ( K L z ,K L ) r-1 ~

K L K~L ; P) ~

Indeed, putting G, := K q ( K L : ~ , K L : l1) I , qI 00, from this relation and from (3.8.15) and (3.8.16) we obtain the following chain of embeddings:

Computations

445

Thus,

Therefore, it remains t o establish relations (3.8.16) and (3.8.17).

The

first of them follows from (3.8.9) since (3.8.18)

(

j

= pi~ ,

~i = 0~, l

.

Indeed, 'p; is l-quasi-concave by hypothesis, and it is also oo-quasi-concave as a fundamental function o f a functor (see Example 3.8.4(a)). From statement (c) of this example it then follows that

'pi

is pquasi-concave. Then (3.8.18)

follows from statement (d) of this example. In order t o prove (3.8.17), we note that in view of reiteration theorem 3.3.11, both sides o f this relation contain functors of the K-method. Since these functors possess t h e maximal property on the couple

im (see Theorem

3.3.4), it is sufficient t o verify the embedding (3.8.17) only on this couple. Here K g ( i m )= (Lrr(see Corollary 3.3.6 and Lemma 3.3.7). On the other hand, if p is concave, the inequality

If1 5 Mp is equivalent t o the inequality

f^ 5 Mp so that (LLr=LL. Therefore, the quasi-concavity of the function 'pi

leads t o the relation

Thus, everything is reduced t o the proof of the embedding

But the functions p; are l-quasi-concave, and hence the operator S is bounded in

L?.

Applying Lemma 3.3.14 with the operator

&(f)

:=

obtain the following relation for the right-hand side o f (3.8.19):

K*((L;por,(Lry)

&(L?$LY')

It remains now t o prove the embedding

*

S(lfl),we

The red interpolation method

446 (3.8.20)

K Q ( L ~ ,) + L~ KQ(L?,LY').

For this purpose, we take f in the left-hand side of (3.8.20). Then according to formula (3.1.39) with p := 00 we have (3.8.21)

K(t; f ;LE)Lz)M sup {min s>o

(-

1

cpO(S)

'

-) t

If(s)j}

.

'pl(S)

In view of our choice, the right-hand side of this expression belongs to Q .

Hence we must show that the K-functional

[also see the same formula (3.1.39) for p := 11 belongs t o

9 as well. We

take s := w(t) under the supremum in (3.8.21) (recall that w := and

910

('pl0)-'

:= cpl/cpo). Then we have

where g E 9. Using the same substitution s := W ( T ) in the right-hand side of (3.8.22), we obtain in view of the definition of the operator S, [see (3.8.8)]

Since S, is bounded on Q and g E Q, the K-functional (3.8.22) belongs to ik. 0

Theorem 3.8.5 displays a peculiar stability of the li-method of interpolation to any other interpolation functors. In order to be able to apply this result, we require one more statement of this type in which the functors of the form KL;;,i = O , l , participate instead of Fi. In order to formulate this statement, we put

447

Computations

(3.8.23)

cp(s) :=

where q and p are defined by

(3.8.24)

1

-

:=

P

1-79 + -19- , Po

Pl

77 := 79p ( 0 < 7 9 < 1) Pl

Further, for given cp and q E [1,00] we put

In this notation, the following theorem is valid. Theorem 3.8.7. (a) If the functions cp; are p;-quasi-concave, pi

< 00, i = 0 , l

and p and 77

are defined above, then

(b) If the functions cp; are q-quasi-concave, 1 5 q

Proof. We shall

5 00,

then

require an auxiliary statement about the L,-functional of

the couple ((Lz;y, (Lz;y). Recall (see Definition 3.1.22) that this quantity is defined by the formula

(3.8.28)

J u t ;f ; (q3’(q;n

:= i n f { ( l l f o l l L ~ )+t(llflllL;;)p~} ~

where the lower bound is taken over all decompositions f = fo

7

+

fi. We denote the left-hand side by L(t). Then the following lemma is valid.

Lemma 3.8.8.

If functions cpi are pi-quasi-concave, i = 0 , l and 1 5 p o , < ~ f E Conv we have

00,

then for

The red interpolation method

448

(3.8.29)

L(t)x

J

min

{ (-)

R+

f(s)

%(S)

,t

2.

(-$$)’I}

S

Proof. In view of

Corollary 3.1.25, the right-hand side of (3.8.29) is equivalent to the L rfunctional of the couple ( L g , L r ; ) so that the left-hand side majorizes the right-hand side. In order to prove the inverse inequality, we shall use Propositions 3.2.5 and 3.2.6 according to which for f E

( L z + 15;;) n Conv and a given

q

> 1, we

have

n

(3.83)

f ( t )I Q

C

f(tzi+l) min(1, t/tzi+l)

.

i=-m

Indeed, in the case under consideration, the last two terms in formula (3.2.20) vanish since pi < 00, and hence ti # Lg+Lp’: (i = 0 , l ) . Let us n o w suppose that E ; ( t ) = 1 if

(

)” I (

f(tzi+l)

and

E;(t)

f(t2i+1)

)’I

9 1 (t2i+l)

~0(t2i+1)

:= 0 in the opposite case. From the fact that cpi are pi-quasi-

concave and that

f belongs to the cone Conv it follows that

Here 7 1 does not depend on i. Further, we put

n

fl(s)

:=

C

(1 - e ; ( t ) ) f(tz;+l)min(l,s/tz;+l) .

i=-m

Since fo, f1 E Conv, we have in view of inequality (3.8.30) and Theorem 3.2.2

Computations

449

From properties (b) of Proposition 3.2.5 and inequality (3.2.21) it follows that

A = 0 , l . Here we put €4 := E ; , ~f = 1- E ; . Taking into account that the functions (p; are pi-concave [see (3.8.6)], we have from these inequalities

for

and (3.8.32)

+

(1 - E i ( t ) )

(

f(t2i+1) cPl(t2i+l)

)")

.

Since the intervals [t2;+1, q t z i + l ] do not intersect pairwise for different i (see Proposition 3.2.5(a)), we obtain from this inequality and (3.8.31) the required inequality

Let us now prove statement (3.8.26) of the theorem. First of all, we apply the reiteration theorem 3.3.11 to the left-hand side of (3.8.26). This gives

where we put

Let us take an arbitrary function in the space XP. Since Kq = K&,we can assume that f E Conv. Then putting for brevity x' := ((Lg&(L,':r),we get

Ilflla

=

I K ( t ; f ;2)I' td

sj'"

The red interpolation method

450

In Chap. 4, we shall prove (see Theorem 4.2.15) that the right-hand side of this expression is equivalent t o the quantity

if 71 and p are defined by the formulas (3.8.24). The latter expression can be computed with the help of Lemma 3.8.8 since f E Conv. According t o this lemma, we then obtain

(f ( s ) p l - m t ) - q

(f(s)pl-pot)’-v

’

cpI(S)p’

In the last equality, we have taken into account the fact that in view of (3.8.24),

P = (1 - V)Po + 77P1 Making the substitution u

:= f ( ~ ) ~ l - mint the inner integral and using

the definition of the function cp [see (3.8.23)). we obtain from the previous relation I

Thus, we have proved that

( K L Z ’ Kq;)9p

KL,..

In order t o prove relation (3.8.27) of the theorem, we apply the reiteration theorem 3.3.11 to its left-hand side. This gives

Computations

451

KL;(KL:o 7 KL:l)

= Kv

7

where we have put

iIJ := Kq-((LFT,(L?T)

.

As before, we henceforth shall take an arbitrary function f in iIJ

n Conv.

Then in t h e notation adopted above, we have

But in the case under consideration [see (3.1.37)],

K ( t ; f ; 2)x L',q(t' ; f ; 2 ) l I '

,

and hence an application of Lemma 3.8.8 with po = pl = q gives

where the function $5 is defined by formula (3.8.25). Thus,

llfllv

= IlfllL$

7

and the relation (3.8.27) is proved. 0

Corollary 3.8.9.

Let us assume that cpo and cpl are quasi-power functions in the cone Conv such that the function cplo := 'p1/cpo is a homeomorphism of

R+which

preserves the orientation. If the operator S, defined by formula (3.8.8) with w := (cpIo)-'

is bounded on

Ly, then

452

The real interpolation method

K p ( K p , K L ; ; ) g KLf .

(3.8.33)

Here the function @ is defined by formula (3.8.25).

Proof. As

was noted in Example 3.5.2, for a quasi-power function w, the

Lr is a quasi-power parameter for any p 2 1. Consequently, the operator S is bounded in L;p',and hence the function cp; is 1-quasi-concave lattice

(see Example 3.8.4(b)). Then t h e conditions of Theorem 3.8.5 are satisfied.

Indeed, only the inequality GK,;;

Pi

7

i = 0,1

7

has t o be established. But this inequality follows from items (a), (c) and (d) of Example 3.8.4. Applying the theorem indicated above t o the left-hand side of (3.8.33), we obtain

Since the functions cp, are pquasi-concave for p := 1,m, they possess this property for p := q as well. Therefore, statement

(b) of Theorem 3.8.7

is

applicable to the right-hand side of the last relation. This immediately leads t o relation (3.8.33). 0

If cp; are power functions, we obtain the following classical result as a corollary. Theorem 3.8.1Q. (a)

(Lions-Peetre). If the numbers Q and 9; E ( 0 , l ) (i = 0 , l ) and do # d l , then

Here 6 := (1- v)d0

+ ~9~

(b) (Peeire). The above relation is also valid for 80 = pi

< 00,

i = 0,1, and

dl, provided that

Computations

453

_1 -- 1-"+1 P

Pl

Po

Proof. (a) In the case under consideration, the operator S , is equivalent (on positive functions) t o the operator S (since w ( t ) = t

1

m ) . Therefore, in order

t h a t Corollary 3.8.9 be applicable, we must only verify that the operator

S is bounded in the lattice L;. For this we should recall the classical Hardy's inequality (see Lemma 1.9.13) (3.8.34)

5 00, X < p

and

( H f ) ( z ) :=

12 j

Here 15 p

f(t)dt.

0

From (3.8.34) it follows a similar inequality (with the same constant) for

the conjugate operator 00

(H*f)(z) :=

J f(W

7

I

where X

>p.

An application o f these two inequalities, taking into account the identity

indicates that for 0

< q < 1the operator S is bounded

in

L:.

Applying now Corollary 3.8.9 and computing the functions @ for the case cpi(t)

:= tBo and

cp(t) := tq, we complete the proof.

(b) In this case, statement (a) o f Theorem 3.8.7 is applicable. The computation of the function cp appearing in this theorem for cpi(t) := t B i ,

90 =

291,

and o f the exponent p satisfying condition (3.8.24) gives the

required result. 0

The real interpolation method

454

There exist other corollaries of the above results, which are useful for applications. We recommend t o the reader, by way of an exercise, t o consider

the case when cp;(t) is equivalent t o the function t9*Ilntla*,0

< iJi < 1 in

the vicinity of zero and infinity, and also the case cpi(t) := min(t9*,tn*) or

y ; ( t ) := rnax(ttP*,t4*), 0 < iJ;,q, < 1. In these and similar situations, t h e following generalization of the Hardy inequality can be used for the evaluation of the operator S,. Proposition 3.8.11.

Let w,v > 0 be arbitrary weights and 1 5 p

5

00.

Then the following

statements are valid. (a) The norm of

H as an operator from L; into Li is majorized by the

quantity

K :=

Here K

SUP r>O

(7 (i wPdx)lip

v-p’dx)lip’

5 IlHll 5 K(p)’lp(p’)llp’ for ‘ < p <

00

and K = lIHl1 for

p = l,0O. (b) The norm of

H’ as

an operator from L s into

Li is majorized by the

quantity

Here Kl 5 llHll p := 1,m.

B. Let

5 K ~ ( p ) ’ l ~ ( p ’ ) ’for / ~ ’1 < p < 00

and KI = 1 1 8 1 )for

us consider some examples illustrating the theorems on the con-

nection between the functions o f t h e

K- and J-methods,

contained in Sec.

455

Computations

3.5. We shall be interested only in the case when w E Conv is not a quasipower function. Otherwise, everything is reduced to Theorem 3.5.3. In order t o formulate the result, we put for a given function w E Conv (3.8.35)

G(s) :=

:}

-1

rnd(t)/w(t)

.

Proposition 3.8.12. (a) If 3

$ Lf, i = 0,1,then

(b) If w $ L:, i = 0,1,then

where3 :=

Sw.

Proof. (a) If Q is a parameter of the AC-method and

then in view of Theorem 3.5.5(a)

where we put

Indeed, the right-hand side of (3.8.38) is equal to IlK(.; f ; & ) l l ~since S-'(9) = K Q ( ~ , If) .now 9 := Ly,the computation of the right-hand side of (3.8.38) gives

s-'(L;) = L?

.

456

The red interpolation method

3.5.5 (3.8.36)are satisfied if

T h e applications of the arguments used i n the proof o f Theorem shows that when Q Lz,

Ly, the

:=

conditions

fz LI, i = 0 , l .

(b) W e now put Q := L&, where b :=

(3.8.39) lim t-0

o( tt)=

Sw. If w is such that

lim ( S w ) ( t )= +m ,

t++m

(3.8.36)is obviously satisfied for 4. Consequently, for such a parameter Q relation (3.8.37)holds. Let us compute S-'(L'&)).

then condition We have

The last equality is based on Lemma

3.4.10.Since according

3.4.5the right-hand side is equal

JL%(&), we get

S - - l ( L L )= JL&) This equality, Corollary

to

to Lemma

.

3.4.6and the equality (3.8.37)yield

(3.8.39)can easily be derived from the condition w $! L i , i = 0,1, and we leave it t o the reader t o carry out this simple verification.

T h e relation

C . Let us consider t w o examples illustrating the theorems on density and relative completeness i n Sec. Let 15 p

< 00,

(3.8.40) T~ :=

3.6.

the function w E Conv and

I min(1, l / t ) w ( t ) l P -

Cornput ations

457

Then the following proposition is valid. Proposition 3.8.13. For any couple

x', the set A(x')

is dense in the space

JL;(~).

Proof.Since in view o f the Holder inequality Ilfllc(z,, 5 X JIlfllL; the space L; -+

C(zl),and hence is a parameter of the 3-method. Further, {fx(,,-i,,,); f E C(i1), n E

it can easily be verified that the set of functions

N }is dense in L; for p < 00,

and hence A(i,) is dense in L:.

It remains

t o apply Corollary 3.6.2. 0

Let us now suppose that 1 5 p

5

00,

that the function w E Conv and

that

Then the following proposition is valid. Proposition 3.8.14. For any couple

x', the space K L ; ( ~ 'is) relatively complete in C(d).

k f . According t o Lemma 3.6.8, it is sufficient t o establish that ( L r y i s relatively complete in c(L,). Let fn -+ f in ~(i,). Since := K ( .; f ; Em) and Ilfllc(t,,= K(1; f ; L,) = f = 1, -+ j uniformly on each interval [a,b]with a > 0 and b < 00. Therefore -0

-

Passing to the limit as a + 0, b

in

-+ +00

1

and taking into account Definition

2.2.15 of relative completeness, we obtain the equality ((L;r>' 0

= (L;y

458

The red interpolation method

D. Let us consider examples illustrating the application of the duality theorem in Sec. 3.7. We suppose (3.8.41) is fulfilled. Proposition 3.8.15. (a) If 1 5 p

(3.8.42)

5 00 and w E Conv,

Ji;

Kp,

,

where G(t) := l/u(l/t).

(b) In the same notation and assumptions,

provided that ( L i r is a nondegenerate parameter of the K-method.

Proof.The first statement followsfrom Theorem 3.7.2, the equality ( L i ) + = L;,, and Remark 3.7.8. The second statement follows from Theorem 3.7.6. 0

Remark 3.8.16.

It can easily be verified that the nondegeneracy of (Lj;'rfollows from condition (3.8.41).

Comments and supplements

459

3.9. Comments and Supplements A. References Th e K-me thod o f interpolation has i t s origin in the classical Marcinkiewicz theory and its generalizations proposed in the work o f Zygmund and M.

1.11). A transition t o arbitrary Banach (see the method o f r-diagrams). spaces is made in the work o f Gagliardo [1,2] Cotlar (see in this connection Sec.

A n equivalent but more convenient approach for applications, based on the concept o f t h e K-functional, was proposed later by Peetre

[l] (see

also

Ok-

[1.21). A predecessor o f the second component of the real method, viz. o f the 3 -method, was the method o f traces worked o u t by Lions [l-31. lander

Later Lions and Peetre proposed the method o f means which was equivalent t o the method o f traces (the first publication on this method is contained in t h e paper by Lions and Peetre

[l]; a detailed account is made i n the fundamental work o f Lions and Peetre [2]).The achievements of these authors will be described in detail later. The concept o f t h e J-functional and the 3 - m e t h o d o f interpolation based on it were proposed in the work by Peetre

[I].

The first complete description o f the real method o f interpoaltion for

power parameters is contained in the book by Peetre [7],based on the course

1963. This material can also be found in the books by Butzer and Berens [l], Bergh and Lofstrom [l], Triebel [2], and in the paper by Peetre and Sparr [l] (see also the review by Brudny'i, Krei'n and Semenov [l], where the contemporary state o f art in this theory

o f lectures delivered by this author in

is outlined). Sec.

3.1. As was noted above, the concept o f the K-functional was

introduced by Peetre and independently by Oklander, while the J-functional and the E-functional were introduced by Peetre. In these papers, most o f the properties o f these functionals (sometimes in a less accurate form) are

3.1.11and 3.1.14taken from the work by Behrens [l] and also Proposition 3.1.10 and Corollary 3.1.13,which apparently were not formulated explicitly earlier. An exact computation o f the K-functionals for the couples (Proposition 3.1.17), also described. The only exceptions are Corollaries

460

T h e r e d interpolation method

3.1.18)and (C, C') (Corollary 3.1.20)is carried out [1,8].Here Corollary 3.1.20is derived in a different way on the basis of a more general proposition 3.1.19proved by Yu.A. Brudny'i. The K-functional of the couple (Proposition 3.1.17)was computed by V.I. Dmitriev. Proposition 3.1.21was obtained in the paper Lions and Peetre [2] (see Aronszajn and Gagliardo [l]). The concept of the L-functional and the (Lll L,)

(Proposition

by Peetre

z,

computation of this functional for concrete couples presented in this section are contained in the paper by Peetre

[lo]. Corollary 3.1.29was obtained by Proposition 3.1.27generalizing it was put

[l], while (21.The main definitions concerning t h e category of Abelian groups are given in the paper by Peetre and Sparr [l]. Example 3.1.35, Holmstedt and Peetre

forth by Sparr

which is important for applications, is presented in the paper by Brudny'i and Krugljak

[l].

Sec. 3.2. The fundamental property of the K-functional (K-divisibility) established in this section was discovered sixteen years after it had been

[2]). The detailed description is contained in the deposited manuscript by Brudny'i and Krugljak [3].One more introduced (see Brudny'i and Krugljak

proof of this theorem was proposed by Cwikel [4] without a prior knowledge of the work o f Brudny'i and Krugljak

[3]. Cwikel's arguments are based on

ideas close t o those used in the first proof in the cited manuscript (in particular, a version of Propositions

3.2.5and 3.2.6was employed). Nevertheless,

he managed t o reduce the estimate o f the K-divisibility constant K from 14 to 8. A slight refinement of the proof in Brudny'i and Krugljak [3] (the sum o f t h e last two terms appearing earlier in formula (3.1.20)had the form cp(O)+cp'(+ca)t)

leads t o the same estimate of this constant. Theorem 3.2.2

was obtained by Asekritova

[l] for

a finite number of terms. In t h e general

case, it was proved in Brudnyi and Krugljak [3]. In the same paper, the theorem on K-divisibility in the case of a finite number of terms was proved for normed Abelian groups; furthermore, an example close t o Example

3.2.11is

considered. This theorem was later proved independently in the paper Nils-

[l]. The equality K ( L ~ , L , )= 1 in Proposition 3.2.13,which remained [l]. For t h e unproved statement contained in item (b) of this proposition see Krugljak [2]and Podogova [l].

son

unproved, can be found in A.A. Dmitriev

Comments and supplements

461

Sees. 3.3 and 3.4. The definitions of the functors

Kc and Ja with ar-

bitrary parameters was proposed in t h e book by Peetre [7].Theorems 3.3.4 and 3.4.3were obtained by the authors in September 1978 (see Brudny’i and Krugljak [2,3]).Thus, an unexpected relation between the “abstract”

[l] and the “concrete” interpolation methods was established for t h e first time. In 1980,Janson [2]independently proved this fact for K t ; and a similar fact for the complex interpolation method. Theorems 3.3.11and 3.4.9,proposed by Yu.A. Brudnyi’ and N.Ya. KruglAronszajn-Gagliardo theory

jak, is presented. These theorems were preceded by the classical reiteration theorem for power parameters, obtained by Lions and Peetre [2] (see Theorem 3.8.10).The method developed by these authors does not allow one t o extend this theorem t o a more general situation. Separate statements generalizing the Lions-Peetre theorem mainly t o the case of quasi-power parameters of special kind are contained in papers by Peetre [7],Bennet [3], Kalugina [l], Gustavson [l], Asekritova

[l] and many others.

Theorem 3.3.15is implicitly contained in the work of Brudnyi‘and Krugljak [3].The nontrivial part of Theorem 3.4.9is also proved there. A new proof of this theorem was proposed by N.Ya. Krugljak. The remaining results presented in these sections are proved (sometimes in an equivalent form) in Brudny’i and Krugljak [3]. The only exception is the fundamental lemma

3.4.10established by Sedaev and Semenov [l] and Theorem 3.4.12,whose nontrivial part was established by Janson (21 for a quasi-power parameter and by Ovchinnikov [7]for the general case. Sec. 3.5.The first result concerning the relation between the functors of

the

K- and 2-methods (the

equivalence theorem) was established by Lions

and Peetre [2].Their statement is equivalent t o Theorem 3.5.3for @ := LE,

0

< 9 < 1. The

method of the proof applied by these authors immediately

leads t o a more general theorem 3.5.3. This is noted, for example, in the paper by Dmitriev and Ovchinnikov

[l]. The necessity of the condition of

boundedness of the operator S in the condition of this theorem was indicated in Brudny’i and Krugljak [3]. In this paper, the proofs of the equivalence theorems 3.5.5and 3.5.9,announced in BrudnyY and Krugljak 121, as well as the proof of all other results contained in this section, are given.

462

The real interpolation method

See.

9.6. The first general theorems on density and relative density

[3].Th e stronger versions of these theorems, contained i n Theorems 3.6.1and 3.6.7,were obtained by Aizenste’in [l] who used for this purpose methods of the duality theory. Theorem 3.6.7is proved by this author under the condition of the nondegeneracy o f the parameter 6.A direct proof o f these theorems was proposed by N.Ya. are formulated in Brudny’i and Krugljak

Krugljak. -+

+

3.7. The classical duality theorem (X,,)’E (X‘),,,, 1 5 p 5 00, was established by Lions [5].The extension of this theorem t o quasi-power parameters of special type is contained in the paper by Scherer [l]. Peetre [16]supplemented the Lions theorem by showing that Z for p < 1. Th e general duality theorems 3.7.2and 3.7.6and corollaries to these theorems given in this section were obtained by Brudny’i and Krugljak [2,3]. We have simplified the proof o f the first of these theorems w i t h the help of Sec.

(zfi,)’(z’),,

the general duality relation for computable functors, developed by Janson

[2](see Theorem 2.5.13). The statements indicated in Remark 3.7.10are contained in Aizenste’in [1,2]. Sec. 3.8. Most of the results described in the section are taken from

[3].The exceptions are Proposition 3.8.6proposed by V.I. Dmitriev (see Dmitriev, Kre’in and Ovchinnikov [l]), the classical result obtained by Lions and Peetre [2](Theorem 3.8.10(a)), the result obtained by Peetre [2] (Theorem 3.8.10(b)), and the generalized Hardy inequality (Proposition 3.8.11),established by Talenti [l] and Tomaselli 111. T h e first version of Theorem 3.8.5was established by Janson [2].Janson’s statement Brudny’i and Krugljak

has the form

where

x

:=

‘po+(’pl/’po)

and (pi := G F , . This relation is valid under

the assumption that (pl/(po. ‘p, and

+ are equivalent t o quasi-power concave

functions. A slightly stronger version o f Janson’s theorem was anounced by Astashkin

[l].

Namely, the assumption concerning the function ‘p := yo/’p1

(:= u-’) is replaced

by the condition

i 9 ds

S

u(t). In this

paper, it

0

is stated that a further weakening of the conditions of Janson’s theorem is

Comments and supplements

463

generally impossible. It can easily be seen that Janson’s theorem can also be obtained from Theorems 3.8.5 and 3.8.7.

The real interpolation method

464

B. Supplements (Computation o f K-functionals and Real Method Spaces) 3.9.1. General Approach The supplement contains the main results connected w i t h the computation of the K-functionals and K-spaces. In this section, we shall consider some general results, while the next sections will be devoted t o the facts referring t o concrete couples important for applications. The computation of the K-functionals i n concrete situations is a complicated nonlinear problem. The cases of exact solution o f this problem are quite rare (the most important o f them are considered i n Sec. 3.1). However, i n applications it is often sufficient t o know the I<-functionals of a given couple only up t o the equivalence. The computations are considerably simplified when it is possible t o linearize the optimization problem emerging in this case. In this connection, it is expedient t o introduce

Definition

A couple

K

:

(Peetre

[+TI).

2 is referred t o as I~-Zinearitableif there exist operator functions

lR+ + L(C(J?),X;),i = O , l ,

such that

and for some constant

y > 0, all z E C ( 2 ) and all t > 0, we have

(3.9.2)

+ t IIVI(t)~llXl5 y w t ; ; x’)

IIVo(t)zllxo

’

It follows from (3.9.1) that, conversely, the K-functional o f z does not exceed the left-hand side o f (3.9.2), and hence is equivalent t o it. A’-linearizable couples are quite rare. The most importnat among them is the couple

(3.9.3)

w;),where k; is a “homogeneous”

(Lp,

llfllw;

:=

SUP lal=k

Sobolev space. Thus,

IlsafllP .

The statement about the I<-linearizability o f this couple is essentially due t o Lions (see also Lions and Peetre [2]). On the other hand, Peetre (14)noted that the couple

(Zp, Z q ) ,

p

# q , is not of this type.

An interesting example is

Comments and supplements the couple

465

( B ( w ; M),LipM). Here M is an arbitrary metric space w i t h a

metric r ,

and

B ( w ; M ) , where w : R++ R+U {+m} is an arbitrary weight, is

defined by the norm

(3.9.5)

I(~~(B(~;M)

:=

> 0;

inf { A

If1

5 Xw}

.

It turns out (see Brudny'i and Schwartzman [l])t h a t the couple under consideration is K-linearizable iff for any closed subset linear extension operator Lip M' Thus, for

M

4

M' c M there exists a

Lip M .

c R"this couple is I<-linearizable.

wrong for infinite-dimensional

However, this can be

M's.

Another useful concept is contained in Definition (Peetre

[If!]).

A couple? is referred t o as a K-subcouple of a couple x' i f f is isometrically embedded into

(3.9.6)

2 and, moreover,

K ( . ; y ; ?) = K ( . ; y ; 2)

for any y E

c@).

0

Let us now consider a few examples. According t o equality (3.1.32), the

x' is a K-subcouple o f the couple X'' if it is isometrically embedded into 2'. For example, the couple (C,C1) is a I<-subcouple o f the couple

couple

(L,,Lip 1). Namely, it was established there tha t if the couple

is regular,

~ ( x is' )a K-subcouple o f the couple x'". 2 c f is a retract of the couple ?, and the retractive map R : p + x' has the norm equal t o unity, then x'

then

Finally, it should be noted that if

is a K-subcouple of

?. This directly

follows fro m the definitions.

Let us consider the following result o f principal importance, which, however, has not found an application so far.

The real interpolation method

466 Theorem (Peetre [12]). Any regular couple

x’

is isometrically isomorphic t o a K-subcouple of the

( B ( w o; M ; X,), B ( w 1 ; M ; XI)) for an appropriately chosen metric space M and weights w ; . couple of spaces of vector-valued functions

0

The space B ( w ; M ; X ) is defined by the norm (3.9.1.5),where the absolute value

If1

is replaced by the norm

Ilfll.

Concluding the section, let us outline a number of questions having their origin in Holmstedt’s formula for t h e K-functional we can obtain the I<-functional of the couple

0 5 Qo < Q1 5 1 and 0 initial couple

(3.9.7)

2 . Namely,

K ( t ;z ;

uniformly in

< p0,pl 5

t > 0.

P)

(l

x

03,

[s-%qs;

?

111.

:=

Using this formula,

( ~ ~ o m r J ? ~ where lpl),

if we know the K-functional of the

”)

1 /PO

z;

z)]”

+

S

Here X

The quantity on t h e right-hand side of (3.9.7)can be shown t o be equivalent t o the K-functional o f the concave function

K ( . ; cc ; 2)in the couple

+

@ := (Lz:,L;;). Therefore, formula (3.9.7)can be written in the form

(3.9.8)

K ( t ; z ; ?)

I<(t ; K ( . ; cc ; 2); 5).

This circumstance was noted by Dmitriev and Ovchinnikov that formula

[l], who showed

(3.9.8)is also valid for quasi-power parameters

A general formula for the I<-functional in the couple arbitrary parameters was obtained by Brudnyl’and Krugljak

a,.

P in t h e case of [2,3]and has the

form

(3.9.9)

K ( . ; cc;

F) = I<(.; K ( . ; 2 ;2);(&o,&l)) .

It is equivalent t o the reiteration theorem for the K-method, and hence also holds for normed Abelian groups.

Comments and supplements

467

3.9.2.Couples o f Banach Lattices (3.9.9)emphasizes the importance of the calculation o f the K -

Formula

4

functional for the couple @ o f Banach lattices over a given measurable space for concrete applications. It can easily be verified t h a t for any exists a measurable subdivision

a+= At U Bt such t h a t

(3.9.10) K ( t ; f ; 5 )

11% + t I l f x B t (1%

t.

uniformly in

fo

+

f1,

Indeed, if

IlfxAt E

> 0,

fi

E

@iare

t > 0,

2 0

such t h a t fi

and

there

If1

=

where

llfolloo

+ t llflll0,

5 (1 + E)K(t ; f ; $1

>

it is sufficient t o put

At := {. E R + ; fo(z) > (1

+

E)fi(Z))

.

(3.9.10)can be made as close to 2 as desired. A more profound is the statement that the sets At and Bt can be chosen

The constant in

t.

so th at they depend monotonically on and Nilsson

This result was anounced in Cwikel

[2)and will be proved below (see t h e proof of Lemma 4.4.30).

Apparently, it is impossible t o obtain more comprehensive results in the general case. In some cases, the sets

At and B, from (3.9.10)can be chosen t o be

independent o f f . Namely, according t o a result obtained i n Nilsson this situation may take place when

Z,such

lattices over

5 is

[l],

a linearizable couple o f Banach

that the basis of &functions is unconditional in each

ipi, i = 0 , l . Thus, for the operators

V;

i n Definition, see

(3.9.1),we can

can take diagonal projectors in the case under consideration. Let now

@pi

be a symmetric space (see Definition

2.6.13).W e can assume

that its norm is defined by the formula

Ilfllo,

= llf*lllyi

t

where Q, is a certain Banach lattice over

R+. Then it follows from formula

(3.9.9) and from the definition o f the decreasing rearrangement f ' that for any

t > 0 there

exists a measurable subdivision

R+= At U Bt, such t h a t

The real interpolation method

468

uniformly in

t.

The cases when

R++ R+is a

At has the form o f the interval (0, V ( t ) ) where , V :

monotonically increasing function, are important for appli-

cations. Th e first result of this kind was obtained by Brudny'i and Krugljak

[l]. In this paper, a general case o f the couple of E-spaces is considered (see Sec. 4.2.C for the definition). (Some methodical refinements o f the proof presented in this paper and new interesting applications are contained in Pietsch [2].) In the paper Brudny'i and Krugljak it is assumed that the function w := (pl/(po, where (pi is a fundamental function' of the symmetric space @i, operator

is a surjection and that the operator

Qn E L(@f,@;),R

E

Z.Here

P,, E L(@&@f),while the

ad is the

Banach lattice @, i.e. the space o f sequences

f

:=

discretization of the ( f n ) n E ~ , defined

by

the norm

P,, and Q,, are the operators o f multiplication by the characteristic i 5 n } and its complement respectively. In functions o f the set {i E Z ; this case, the function V i n (3.9.11) is defined by the formula Further,

(3.9.12)

V ( t ) := SUP{^; IIP,,llqot 5 t } ,

In particular,

(L,9)d

Zz, where

Ilfll,;

:=

t E R+ .

(c,,~ (2-"sfn(p)1'p,

and an

application o f the inequalities of Jensen and Holder allows us t o obtain a formula for the K-functional of the couple

(Lwpo,L P l q 1 )derived , by Holmstedt

[l]fro m the statement formulated above. Using the concepts of pconvexity and q-concavity (see Lindenstrauss and Tzafriri [2]), it is possible to establish similar results for other couples of symmetric spaces (such as Orlicz spaces) as well. Finally, it should be noted that under the assumption t h a t the function w is monotone and its lower index is strictly positive, the function V in

(3.9.12) is equivalent t o w - I . 2This means that pi@):= Ilxsll*,, where m e s s = t (0 < t

< m).

Comments and supplements

469

Other results of this type were obtained in Milman Nilsson

[l], Maligranda [2],

and Mastylo

111. A

[l], Torchinsky [l],

generalization of the result

described above which is contained i n the paper of Brudny'i and Krugljak, t o the spaces o f the K-method is contained in Asekritova

[l].

A new approach t o the proof o f the formula

is contained in the paper by Arazy

[l], where the function V is defined as

the inverse function o f the fundamental function o f the space o f multipliers

M(@o,@ I ) .

Here

It should be noted, however, that all the results described here do not differ much fro m the results obtained by Holmstedt. example o f the couple o f Orlicz spaces that for a convex function

M

:

This is confirmed by the

( L M ~ , L M It ~ )should . be

recalled

R++ R+,

Another important class o f the couples o f the Banach lattices, whose

K-

and J-functionals are important for applications, are the couples o f weighted L,-spaces.

We shall briefly describe the results obtained in this direction.

3.1.24contains a formula for the L,,,,,-functional of the couple (Lpo(wo), L p l ( w l ) ) while , in Sec. 3.8 Peetre's theorem is formulated, which

Proposition

describes the relation between the spaces o f the G m e t h o d and similar spaces defined by replacing the A'-functional

by the L,,,,-functional

result o f this kind is described in Sec.

4.2). The result obtained by Peetre

(the general

[lo] has the form

Here 6 E ( 0 , l ) and r

E (0, +a31 are arbitrary and

(3.9.15) p := (1 - U9)po + 6 p l

,

q := 19p/p1

,

q := pr .

470

The real interpolation method

Thus, i n the case under consideration the knowledge o f the L-functional is sufficient for computing the space

Ka

with power parameters

0. The

situation for non-power parameters has not been studied sufficiently, the only exception being the case when po = p l . Indeed, i n this case

(3.9.16)

K ( t ; f ; L p ( w o ) , L p ( w ~ x) ) 11 f i n ( w o ( t w i ) f I I ~ ,

[see formula (3.1.39)]. It is useful t o note that it is one of a few K-linearizable couples of Banach lattices (see Gilbert [l]). Returning t o formula (3.9.14), we note that, together w i t h Proposition

3.1.24, it leads immediately to the classical result of by Lions and Peetre [2]:

Here p

:=

p ( 6 ) is defined by formula (3.9.15) and L p ( w ;X) stands for

the space o f strongly measurable X-valued (classes of) functions, w i t h the norm

In the “nondiagonal” case p

# p(t9), formula (3.9.17)

does not permit a “na-

tural” generalization. Namely, Cwikel [5] showed that in this case the space on the left-hand side of (3.9.17) is not isomorphic t o any space of the form

{f;JJf(z)JJa,, E S}, where S is an arbitrary

class 0s measurable functions.

Nevertheless, the spaces o f this type are often encountered in applications. The description of their structure for

X o = X I = nt is given in Peetre (51, in

which it is established that the space (Lp(wo),L p ( w I ) ) coincides with the 89

]connection w i t h problems of spectral spaces introduced by Beurling [lin synthesis. A further advancement in this direction was made by Gilbert [l] who analyzed the case of spaces ( L p ( w ~ ) r L p ( q )and ) established their 89

relation with a generalization o f the Beurling spaces, which was proposed and also by Lizorkin [l] and Freitag (1,2]. Th e Lizorkin-Freitag by Herz [l], formula has the form

Comments and supplements

471

where the right-hand side contains the Lorentz weighted space

(Ilfll~,,(~)

:=

and

IlfWllL,,)

Here it is assumed that po

# p l , p0,pl

E (O,+w), p

# q and p

:= p(19).

Finally, Persson (11generalized the cited result by Peetre and Gilbert. In particular, he proved that

Here p :=

p ( 9 ) , the weight w y is defined by the equality

and Q is a set o f functions 'p i n the unit sphere o f the space for which

'p(t)t' increases,

number

> 0 is not fixed).

E

while

cp(t)t-'

Ll(R+;

decreases for a certain

E

dt -) t

> 0 (the

This author also considered the space o f weight

couples for vector-valued Lp-spaces. These results can be used for computing "concrete" spaces o f the real

method for Banach lattices (Lorentz spaces, Marcinkiewicz spaces, Orlicz spaces, and so on) (in this connection, see, for example, BrudnyT, K r d n and Semenov [l]as well as Merucci [l)and Person

111).

W e note here only a few

results which are beyond the framework of this approach. For this we define the Lorentz space, see Lorentz [l],A(p), where cp E Conv and p(0) = 0, by putting

(3.9.20)

)IfllA(y)

:=

/

f*(s)dp(s) *

R+ T h e right-hand side of this formula contains a Lebesgue-Stieltjes integral. Then the following equality is valid (see Lorentz and Shimogaki [l]and Sharpley [l]):

The real interpolation method

472

The duality formula in Sec. Marcinkiewicz space

3.8 allows us t o obtain a similar result for the

M(cp), where cp E Conv and cp(0) = 0. It is defined by

the norm

and is the associated space to the space h ( y ) and isometric t o A(y)* (see Lorentz [2]). In this way we have (3.9.24) where

KLY, (M(cpo),M(cp1))

= Wcp) 3

y is defined by the same equality (3.9.23).

Another exact result was obtained by Bennet [l]and refers to the couple

(JL, L1). Namely, K ( t ; f ; Ll,,L1)

=

sup

sf*(s) .

o<s
Finally, the same author (see Bennet [2]) calculated the spaces ( L ,Lln'

L)s,

and ( L In+ L , Loo)8,, where L(ln+ L)@stands for the Orlicz space L M w i t h

M ( t ) := t [max(0,1nt)18, consisting o f functions defined ( 0 , l ) . For example,

( L ,L ( h +L ) )

89

=

L ( h + L)ff for q := 1 ,

on the interval

o < 19 < 1

for 9 := I / q ,

q E

[I,+.)

Comments and supplements

473

Similar results can be obtained from the general theorem o f reiteration. For this purpose, it is sufficient t o make use o f the following fact which is equivalent t o a result obtained by Hardy and Littlewood [2]:

L(hfL) where

Ka(L1,L,) ,

@ := L(0,l).

Since the latter statement can easily be verified to be valid if placed by

L,, 1 < p < 00,

L1 is re-

the statements made i n Bennet’s paper can be

obtained in this way for the spaces

L,(lnt L ) as well (see in this connection

one more paper by Bennet and Rudnik [l]).

3.9.3. B M O and H, Many problems i n harmonic analysis and the theory of operators acting in function spaces necessitate a “correct” extension o f the scale o f spaces

(Lp)l
03.

The natural extension of this scale involves large losses. Th e following three circumstances should be noted. (a) The lack o f nontrivial continuous functionals on the space L, for 0

<

p < 1.

(b) The unboundedness in L1 and L , of many important operators which nevertheless act boundedly i n the spaces

L, for 1 < p <

03

(see, in

particular, Example 1.10.17). (c) The lack of a useful analog o f the “weak” space L, for p

:=

00

and

the fact that this space i s not normed when p := 1. This circumstance prevents us from applying the Marcinkiewicz interpolation theorem to many interesting cases. Example o f this kind have led mathematicians t o the conclusion that

L, scale is associated w i t h the Hardy spaces H p ( D ) (0 < p 5 M) likewise introduced by F. Riesz in 1923. W e recall that the space H , ( D ) consists o f those functions F analytic in the unit disk W := { z E C ;1.1 < 1) for which

a “correct” extension of the

The real interpolation method

474

The main idea is that t h e set of real parts of the boundary values of functions H p ( D ) should be taken for t h e space o f the required scale.

from

When 1 < p

< 00,

the function

(3.9.26)

f(8) := lim ReF(re")

belongs t o

Lp(T) and

(3.9.27)

llfll,

r-1

( F E H,(D))

N llFllHp(D)

(M. Riesz [2]).3 Thus, the realization of this idea may indeed lead t o the required extension o f the scale ( L p ( T ) ) . The spaces o f this new scale l
The theorem under consideration states that for certain constants C1,Cz > 0, which depend only on u and p , and all F E H,(ED) (0 < p < m ) we have4

3We recall that for any p > 0, the limit on the right-hand side exists for all 0 E T. Unfortunately, for p < 1 this limit c a n be equal to zero alrnast everywhere even when F # 0. In this connection, we assume for p < 1 that the limit in (3.9.26) is taken in the sense of the distribution theory, and that the left-hand side of (3.9.26) is an element in ?)'(TI. 4The simpler left inequality (3.9.29) was proved by Hardy and Littlewood [2].

Comments and supplements

475

When f in (3.9.26) belongs to the space L l ( T ) , the function u :=

ReF

is defined with the help of the convolution o f f with the Poisson kernel 1- r L P,(9) := (0 5 r < 1, 9 E T). Thus, it becomes 1 21. cos 9 rz possible t o determine the space H p ( T )directly in the n-dimensional case in

+

-

the following way proposed by Fefferman and Stein [l]. Let f be a tempered distribution (f E S', see Schwartz [l]),and for all z E R", t > 0, there exists the limit

Here P,(z):=

ent

(t2

+ Iz(z)("+W is the Poisson kernel corresponding t o the

domain El:++'. Let us define the maximal function (3.9.31)

f i x ) :=

SUP

!(Pi*

f)(z)l

I+-vl
and with i t s help define the space Hp(Bn)by

IlfllH,(lR~,

(3.9.32)

:=

Ilfll,

(0 < P < 00) .

Then the analog of the classical theorem by M. Riesz allows us t o conclude that

H p ( B " )r Lp(El") ,

1< p

< 00 .

Therefore, we indeed obtain an extension of the scale

(Lp)l
Instead of the maximal function (3.9.31), it is often convenient t o use another maximal function (like the Luzin S-function of the Littlewood-Paley g-function).

In the paper by Fefferman and Stein mentioned above, the

equivalence of definitions of this type is proved and one more equivalent definition based on the "grand" maximal function is given: (3.9.33)

f # ( z ) = sup VSA

sup Ipt Iz-~l
* fl .

Here cpt(z) := t - " p ( z / t ) and the class A consists of those test functions5 cp for which 51.e. such that supI ((1

+ Izl)' I(~'"V)(Z)~) <

00

for all k E N and all a E

Zq.

476

The red interpolation method

for a certain quite large but fixed

No.

For some reasons (one o f which associated w i t h duality considerations will be given below), the space the John-Nirenberg

[I] B M O

H,(BT)

should in this scale be replaced by

space. Recall that for

Here Q is an n-dimensional cube and

f~

:=

& QJ

f

E L'p"(B2")

fda:.

One of the convincing arguments in favour of such a replacement is the following fundamental fact (due to Fefferman):

BMO

(3.9.35) H i ( R " ) *

To be more precise, for each linear functional f E H1(Rn)*,there exists the unique function q E B M O , such that =

(f7s)

J

W d X

nfor all functions for which the integral on the right-hand side is defined.6 In

addition llfllH3(n")*

uniformly in

= ll(PJIBMO

f.

After this preliminary information,'

let us pass to the subject under in-

vestigation. Concerning the first results on interpolation of spaces

Hp(43),

obtained by Thorin, Zygmund, Salem and Calderbn, see the works mentioned i n Sec. 1.11.5.We consider here only results involving the real method. One o f the first was the result by Riviere and Sagher [l],who proved that 61f f E H l ( R " ) and p E B M O , then f p generally does not belong to L l ( R " ) . 7A detailed discussion of the theory of the H p spaces is contained in the review by Coifman and Weiss [l].

477

Comments and supplements

In particular, it follows from this relation and from t h e reiteration theorem

that

= LP 1 for - := -+--. 1 (Hl,L P l ) 9 P 1-29 29

Pl

+he next important advancement was made in the work by Fefferman, Rivikre and Sagher [l],where it is shown, in particular, that

Here 0

< po < pl < 00, 0 < 19 < 1 and

1

-

1-29 29 =Po Pl

+-.

A significant feature o f this work is t c e computation o f the K-functional ( H p o ( R n )H, p , ( R n ) ) .Namely, we have

of the couple

(3.9.38)

K ( t ;f ; H , , H p , )

M

K ( t ;f # ; L p o , L p l )

f and t > 0. Here f # is defined by formula (3.9.33). This formula remains in force also for pl := 03 if we replace Hpl and L,, by L,. This paper established t h a t equality (3.9.37) is valid also for pl := 00 if we replace H , by L,. A stronger version of this result was obtained by Hanks [l].According t o the theorem obtained by this author, uniformly in

(3.9.39)

(H,, BMO)s*

< 00

1

Hp . 1-19 -

. Relation (3.9.39) remains in force also and - := P Po when Hpo is replaced by L, and H p by L, (see Hanks [l]). Here 0 < P O

It was shown by Peetre [20] that t h e proof of statement (3.9.37) can be considerably simplified if we use the "atomic" representation o f the space

H p ( R n ) ,proposed by Coifman (11for n := 1. A generalization for the case n

> 1 is contained

in the paper by Latter [l].In view o f the importance of

this result and i t s close relation t o the theory o f interpolation o f the space

H p , we shall consider it in greater detail. Let us define a p a t o m (0 < p 5 1) as a function a on E2", whose support lies in the cube Q c E2" and which satisfies the conditions

The r e d interpolation method

478

1

for all multi-indices cy for which la(I [n(- - l)]

(integral part).

P

Theorem. A tempered distributation

f belongs to H p ( R " ) (0 < p 5 1)

iff there exists a sequence ( a i ) ; € p o~f p a t o m s and a sequence of numbers (X;)*€N

E ,Z such that

f =

c

X;a;

idV

(convergence i n the sense of distribution theory). In addition

uniformly i n

f

0

It remains t o note that for p := 1, the above theorem is exactly equivalent t o the Fefferman duality theorem formulated earlier, i.e. to t h e identity

H; = B M O . Continuing t h e account, let us consider the result obtained by Bennet and Sharpley (11 who calculated the K-functional of the couple (L1,B M O ) . Namely, if in the notation used in

(3.9.34)we put

I

\

If,

then

(3.9.40) K ( t ; f ; L1,B M O ) FZ t(f?)*(t) uniformly i n f and t of g.

> 0, where,

as before, g* is the decreasing rearrangement

Comments and supplements

479

Finally, in Devore [l]and in Jawerth [l]the K-functional of t h e couple ( H I ,B M O )

is computed.

The results obtained by these authors provide the answers in different terms and are rather cumbersome. We shall limit ourselves t o the discussion of t h e formula obtained by Jawerth and associated with the Luzin S-function (area integral). In order to formulate this result, we put

f(z,t) := (Pt

*

f)(x)

(x E BR", t E R,),

see (3.9.30). Then the (truncated) S-function has the form

Here

I v fI2

:=

af 2 ( baft ) 2 + Cbl (%)

displacement by z of t h e cone {(y,t); Iy[

and I'h(x) is obtained by a

< t, 0 < t < h).

Further, we

shall require t h e concept o f "median" MQ[q]. Namely, we put (3.9.42)

MQ[g] := inf

{ A ; mes,{y E Q ; Ig(y)I > A } 5

$ mes,

Q}

.

Using thse concepts, we finally define the maximal operator S# by putting

(s#f)(z) :=

MQISr(Q)fl. Q 31

The upper bound here is taken for all n-dimensional cubes Q containing 2 , and r(Q) := (mes,Q)*/". Then t h e result obtained by Jawerth has the form (3.9.43)

K ( * ;f ;H1,BMO)

= K ( * ;S#f; LI,L,) ,

where the constants are independent o f

f.

The computation o f the K-functionals of the couples p

< 00, 0 < q 5

m),

(Lo,B M O ) , ( B M O ,L,)

(Lw, B M O ) (0 <

are contained in Jawerth

and Torchinsky [l]. In particular, the results o f this work lead t o the following generalization of the result (3.9.40) obtained by Bennet and Sharpley: (3.9.44)

K ( t ; f ; L,,BMO)

= t(f,+)*(t")

(0

< p < 03) .

The real interpolation method

480

In the cited paper also is stated without proof validity of the formula

M ~ ( f , a is) defined by (3.9.42)with the substitution 1 := f dx.

Here fQ

IQI

1

(Y

for 1/2 and

Really, the relation (3.9.45)needs a correction. It becomes t o be correct if we change the right part by its majorant. This result implicitly contains in the paper o f Garnett and Jones

[l].

The results on the K-functionals formulated above, in the case of the

az" space

with the Lebesgue measure lead t o a refinement o f the classi-

cal Marcinkiewicz theorem, making it possible t o use "weak"

L,

space and the

general measure space

H1space instead of L,.

(R, p ) (the

BMO

instead of the

Unfortunately, for a more

Marcinkiewicz theorem was proved just in

this situation) the above concepts (and results) become meaningless. In this connection, we should mention the paper by Bennet, Devore and Sharpley where the "weak"

L,

space (denoted by

L,,)

[l]

is introduced i n the general

case. This concept is implicitly contained in the work o f Hertz (21. In order t o formulate the results obtained by these authors, we put

(1.9.19)on the definition o f f"]. It can easily be seen that L,, is not a linear space. Therefore, L, c Lm,. For the measurable space Q := [ O , l ] " (with the Lebesgue [see formula

# measure), there exists an elegant relation between space "space" L,,(Q).

B M O ( Q ) and the

Namely, i n the above-mentioned paper o f these authors

it is proved that L,,(Q)

is the "symmetric envelope" of

Lmco(Q) = {f E

L ( Q;)3 9 E BMO(Q), f'

B M U ( Q ) , i.e.

= g*J

A simple proof of this fact was proposed by Milman [2]. It is worth noting that the space Lp(lRn)is t h e symmetric envelope of the space HP(Ei!")(see Krantz

[l]).

Comments and supplements

481

The paper of these three authors, as well as the paper by Hertz [2], contains a corresponding extension of the Marcinkiewicz theorem t o the

We shall consider only a typical special case. For this purpose, we denote by S an arbitrary subset of the set { X E ; p ( E ) < m} whose linear envelope is dense in Ll(Q,p ) . Further, let T be a linear operator whose domain contains S and for which case of p

:=

00.

IITfllP I MP I l f l l P (P for all

:= 1 7 m)

f E S. Then the following theorem is valid.

Theorem.

The operator T can be extended by continuity t o the operator T acting from L, into L, (1 < p < m). In addition

0

In order t o estimate the use of this result, we shall apply it t o t h e Hilbert transform 'H (see Example 1.10.17).

If

s

:= { X ( a , b ) ; a , b E

R}, then

the conditions of t h e theorem are fulfilled since a trivial calculation yields ('HX(a,b))($)

I

1 b-x =; h

1.

Thus, 'H is bounded in L,(R) for 1 < p

< 00.

Concluding the discussion, l e t us mention an important paper by P. Jones

[l]containing, in particular, the computation of the K-functional of the couple of spaces of analytic functions ( H , ( D ) , H ~ ( D ) ) (0 < p

<

m>.

Namely, the following formula is valid: (3.9.46)

K ( t ; f ; H,,H,)

x

(f

)'"

(N,(f)*)Pds .

Here N,(f)is the maximal function defined by (3.9.28). 3.9.4. Differentiable and Smooth Functions Let us begin with the formulation o f the statement which, together with a result obtained by J. Peetre (see Corollary 3.1.20) and by Yu.A.

The red interpolation method

482 Brudny'i (see Proposition

3.1.19),are among a few facts known about the

exact values of the K-functionals in the situation under consideration. Na-

Lip(M,r) be the space o f Lipschitz functions on a metric space M with the metric T [see formula (3.1.28)]. Let us consider the couple Lip(M,q := (Lip(M,ro),Lip(M,rl)) and assume that the space M is separable relative t o the metric r0 + rl. With the help of the given metric T , we define the "distance" between two finite equivalent subsets S, T c M mely, l e t

by the formula

S -+ T is a bijection. we put For t h e function f : M -+ R,

where

f

where

f(S) :=

:

xzESf(z).

Then the following formula is valid (Schwartzman .[2]):

q; f ; LiP(3) =Go,71(f; t>. One more exact result refers t o the Sobolev space. Recall that for t h e domain

SZ c

R", the space W,"(C2)is defined

(3.9.47)

llfll

:=

SUP

Il?fllLp(Q)

by the norm

.

la19

If the upper bound is taken only for multi-indices with = Ic, we obtain the definition of the "homogeneous" Sobolev space i&':("). If := Hz" there also exists a different definition o f these spaces in which the Fourier transform is used (see, for example, Bergh and Lofstrom

[l], Secs. 6.2 and

6.3). This definition is valid for any k E R k . In the case of noninteger or negative k's, by the space W,k(fl) we mean the space of traces W,kln. It should, however, be borne in mind t h a t for natural k's and R := En, the second definition gives the same space as the first one only for 1 < p < 03.

Comments and supplements

483

But even for such p’s the space o f traces

W:~Qis generally narrower than

the spaces W:(s1).

Finally, l e t us consider an obvious generalization of definition (3.9.47),

L,(s1) by an arbitrary Banach lattice 9 over (8,dz). The corresponding space will be denoted by W,k(Cl).

which is obtained by the replacement of

After these remarks, we shall formulate one more exact result. It concerns the couple

( L 2 ,*;)and the K2-functional (see Definition 3.1.23). Namely,

the following formula is valid (Peetre[3]):

where 1x1 is the Euclidean length o f the vector z E

R”.

We have all grounds t o assume (see Holmstedt and Peetre [l]) that a similar formula with an appropriate kernel is valid also for the K,-functional of the couple

(L,, *;).

Let us now consider t h e general situation. The first result in this direction was obtained by the method proposed by Lions [1,3] (see also Lions and Peetre (21). The following formula was indicated explicitly by Peetre [7]: (3.9.48)

K ( t ; f ; L,,

W;) FZ ~ k ( f ; t ” k ) L p ( ~ n ) .

The right-hand side of this expression contains the k-th continuity modulus,8 and the estimate is uniform in

t

and

f.

A complete description o f the class o f domains s1 for which an analog of formula (3.9.48) holds has not been obtained so far. It is proved in the paper by Brudny’i 12) (see also BrudnyT 141) that this formula remains in force for R satisfying the strong cone condition, and also upon a replacement of L p ( R )by any space 0 which is an interpolation space in the couple

(Ll(sl),Lm(sl)).Therefore, we can take for 0 any symmetric space possessing the Fatou property (in particular, an Orlicz space, a Lorentz space, and so on). A special case o f the result formulated above (0 := Lp(Cl)) was obtained independently byt later in a paper by Johnen and Scherer 111. ‘1.e. the function (f,i)

+

sup IhlS:

k-th difference of step h E R“.

IlAkfll~~ where , A; is the operator of taking the

484

The real interpolation method

In Adams and Fournier [l], the latter result is extended t o domains o f a somewhat more general form. Yu.A. Brudnyi showed that formula (3.9.48) is generally invalid for domains satisfying the Lichtenstein-Jones

- b con-

"E

dition" (see Jones [l]for its definition) and indicated a correct analog o f formula (3.9.48) in this case. The proof of the corresponding result is given in the dissertation by Schwartzman [l]. Concluding the discussion about the computation o f the K-functional of this couple, we must mention two more results. The first of them (see Ciesielski [l])refers t o periodic function defined on the circle of periods

T. It has the form

~ (; f t;L~(T), W,"(T))

C

M

/I

(fin(l,tInIIk)fnen

/Inez

< 00

where 1 < p

and

7

En fnen is the Fourier expansion of the function f .

The second result concerns the case p

<

1. It should be noted a t the

outset that the "natural" definition of the Sobolev space i n this case (the completion of the space C r in the norm

sup

Ila"fllp)is not of

much

bl
interest (see in this connection Peetre [13]). Perhaps, a more suitable generalization is t h e space

SUP t>O

Wk(f

'

t)Lp(Rn)

tk

m;

W:

.

which is defined by the finiteness o f the quantity

For 1 < p

5

00,

the equality

Wi

2

holds. For

~:

p := 1, coincides with the relative completion in the space L1 (see BrudnyT [I]). In lrodova [l] (the case when n := 1 was proved earlier by Yu.A. Brudny'i), it was shown that formula (3.9.48) is also valid for the couple ( L p ,W i ) . The problem o f the computation of the K-functional for the couple

,::&I(

*;;) remains unsolved. We shall describe a few results obtained

in this direction. Peetre [4] in fact showed that for 1 < p

Devore and Sherer [l] considered the case of the cople

< 00,

(k:,@,).

A new

approach to their result and a certain generalization is contained i n the paper

Comments and supplements

485

by C. Calder6n and Milman [l].We shall give the most general formulation of t h e result obtained by the first two authors (see Nilsson [l]):

K(t; f ;

M

sup K ( t ; P f ; OO,@l) . lal=k

Here Oi E Int(L1,L,),

i = 0,1, and the equality is satisfied uniformly inf

and t .

The above formulas for the K-functionals allow one t o calculate the spaces (W:, k:;)~ and the corresponding inhomogeneous spaces for some values of the parameters. The answer usually contains the (Nikolskii-) Besov space which is defined in t h e “homogeneous” case with the help of the seminorm

Here 0

< p,q 5

depend on

k

00

and 0

< X < k. The sapce B2

up t o the equivalence o f

in this case does not

semi norm^.^ Let

us first consider a

result which is mainly due t o Lions:

(3.9.49)

(*$ *2)dP

where X := (1- 6 ) k o

B? ,

+ 9kl and 1 5 p 5

00.

Further, l e t us mention a result obtained by Peetre [4]

( P q ,w;;)&, w;; where pi E (1, m), ki

# 0, i = 0, 1, and, as usual,

p i ’ := ( l - q p ; * For

k

,

+6p;’,

ks := ( 1 - 6 ) k o + 6 k ,

:= 0, this result does not hold any longer, and the following embed-

ding i s only known: +l p

c #

(Lm,*;;)B,Pr

,

gThe book by Bergh and Lofstrom [l], Sec. 6.3, contains Peetre’s definition of the Besov space for negative A. Denoting this space by E?, we note that for A > 0, the equality B i g Y B i g takes place only for A 1 n( l/p - 1) (see Peetre [19]).

The real interpolation method

486

which is valid for pi E [l,+m],i = 0 , l as well.

it;q1)

A similar scope o f problems can be considered for the couple (h2q0,

as well. The K-functional for this ocuple for po = pl can easily be determined

with the help of the result obtained by Lions (3.9.49) and the Holmstedt formula. However, for po

# pl the answer is not known any longer.

In some cases

it is possible, nevertheless, t o compute the spaces of the K-method with power parameters. First o f all, let us consider the simple result

which follows from (3.9.49). A more profound is a result obtained by Peetre

[15], according t o which this formula remains valid for 0 < p < 1as well (for the spaces @?). The above-mentioned result obtained by lrodova [l]leads t o the validity of a similar result for t h e Besov spaces defined with the help of the modulus o f continuity. When po

# p l , it is

possible t o identify the

space appearing on the left-hand side of (3.9.50) only in rare cases. In this connection, let us consider the following general result obtained by Peetre

[6],in which Bi,?denotes the Besov-Lorentz space obtained by substituting Lpr(ntn)for L p ( n t n )in the main definition. Namely, putting s := min(qs,r)

,

t

:= max(qs,r)

,

we have

In particular, for r = qs, we obtain the equality

For pi = q i , i = 0,1, this result was obtained for the first time by Grisvard

[l]. The Grisvard result can be reinforced for Xo := 0 and pa := Namely, the following equality is valid:

00.

Comments and supplements

487

[l], and in the general case, by Peetre and [l] and by Janson. Let us also indicate the dual [see (3.9.35)]

which was proved by Peller Svensson

[I]:

relation obtained by Bui

(recall that

Bi := B F ) .

Concluding the section, we note that the analog o f result valid for the couple of spaces analytic i n the circle boundary values belong to

W

B M O ( T ) and, respectively,

(3.9.51)is also

of functions whose ii;ql(T)

(the se-

cond o f these spaces is referred to as the Bergman space and is denoted by Ah21). PI

3.9.5. Interpolation o f Operator Spaces Let us begin with an analysis o f operators acting from the Hilbert space

Hl into the Hilbert space H2.l' If the operator T belongs t o L(H1, H,) and , ~ ~ + the nonincreasing set o f eigenvalues is compact, then ( S ~ ( T ) ) denotes of the operator (T'T)1/2(singular or s-numbers o f the operator T).The s-numbers also allow the following description:

(3.9.52) s n ( T ) = inf {[IT- R I / H ~;, rankR H ~ 5 n} . W e recall that a compact operator T belongs to the class S,, 0

if the sequence of its s-numbers belongs t o /

\

I,.

< p < 00

Here we assume that

U P

Substituting in this definition the Lorentz space

Zpq for Zp, we obtain the

class S p q .More generally, if S is a symmetric space of sequences, then S s is defined by the (quasi-) norm

'OA detailed description of all statements concerning s-numbers of compact operators is given, for example, in the book Gokhberg and Krein [l].

The red interpolation method

488

It is natural t o say that S , coincides with the subspace ,C(Hl,H2), consisting of compact operators. It should also be noted that S , coincides with the class of nuclear operators, i.e. such T E S,, for which

The lower bound here is taken over all families of one-dimensional operators for which

T = C n ET,,.~

The norm (3.9.55) coincides with the norm (3.9.53) for p := 1. The K-functional of the couple (S1,S,) can easily be computed (see Miroshin

[l])due t o t h e following simple method of reconstructing the operator T from the sequence of its s-numbers: (3.9.56)

TX =

C

sn(T) (5,xn) Yn

7

n>O

where (xn) and (y,)

are orthonormal sequences in

H1 and Hz respectively

(the Schmidt series). Namely, the following formula is valid:

1 t

K ( t ; T ;S1,S,) =

sT(x)dx

,

0

where

ST(IC)

:= s,(T)

for n

5 z < n + 1, n E Z+.

Hence we obtain the well-known result

Using the series (3.9.56), it can easily be seen t h a t ( S , , S , ) is a retract (orbital) of the couple ( l , , ~ )with retraction maps equal t o 1 in norm. Hence for any functor F we have

More generally,

Comments and supplements

489

This result is from the realm of mathematical folklore, and it is difficult t o indicate exact references (see, however, Sec. 9 of the survey Brudny'i, Kre'in and Semenov [l]).

Let us consider now a result concerning the interpolation of spaces of the Hunkel operators due t o Peller [l]. Recall t h a t if f is a bounded function on the unit circle

:=

dW,then

the Hankel operator

r,

acts from the

Hardy space H z ( D ) into the space H i - ) ( D ) (the orthogonal completion of

H2 in L,) in accordance with the formula

where P is the orthoprojector of L2on H z . For the definition of Hardy space, see formula (3.9.25).

If we denote by

r(S,) the space o f the Hankel operators belonging to

S, then ( p := p s = -)

1

1-6

The proof can easily be reduced t o the case of the couple (S1, S ),

with the

help of the following profound result due t o Adamyan, Arov and M. Kre'in

If I?f E ,S , then for each n E ZZ+ there exists a rational fraction r := r,, of degree not exceeding n such that

Comparing this formula with (3.9.52) and taking into account the fact that

rankr, 5 n , we find that the lower bound in (3.9.52) i s attained in this case by a Hankel operator. It should be borne in mind that the right-hand side is equivalent t o

1l.f - r ( ( B M O A . where the

the functions analytic in

W

space B M O A consists of

and such that their boundary values belong to

BMO(-(T) (a corollary of the theorems o f Nehari and Fefferman-Stein; see, for example, Peetre [22]). Concluding this section, let us consider a result on the interpolation of the operator couple

Z(2,f) :=

(L(Xo,yO),L(Xl,&)), where 2

are Banach couples. If the couple x" is regular, then

- - + -

and

L ( X , Y )is a generalized

The real interpolation method

490

Banach couple in the sense of Definition 2.1.24 (see Example 2.1.25). Therefore, we can consider the computation o f the spaces

(.)$p

for this couple.

We shall describe whatever little is known in this field.

It can easily be noted that

The author of this remark, Peetre in [14], obtained the following partial inversion of this embedding. Let

where

X o = X1, and the couple

y depends only on

? be K-linearizable (see 3.9.1).

Then

P.

This statement remains in force also when the weight

t*

is replaced by

any weight w E Conv. In this case, the condition o f the K-linearizability generally cannot be removed. Namely, if the statement formulated above is valid for any couples

x’ with X o = XI and any weights w, then the couple

? is I<-linearizable. 3.9.6. Some Unsolved Problems (a) Determine the exact constant in the interpolation inequality

where

Mi := IITIIL~~~,, 0 5 cro < cq I 1.

It follows from Proposition 3.1.19 that for cro := 0 and crl := 1 this constant is equal to unity.

(b) Calculate the exact value o f the K-functional o f the couple (C, C2)(T). (c) Characterize the domains 52

is valid.

c R”for

which the Lions formula

Comments and supplements

491

Determine the order o f the K-functional of the couple (d o t W 2 ,i@j;)(T)

for ICo

# Icl

and po

# pl. A n

"easier" question: calculate the space (.)ap

for this couple. Calculate the K-functionals of the couples of weight spaces

(Lm(uo),Lpl(ul))to within the order ot magnitude. Solve the same problem for a couple o f the Orlicz spaces. Let GJ be a couple of symmetric spaces on R+and let ( D J ) ( t ) := f ( t / s )

( t E R+) be the dilation operator

(s

> 0).

We assume that the function

increases monotonically. Is the following generalization o f the Holmstedt formula correct

(see Arazy

[l])?

Describe the extreme rays of the cone Conv(Rn) o f concave non-negative functions defined on

R;.It should

be noted that for n := 1, Corollary

3.2.4 states that each such ray has the form arbitrary nonempty convex set for n

>

(xErR+,

R+. There exist

where

E is an

examples showing that

1, the set Conv(nt",) contains an extreme ray difFering from

those indicated above. For what couples is the following stronger version of Theorem 3.2.7 on I<-divisibility valid?

If cp0,cp1 E Conv and

there exists a representation

K ( . ; I i; d ) x

(pi

I = 10

+ zI

such that

, i =0,l .

The red interpolation method

492

(i) Determine the exact value of the K-divisibility constant There are sound reasons to believe that

(j) Determine the exact value of

(k)

Let

x’

K.

K..

54

.(C,C’).

2 be an Z-couple (see Definition 3.1.33).

Under what conditions on

is the K-divisibility theorem valid in its general form (the validity o f

this theorem for a finite number o f terms was proved for any a-couple, see Theorem 3.2.12).

(I) Let cp be a quasi-Banach lattice over 23,which is a parameter of the discrete J-method

(see Sec. 3.4.C).

Is it true that

Here cp# is a “Banach envelope” o f cp, i.e. the Banach lattice defined by the norm

Recall that cp+ denotes the Banach lattice associated with cp (see formula (3.1.7)). In the power case, this statement was proved by Peetre [16]. (m) Is formula (3.4.41) correct without the assumption about the regularity o f the couple? In other words, is the relation

+

where @ := K * ( L 1 ) ,true?

493

CHAPTER 4

SELECTED QUESTIONS IN THE THEORY OF THE REAL INTERPOLATION METHOD 4.1. Nonlinear Interpolation

A. In this section, we shall establish the invariance of the functors of the K-method under the action o f a broad class of nonlinear operators. This class includes quasi-additive and Lipschitz operators considered above as well as many other operators which have important applications. In particular,

it also contains continuous linear operators. It will be shown below that the "basic problem" in operator interpolation theory (see Sec. 2.6.8.8) can be solved completely by replacing the linear operators by operators in the class under consideration. Let us first formulate basic definitions. Recall that an operator'

from a Banach space X into a Banach space

T acting

Y is called bounded if

We denote the Banach space o f such operators by B ( X , Y . A natural generalization of the concept introduced above t o the category of Banach couples is Definition 4.1.1 (Gagliardo, Peetre). The operator T acting from C ( 2 ) into C(?) belongs t o the space B ( 2 , ?)

if there exists a 7 > 0 such that for any xi E X i and any y, E yi for which

0

'Such an operator is generally nonlinear.

E

> 0 there exist

Selected questions in the theory of the r e d interpolation method

494

We leave it t o the reader t o verify the validity of the following simple fact. Proposition 4.1.2. The set of bounded Operators

B ( i , ? ) with the naturally defined addition

and multiplication by scalars is a Banach space in norm (4.1.3). Moreover,

L(z,?)

(4.1.4)

4

B(-f?,?) .

0

The relationship between the concept introduced above and the concept of t h e I--functional is revealed in the following criterion for the operator

T

t o belong t o the space o f bounded operators. Proposit ion 4.1.3. The operator T belongs t o the ball of radius y in the space B(r?, ?) iff for any z E C ( x ' ) we have (4.1.5)

I { ( . ; T ( z ); 8)5 TI<(.; z ; 3 ) .

Proof. In

view of Corollary 3.1.29, inequality (4.1.5) is equivalent t o the

inequality

{max(IlYollYo,t IIYlIIYl)> 5

inf

(4.1.6)

W)=Yo+Yl

where t

> 0 is arbitrary.

This inequality is equivalent t o the fact that

T

belongs t o

B,(z,p).

Indeed, it follows from Definition 4.1.1 that inequality (4.1.6) is satisfied. E > 0 I1zoJIxo/l(zlJIxI there exist yi E Y , such that

Conversely, suppose that inequality (4.1.6) holds. Then for given and z i E X ; and for

T ( z ) = yo

+ y1 and

t

:=

Nonlinear interpolation

495

Hence it follows that

Corollary 4.1.4. Let T E B(Z?,f) and assume that the series the space C(r?). Then for a given E

C

z,

converges absolutely in

> 0 there exists a sequence (y,) c C(?)

such that

T(Cz n ) = C Yn and, moreover,

Proof.In view of the statement

formulated above,

K ( *; T(CI n ) ; F) L IITIIB(C K ( . ; z n ; 2)). We put

(P,

:=

llTllBK(-;z,;

C Pn(1)

:=

2).Then

IITII~C ~

( 12,; ;

2)= IITIIB C I I ~ ~ I I <~ B ) ~0

by hypothesis. Consequently, i n the situation described by inequality (4.1.7), Theorem 3.2.7on K-divisibility is applicable. According t o this theorem, for a given

E

> 0 there

exists a sequence (y,)

c C ( f ) which satisfies the con-

ditions o f the corollary. 0

Let us consider some important examples of operators from the class 0.

496

Selected questions in the theory of the red interpolation method

Example 4.1.5. (a) In view of (4.1.4), any linear operator in

, C ( X , Y ) belongs t o t h e space

B ( 2 , ?). T : C ( 2 ) --f C(?) have the property that Tlx, belongs t o t h e space of strongly Lipschitz-type operators ,Cip(")(Xo, Yo) (see Definition 3.5.21) and that Tlx, belongs t o the space of bounded operators B(X1, Yl) [see (4.1.1)]. Then T E B(x', ?); indeed,if E > 0 and z; E X i are given, i n view o f Definition 3.5.21 vo := T ( Q 21) T(zl) belongs t o Yoand

(b) Let the operator

+

Moreover, in view o f (4.1.1), for

y1

:=

T(zl) we have

Thus, the conditions of Definition 4.1.1 are satisfied. In addition, we find that

(c)

The operator T : C ( x ' ) + C(p) will be called a quasi-additive o p erator if for a certain constant y > 0 and any z; E E ( d ) (i = 0 , l ) we have

In particular, if

9 is a couple of Banach lattices,

and if

Nonlinear interpolation

497

almost everywhere, then in view o f Proposition 3.1.15 inequality (4.1.8) is satisfied. Thus, the concept introduced above generalizes the concept of quasi-additive operator which was used in the Marcinkiewicz theorem 1.10.5. Let not Then

T be a quasi-additive operator and Tlxi E B(Xi, E;.), i = 0 , l .

T belongs t o B ( d , f )and

Indeed, if I = 10

+

21, where ; I

E Xi,then

where Mi := llTlxiIla(xi,x).Taking in this inequality the greatest lower bound over xi, we get

It remains for us t o use Proposition 4.1.3. (d) The following simple example, which will be important for the further analysis, shows that the class

B(d,?)contains discontinuous

maps as

well. Namely, suppose that the inequality (4.1.9)

K ( . ; y ; f )I y K ( . ; a:; d )

x E C ( d ) , y E C(P). Let us consider the operator Tz,y : C ( 2 ) + C(p) defined by the equality is satisfied for the elements

y

for z :=

0

forz#z.

I ,

T2J.Z) :=

Selected questions in the theory of the r e d interpolation method

498

Since in view o f (4.1.9) inequality (4.1.5) in Proposition 4.1.3 is satisfied for

Tz,y,then T,,y E B,(r?,p).

Let us now verify that the invariance of the relative action o f operators

B

from class

completely characterizes the functors of the K-method. For

formulating the corresponding result, we shall require Definition 4.1.6.

A functor F is termed B-invariant on the couples x',? (the order is important!) if for any operator T E B(d,f)and any x E F ( 2 ) the inequality

is satisfied. 0

Remark 4.1.7. The class

B

forms an operator ideal in the sense that if T E

B(2,?)

and

R E L(fi,Lf), S E L(?,q) are arbitrary linear operators, then S T R E B(d, This follows directly from Proposition 4.1.3 which also leads t o

c).

the inequality

It would be very interesting t o study the functors which are invariant under the action o f operators of certain operator ideals. Theorem 4.1.8.

If the functor F is B-invariant on the couples (4.1.10)

iw(x'),then

F ( Z ) = KF(zrn)(T) .

Proof. In view of

Lemma 3.3.7 4

F ( L )= I ( F ( L r n ) ( L ) . 4

This equality and the fact that I(F(L,) is maximal on the couple L , (see Theorem 3.3.4) lead to

499

Nonlinear interpolation

F

(4.1.11)

A

KF(t,) .

In order t o prove the inverse embedding, we take an element z E and construct an operator

T, E B(z,,T) such that

l l ~ z l l q ~ mI, q1

(4.1.12)

For this we put

KF(em,(z)

'T

9

f := K ( . ; z ;

W(*; 2 ; @I

=2 .

2).The function f E Conv, and in view of

Proposition 3.1.17,

K(.;f;E,)=K(.;z;Z). Then the operator

'T

:=

Tf,' constructed

in Example 4.1.6(d) satisfies

conditions (4.1.12). From the B-invariance of the functor

T, [see (4.1.12)],

F and the properties of the operator

we obtain

Thus the embedding

which is inverse o f (4.1.11) is proved. 0

Corollary 4.1.9. For

F

E

JF

t o be a functor of the K-method it is necessary and sufficient

that it be B-invariant (on any couples).

Proof. The sufficiency

follows from the previous theorem, and the neces-

sity from Proposition 4.1.3. Moreover, we obtain the following interpolation

inequa Iity

500

Selected questions in the theory of the r e d interpolation method

B.

Let us n o w verify t h a t when t h e category

extended by t h e replacement of t h e class

B

L of

2

of Banach couples is

linear operators by t h e class

of bounded operators, t h e “basic problem” in interpolation theory can

be solved completely (concerning t h e problem above mentioned, see Sec.

2.6.B). Thus, we shall solve here t h e following Problem ( E . Gagliardo). +

X , describe t h e intermediate spaces X w h i c h are invaria n t to t h e action of operators f r o m B(r?). For a given couple

Obviously, all such spaces belong to

Int,(r?).

However, it will b e shown

later t h a t t h e y generally do not exhaust this set. T h e general result presented below contains, i n particular, a complete solution of t h e Gagliardo problem.

To formulate t h i s result, w e shall use Definition 4.1.10.

2,X is called B-in~ariantrelative to the triple ?,Y operator T E B(g,?)we have The triple

-

T(z)

if for any

Y

If, i n addition, t h e inequality

is satisfied, t h e epithet “exact” is added 0

Theorem 4.1.11 (Brudnyz’-Krugljak). The triple

r?, X

-

is B-invariant relative to t h e triple

meter Q of t h e &method,

Kq(2),

(4.1.14)

X

Proof.W e

require

Y

c--’

Kq(?) .

?, Y

iff for some para-

Nonlinear interpolation

501

Lemma 4.1.12. Condition (4.1.14)is equivalent t o the condition

(4.1.15) K ( . ; y ; ?) 5 K ( . ; I ;x'),

IE

X

+y E Y .

If z E X c K o ( z ) ,then K ( . ; I ; x') E Q, and it follows K ( . ; y ; f )E Q as well. Consequently, y E K*(?) c Y . (suficiency). For an element I E C(x')\{O} and a couple ,?, we define the intermediate space KO,(,?) with t h e help of the norm

Proof (necessity).

from (4.1.15)that

KO, = Ka with Q := L:(';,;'). Therefore, from the identity I L ( J L )= Ac (see Corollary 3.1.11 and Proposition 2.2.20)we obtain Obviously,

(4.1.17) A'

c KO, .

Further, condition (4.1.15)signifies that

(4.1.18) KO,(?) C Y (as a linear space).

However, each of the Banach spaces in (4.1.18)is

continuously embedded in

C(d)so that the embedding operator in (4.1.18)

is closed and hence continuous. We denote the norm o f this operator by

7(1).We put n(z) := max(l,y(s)) and suppose that'

71(~)K0, .

(4.1.19) F := Ilzllx=l

Let us verify that the sum is defined in a consistent manner and hence that

F

is an interpolation functor. Since -yl(z) 2 1, in view o f (4.1.16) we have -+

IlYllqZ, := K(1;y ; 2) 5 IIYIIKO,(Z) IlIllc(a)

5

5

.

11~11Xll~l171(,).Ko~(Z)

Here 0 is the constant of embedding of X into C ( x ' ) . Therefore, for

1 1 ~ 1 =1 ~ 1 we have 'It should be recalled that

11 . Jltx :=

111

. llx, t > 0.

502

Selected questions in the theory of the real interpolation method

uniformly in

2.

Thus, the sum

(4.1.19)is well-defined (see Definition 2.1.35 3.3.15shows that F = Kw for some In addition Q = Kw(Z,) = F(f,) (see Lemma 3.3.7)so

in this connection).

\k E Int(E,).

Then Theorem

that

(4.1.20) F = KF(eml. The expressions (4.1.19)and (4.1.16)lead t o the following set-theoretic embedding

(4.1.21) X

L)

F(2).

Applying, as it was done above, the theorem on closed graph, we see that embedding

(4.1.21)is continuous. Further, from (4.1.18)and the definition

of $2) and

TI(.)

we have

and therefore, by the definition of

F(f)

(4.1.22)

F [see (4.1.19)],

Y.

Combining (4.1.20),(4.1.21)and

(4.1.22),we obtain (4.1.14).

L e t us now prove the theorem. By the lemma, it is sufficient t o verify

(4.1.15)is equivalent t o the following statement. For any operator T E 2?(2,?),

that condition

(4.1.23) T ( X ) c Y

.

Let us first verify that this statement leads t o condition (4.1.15).Indeed, if

I<(.; y ; P) 5 I{(.; 2 ; 2),

2 E

x,

4.1.5(d) there exist an operator T := Tz,y E B1(x', for which y = T x . But then by (4.1.23)y E T ( X ) c Y . Conversely, if condition (4.1.15)is satisfied and T E then by Proposition 4.1.3we have according t o Example

P)

B(z,?),

Nonlinear interpolation

503

K ( - ;T ( z ) ;P) I lITlI,(,,p)K(. ; 2 ; 2) f

If 2 E X ,from this inequality and condition (4.1.15) it follows that T ( x ) E Y. 0

Corollary 4.1.13.

The family ( K @ (21) OEht(E,)

exhausts the set of all intermediate spaces

invariant under the action of operators from S ( 2 ) . Ll

This result obviously gives a complete solution t o t h e Gagliardo problem. These arguments make it possible t o prove one more important fact. For its formulation, we require Definition 4.1.14. The triple

d ,X

is called K-monotonic reZatiwe to the triple

f ,Y if condi-

tion (4.1.15) is satisfied. 0

Thus, for any

2

E X , y E C(?)

from the inequality

K ( . ; y ; ?) 5 -yK(.; 2 ; 2) it follows that y E Y .

If, besides, the inequality llylly

5 -y 11zllx is satisfied for this element, the

epithet “exact” is added in the definition. In t h e case when the triple f , Y coincides with the triple

d ,X , the interpolation space X

in this definition is

called K-monotone space (or ezact K-monotone space if the corresponding

inequality for the norm is satisfied).

(S.G. Kret’n). If the triple 2 , X is K-monotone relative t o the triple ? , Y , then for some parameter 3 of the K-method we have Conjecture

x

Lf

K @ ( 2 ) , Y r K*(f).

504

Selected questions in the theory of the r e d interpolation method

Corollary 4.1.15.

The conjecture formulated above is correct.

Proof.The statement

follows from Lemma 4.1.12 and Theorem 4.1.11.

0

In particular, we obtain the following description of all K-monotone spaces of a given couple. Corollary 4.1.16.

The family

(Ko(2)) ocht(Em)

the couple

2.

exhausts the set of all K-monotone spaces of

0

C . All definitions formulated above can be nzwrally extended t o the o f couples o f normed Abelian groups (see Sec. 3.1.C for the definition of this category). The above results whose proofs do not involve the category

A

theorem on K-divisibility remain in force as well. Then Corollary 4.1.4 will nevertheless be valid for finite families

(2,)

with a constant appearing in ine-

qualities for K-functionals y,,, which depends now not only on the couple but on the number card(z,) as well. The proof is based on Theorem 3.2.12. Another modification should be introduced in the formulation of Proposition

4.1.3 since in the case under consideration there is no equivalence:

K(.;y ;

P) 5 K ( . ; 2 ; 2)e K,(.;

y;

P) 5 K,(.;

2;

2).

Red interpolation functors

505

4.2. Real Interpolation Functors

A. The relationship between the functors of the 3- and K-methods established in Sec. 3.5 is not accidental. It turns out that there exists a broad class of functors (including .la) which essentially coincide with functors of the K-method.

However, the definitions of functors of this class

often differ considerably from the definitions of functors of the K-method. In particular, this explains the striking diversity of the real method and the variety of its applications. The main property of functors of the class under consideration is outlined in Definition 4.2.1.

F E 3.F is called a real interpolation functor if it coincides on t h e couples 31 and with a certain functor Ka.

z,

0

Let us denote the new class o f functors described in t h e definition by

R.

If this coincidence takes place t o within equivalence of norms, we obtain the wider class

R,.

Below we shall describe some functors of this class which have important applications (e.g., the functors o f the methods of traces, of constants, of averages, of t h e t-method, and o f the &-method etc.). Here we shall establish the basic statement relating the functors in R with the functors of the K-met hod. Theorem 4.2.2.

If F E R.then (4.2.1)

F

+k

Ka ,

where @ := F(z,).

Proof. In view of Lemma 3.3.7 (4.2.2)

and the definition of @,

F(E,) = K~(J?,) .

Selected questions in the theory of the real interpolation method

506

Moreover, in view of the relative completeness of the couple

i-

and the

identity A"(X')= A(X") (see Proposition 2.2.20), we have

a"(Z,)

= A(Z,)

Therefore, t o within equivalence of norms, the functors on both sides of relation (4.2.1) coincide on the couple of t h e functor

F

(4.2.3)

+

L,.

In view of the maximal property

Ka (see Theorem 3.3.4), it follows hence that

+ A"

Ka

,

In order t o prove the inverse embedding, we use Theorem 3.5.9. According t o this theorem,

KQ L+ J q

(4.2.4)

+ A'

,

9 := Ka(Z1). We recall that @ := F(Z,). Since, by hypothesis, the functor F E R,we have for some parameter 6 E Int(i,)

where

F ( i p )= K&(ip) , p := 1,oo . In view of (4.2.2) and Lemma 3.3.7 (4.2.5)

6 = 0 , whence

\Ir = F ( i 1 ) .

From this equality and from the fact that J q is minimal on the couple i 1 Jlp A F ,while (4.2.4) leads to

(see Theorem 3.4.3) it follows then that

the embedding inverse t o (4.2.3). 0

Corollary 4.2.3.

If the functor F is in R and if one of the following conditions is satisfied: (a)

F(Zoo)is a nondegenerate parameter of the K-method 3.5.4); +

(b) the couple X is relatively complete; then the following relation holds:

(see Definition

507

Red interpolation functors (4.2.6)

F ( 2 ) S KF(j-,,(2).

Proof. In view of Theorem 4.2.2, (4.2.7)

i n both cases it is sufficient t o prove that

Ac(x') ~t F(x')

In case (b), this is obvious since by Proposition 2.2.20 and the equality

x''

= x', the left-hand side of (4.2.7) is equal t o A(??). In case (a), we use the first statement of Theorem 3.5.9. It leads t o the equality

K~zJly, where Q :=

K @ ( i 1 )Using . this relation instead of embedding (4.2.4), we

obtain, as in the previous theorem,

Together with embedding (4.2.3), this proves t h e statement of t h e corollary in this case also. 0

It should benoted t h a t in the nondegenerate case, the functors in R are expressed in terms of functors o f the 3-method. We state the corresponding result, leaving i t s proof t o the reader. Theorem 4.2.4.

If the functor F is i n

R

and

F(E1)i s

a nondegenerate parameter of the

3-method (see Definition 3.5.4), then (4.2.8)

F n Co 2 J,(ti)

In particular, if F

~f

.

Co, then

U

B. Let us consider some simple methods for verifying the fact that F belongs t o the class R.We shall begin with the following simple remark.

Selected questions in the theory of the real interpolation method

508

Proposition 4.2.5.

If from the fact that S(lf1) belongs t o

F(2,)

it follows that f belongs t o

F(Zl), then F E R,. Here f E C(&). Proof.As was noted above, see (4.2.2), (4.2.10)

F(L',) = K F ( ~ , ) ( Z m.)

Further, in view o f Proposition 3.1.17 it follows that

Therefore, the fact that that

S((f1)belongs t o

f belongs t o K F ( ~ , ) ( &is) equivalent t o the fact F(Z,). But then f E F(&) by hypothesis, and

therefore we have established that

c F(L',) . In view of the theorem on closed graph, this embedding is continuous. On the other hand, (4.2.10) and the maximal property of

L',

KF(z,) on t h e couple

lead t o the inverse embedding. Thus,

which together with (4.2.10) completes the proof. 0

Remark 4.2.6. Obviously, the condition o f Proposition 4.2.5 is just necessary for F to belong t o

R,.

A more informative sufficient condition is contained in Theorem 4.2.7.

If the functor F is such that

609

Real interpolation functors then

F belongs t o R, and i s n~ndegenerate.~ In particular

(4.2.12) F E KFcz,) .

Proof.Since the function m(t) :=

min(1,t) belongs t o A(Z,) c F(Z,), we get in view of (4.2.11) m E F(zl). Further, the operator S belongs t o L(Ll,e,), so that Sm E F(i,). Finally, Sm = K ( . ; m ; z1) E Conv. Thus, the function Sm is in F(z,) n Conv. With the help o f a direct computation it can be easily verified t h a t

lim ( S m ) ( t )= +oo

(4.2.13)

t-+m

,

- +oo . lim (Sm>(t> t ~

t++O

The presence of such a function inthe space F(i,)

F(L)\(LO,

signifies that

u LL) # 0 .

Thus, we have established that

F(z,) is a

nondegenerate parameter o f the

K-met hod.

It remains t o verify that the condition o f Proposition 4.2.5 i s satisfied. h E C(z1) and S(lh1) E F(z,), then h E F(Zl). For this purpose, we use the statement from Proposition 3.5.6 [see (3.5.11)], putting there x' := 21, Q := F(z,), g := Sm, 5 := h and E := 1. Then in view of (4.2.13), the function g satisfies condition (3.5.9). Further, S((h1)= K ( . ; h ; so t h a t Therefore, we must establish t h a t if

zl)

z

:= h E K Q ( ~ := )

K F ( z w ) ( z l )Consequently, .

the statement cited

above is applicable t o the situation under consideration. Thus, there exists a function

f E K F ( z w ) ( x land ) an operator T E L(&) such that

(4.2.14)

h = Tf .

Here the function

(4.2.15)

f

f= C

has the form v ( t z i + l ) ~7 i

i&3

where xi is the characteristic function o f t h e interval

q t z i + l ) and cp is

defined by the equality 31n other words, F ( I ? ~ is ) a nondegenerate parameter of the K-method.

510

Selected questions in the theory of the r e d interpolation method 'p :=

K ( . ; h ; E ) + Sm

.

A(E,) and h E KF(~=)(il), the function 'p E F(i,). It follows from the definition o f f , see (4.2.15) that 0 5 f 5 q'p SO that f E F(Z,). But according to the condition of the theorem, F ( i m ) C F ( & ) , and hence f belongs t o F(&). From (4.2.14) and the interpolation property of F it now follows t h a t h E F ( i 1 )as well. Since Sm E

Thus, we have established that S(lh1) E F ( t 1 )

+ h E F(E1).

0

In order t o formulate the corollary, l e t us agree t o call the functor F in 4

R nondegenerate if F(L,)

i s a nondegenerate parameter of the K-method.

Henceforth, we shall call functor F regular if F(i,)

c F(il).

In view of

the theorem that has been just proved, a regular functor belongs t o

R.

Corollary 4.2.8.

cR

(a) If ( F c r ) c r E A

and all functors of this family are nondegenerate, the

functors C(Fa)crEA and

also belong t o

R

and are nondegen-

erate.4

(b) If Fo, Fl and F are nondegenerate functors and belong t o R,the functor

F(F0, F1) also has the same property. (c) If Fo and F! are regular functors, and F is an arbitrary functor, then

F(F0,Fl) is also a regular functor.

Proof.Statements (a)

and (b) follow from the equality

(see Corollary 4.2.3) and from the relevant facts for the functors of the

K-method (see Theorems 3.3.11 and 3.3.15). Condition (c) follows from the embeddings

F(F0, Fl)(Lo) := F(Fo(E,), Fl(i,))

c

4 0 f course, provided that the sum and the intersection of this family exist.

Real interpolation functors

c

511

F((FO(il),Fl(il)):= F(FO,Fl)(il),

0

C. Let us now consider concrete functors in R which can be met in applications. We begin with t h e (generalized) Lions-Peetre method of constants. For

this we consider a cople o f Banach lattices 5 := (@0,@1) defined on a measurable space ( Q , C , p ) . We shall say that 6 is a parameter of the method of constants if the following condition is satisfied: (4.2.16)

xn E C(6) .

Further, for a given Banach space X and a Banach lattice @, we denote by

@ ( X )the space o f strongly measurable vector-valued functions f

:

+X

for which the norm

is finite.

Definition 4.2.9.

If

6 is a parameter of the method of constants,

then the space Kg(x') is

defined as a set o f elements z E C ( x ' ) for which the norm (4.2.18)

l141K6(2) := inf {Ilfolleo(xo,

+ llflIlal(x,,~

is finite. Here the lower bound is taken over all pairs of functions f; E @ ; ( X i ) ,

i = O , l , for which 2

= fo(.)

+ fl(.)

almost everywhere. 0

Then the following proposition holds.

Selected questions in the theory of the red interpolation method

512

Proposition 4.2.10. Formula (4.2.18) defines a functor.

Proof. The fact

that

K6(x') is, in view of definition (4.2.18).

a subspace

of the space C ( @ o ( X o )QI(Xl)) , consisting of functions which are constant

C ( 2 ) . Since the spaces

almost everywhere and have values in

@;(Xi)

are

known t o be complete, their sum is also complete (see Proposition 2.1.6). Therefore, the completeness of

K s ( 2 )follows from the fact that

it is closed

on this sum. Further, let us verify that (4.2.19)

A

L)

Km

L)

C

.

Indeed, it follows from (4.2.16) that

xn = fo Therefore, for I E

+

fl

(f;E a;, i = 0,1)

A(Z), we have

II4IK,(2, I112

*

fOllOa(X0)

+ 112 . f l l l O l ( X l ) I

I II"Ila(R)(llfolloo

+ IlflllOl) .

This proves the left embedding in (4.2.19). Further, suppose that z E

K g ( 2 ) . Then for a given E > 0 there exists a

representation

such that the following inequalities hold:

Hence it follows that II"IlC(X)

5

II

Ilxnllc(6) IIlfo(m)llxo

IlfOllOo(X0)

+ Ilf1(m)llx1

llC(6)

+ I l f l l l ~ l ( X l )5 2(1+ E ) 114lK&(R)

5

*

Thus, the right embedding in (4.2.19) is also valid.

The interpolation property in (4.2.19) is also valid. The interpolation property of K6 immediately follows from the equality

Red interpolation functors Tx = Tfo

513

+ Tfi , -+-+

L ( X , Y ) and any representation x =

which is valid for any operator T E

fo(*)

+

fi(.)# fi

E

Oi(Xi),

i = 0~1.

0 Let us verify that under natural restrictions, the functor Ka; belongs t o

R.This follows from a

more exact statement contained in

Theorem 4.2.11.

If the lattice @i contains a generalized unity i = 0,1, then

ei

(i.e.

ei

#

0 almost every-

where),

K3=K*, where Q := Ka;(Zm).

Proof. Let us verify that the functor

2 and F.

Then an application o f Theorem 4.1.8 will complete t h e proof.

Thus, suppose that T (4.2.20)

K6 is B-invariant on any two couples

E B ( Z , ? ) . We must prove that

llT(X>11K6(i), I llTllB(2,i)) ll~IlK(((2)

for any z E

K&(Z;>.

For this we put e := min{Ilol, l e l l } . Then e

> 0 almost

everywhere.

We represent x in the form

x = fo(')

+

f i ( * )7

(fi(.)

E

@i(xi),

and suppose that for a given 7 > 0, the number

i = 071) E

:=

qe(t).

Here t E

s2 is

> 0 and x = f o ( t ) + fl(t). In view of Definition 4.1.1 of t h e class B,for a given E > 0 and a given representation x = fo(t)+ fi(t) a point at which e ( t )

there exists a decomposition (4.2.21)

T(x) = yo(t)

+ yi(t) ,

(y;(t) E E;., i = 0,1) ,

such that the following inequalities are satisfied: (4.2.22)

l l Y i ( t ) l l ~I 7 Ilf;(t)llx + v ( t ) .

Selected questions in the theory of the real interpolation method

514

Here we put

7 := IITlla(n,p,. > 1 be fixed. Let us define for given r n , n , p E 23 the sets

Further, let q

Rmnp := {t E R ; qm qn

I Ilfo(t)ll~o < qm+l,

I Ilf1(t)llXl < qn+l,

qP I l ( t ) < q p + l )

.

These sets are obviously measurable, and their union coincides w i t h refore, choosing arbitrarily a point tmnpe valued strongly measurable functions

Gi(t)

:=

by the formulas

(4.2.21), the equality

T ( x ) = Go(-) Besides, in view of

& (i = 0 , l )

defining countable-

for t E Qmnp ( m , n , p E 23) 7

Yi(tmnp)

we obtain, in view o f

RmnP and

R. The-

+ Gl(.)

*

(4.2.22) and the definition of Rmnp,we have for t belon-

ging to this set

+ Ve(tmnp) I q [YIlfi(t)llx, + ~ ( t ). l

IlSi(t)llx, I Y Ilfi(tmnp)llX,

I

Consequently, taking the @i-norm of both sides, we obtain

llGill*n(ys)5 q [YllfiIl*,(~,) +

IlelIq6)l

*

Adding these inequalities and taking the lower bound over all representations

of z i n the form of the sum

fo(.) + fi(.),

IIT(~>IIK*(P) L 4(7 As q

--+

1 and

r] -+

we get

ll4lK*(n)+ 77 IlellA(6)) .

0, this leads t o (4.2.20).

0

L e t us consider now Peetre's L-method and verify t h a t the functors o f this set also belong to the class the LG-functional [see Remark

R. For

this we shall use the definition o f

3.1.281. It should be recalled t h a t here w i are

R+into itself and vanishing a t zero. Banach lattice over R+.

convex functions bijecting

@ be a

4.2.12. The space L,-,*(z) consists o f elements z E

Further, let

Definition

E(d)for which the norm

Red interpolation functors

515

(4.2.23) is finite. Proposition 4.2.13.

If t h e function m(t) := min(1,t) belongs t o the space 0 , then formula (4.2.23) defines a functor.

Proof.Obviously, when x

= 0 . The converse statement follows

= 0, llxll,+

from inequality (4.2.24) which will be proved below. It should be noted t h a t since w' is convex,

where L :=

L; and

(Yk

assumed that t h e series

:= X k ( c

Xk)-',

the numbers

Xk

c X k < +oo, while the series c

> 0,

21;

and it is

converges in

C(2). Applying this inequality for the case o f two addends and taking X k equal to

11zkll

+ E with an arbitrary E > 0, we obtain

Thus, we have established t h a t

which proves the triangle inequality. Then the homogeneity o f the norm immediately follows from (4.2.23). Now suppose that

p := min(w0,wl) and

M :=

rnax(w0,wl).

Then we have

Further, since

L

is concave as a function o f t , we have

L ( t ; 2 ; 2 )L m ( t ) L ( l ;z ; so that putting 7 :=

Ilmlla, we get

2 ),

516

Selected questions in the theory of the real interpolation method

Together with (4.2.23), this inequality gives

1

(4.2.24)

II~IlC(f)

I P - q ); llzll .

Similarly, for x E A(Z), we have

L ( t ; 2 ; 2)L M(ll4la(a))m(t>> which leads t o the inequality

1

11415 M-' ( y ) 11~11a(a,

(4.2.25)

*

X

In order t o prove t h a t couple

:=

3,it remains t o verify llxkllX

<

5.

z k

It remains t o show that

+5 E

=

converges in

11zkIIX

With such a choice of

xk,

(Zk)k&J

2

c(2). Let

cx

and

us denote its

E X . We shall use the inequality for

established above, choosing for it

the L;-functional xk

is an intermediate space for the

.

In view of (4.2.24), t h e series sum by

L;,o(Z)

i t s completeness. Let

in the formula

( k E m)

we have

5

x f f k = 1 .

Thus, we have established that x E X , and the completeness is proved. It remains t o establish the interpolation property of the functor L;,o. This follows from inequality (4.2.26), t h a t will be proved below, and from embedding (4.1.4). 0

Let us now verify that the functor L;,o belongs t o the class follows from a more exact result formulated below. Theorem 4.2.14. The following equality holds:

R. This

Real interpolation functors

517

L;,o = KG , I

where 9 :=

L;,a(L,).

Proof.As in Theorem 4.2.11,

it is sufficient t o verify that the functor under

consideration is &invariant on any two couples. Thus, let

T E B(J?,?) and z = zo + zl,where

fix a number q

> 0 and

put

E

:= q min

llz;llxi

2; E

X i , i = 0 , l . We

if this minimum differs

k0,l

# 0, while zl-i

= 0.

Then in view of Definition 4.1.1 o f the class B,there exist elements y;

E Y,

from zero. Otherwise, we put

E

:= q

11z;llx,, where z;

such that

Here Y := llTllB(Y,P). In view of the choice of E and the definition of the L;-functional,

we then

obtain

As q + 0, we get

so that using definition (4.2.23), we have

Let us verify that all the results given above can be extended t o the category

A of couples of complete Abelian groups (a-couples).

Let us first indicate

the modifications that should be introduced in the definitions in this case. In Definition 4.2.9 o f the method o f constants, the concept of strongly rneasurable vector-valued functions with values in a Banach space appears. If

518

Selected questions in the theory of the r e d interpolation method

the Banach space is replaced by an Abelian group, t h e use of this concept is connected with difficulties which are irrelevant for our discussion. For this

reason, it is more expedient t o make use o f the following modification of Definition 4.2.9, which is equivalent t o the initial definition in the Banach case. Let @; be a quasi-Banach lattice over a space with the measure (R, C, p ) ,

i = 0,1, such that the couple

6 has the following property:

xn E C(6). For the given quasi-Banach lattice @ over 0 and a complete Abelian group

X , we consider the set o f functions f : R + X of t h e form

where ( E ; )is a family of disjoint subsets of finite measure. We denote by

a($) the set of functions o f this form for which

It can be easily verified that @ ( X )is a Abelian group (in general, incomplete). + Having now the couple @ of quasi-Banach lattices and an a-couple, we define the functor K6 by the same formula (4.2.18). We leave it t o the reader t o veify that all t h e statements proved above and concerning K6 can be extended from t h e Banach case t o the cateogry

A.Some difficulties are

associated only with K 6 ( 2 ) . However, we can use here the completeness criterion formulated in the book Bergh and Lofstrom [l],Sec. 3.10.

Let us now consider the corrections that must be introduced when the definition of the functor L G , is ~ extended t o the category 2. First of all, it is now inexpedient t o assume that the functions w; :

R+ -+ R+ is

a

surjection, with w;(O) = 0, satisfying the Az-condition:

(4.2.27)

SUP

w ; ( 2 t ) / w ; ( t ) < 00

(i = 0 , l ) .

t>O

Further, we now assume that @ is a quasi-Banach lattice over p t + . The &-functional

is defined by Remark 3.1.28. As t o the formula for the

functor, we shall use a modification o f Definition 4.2.12:

LG,~

Red interpolation functors (4.2.28)

llxllLa,+(y):= inf { A

519

> 0; llLsA(-;x ; r?)llo I 1) .

Here w'x(t) := (wo(A-'t),w~(X-'t)),

t > 0.

Since in the Banach case +

L;,(.; x ; X ) = L;(.; A-%;

-8

X),

definition (4.2.28) indeed generalizes Definition 4.2.12 for the category

2.

We leave it t o the reader as an exercise t o prove that L3.0 is a functor on the category [while proving the embeddings A L) L;,o L) C , it is sufficient

A

to use the Az-condition (4.2.27) instead of convexity]. In view of what was said in Sec. 4.1.C. it follows t h a t Theorem 4.2.14 is valid in this situation as well.

The validity o f the formula

for the functors o f the method o f constants and of the L-method is not sufficient for their computation, although it reduces this computation t o the 4

case of the couple L,.

For the sake of completeness, we shall show how

these functors can be computed in the case of power parameters (the proof will be carried out only for a functor of the

L- method).

case, see, for example, Bergh-Lofstrom [l]. For w,(t) := tri and

9 := Lp",we put

Then the following theorem is valid. Theorem 4.2.15 (Peeire). (a) If TO # r1, 0 < 9

Here we put

< 1 and 0 < p 5 00,

then

For the remaining

Selected questions in the theory of the red interpolation method

520

17 := d r l / r ,

q := r p ,

(4.2.30)

r := ( 1 - B ) r o - t - B r l .

(b) If a; := L:,:', where 0 < B

< 1 and 0 < pi < 00 (i = O , l ) ,

then

Here we put

1 P

:=

29

1-9 Po

-+ - . Pl

Proof.We shall require (4.2.31)

L ( s ; I ; r?) M K ( t ; z ; x')'"

uniformly in

t > 0 and

I

E

,

,E(d),where

s and t are connected via the

relation (4.2.32)

s := trlK(t;I ;2)"o-r1 .

Here L := L; with

q(t)

:= t'l.

Proof.We put

Similarly, we put

k(t):= K,(t ; I ; x") . The quantity i ( s ) is obviously equivalent t o the left-hand side of (4.2.31) and

I?

k(t)x K ( t ; I ; 2).Therefore,

and

i.We choose 10

it is sufficient t o prove t h e lemma for and x1 so that for a given E > 0,

Then a t least o m of the numbers

Red interpolation functors t Il~lllxllm

I l ~ o l l X o l m, lies in the interval [l,1

where

6 -+ 0 as

521

+ €1.

Consequently,

E 4 0.

This leads t o the equality

which is equivalent t o the statement o f the lemma.

Let us complete the proof o f item (a). For this purpose, we observe that in this case

Hence, after an appropriate change o f variables, we get the equality

where r is defined by formula (4.2.30). On the other hand, for p :=

00

we have, in the notation introduced in

the proof of Lemma 4.2.16,

where r and

are defined by the formulas (4.2.30). Comparing this expres-

sion with (4.2.33) we obtain result (4.2.29) for this case. Let now p

< 00.

Since

act)is an increasing function o f t , we have

522

Selected questions in the theory of the real interpolation method

Let us substitute the variables on the right-hand side with the help of formulas (4.2.31) and (4.2.32). Then according to the lemma, it will be equivalent t o

the quantity

J

t-drlpl;r(t)--dp(ro--rl)

d(k(t)'OP) w

&

Using t h e last two relations, we get

U

C . Let us consider in greater detail the E-interpoZation method which is important for applications (it was used in Chap. 1 in the proof of Theorem

1.1.5). The definition of the functors of this method is based on the concept of the E-functional [see formula (3.1.3)] and will be given straightaway for the category o f t h e a-couples. The properties of the E-functional in this situation were described in Sec. 3.1.C.

A

In order t o define the E-functor, we shall first introduce the concept of

parameter of the &-method. For this purpose, we consider the cone M of nonincreasing proper functions5 f : Sec. 3.1.6) Definition

MC

cM

lR+ + lR+ U (+m}.

Recall that (see

is a subcone consisting of convex functions.

4.2.17.

The function v : M + El+ U { O , + m }

is called a monotone quasi-norm

if (a) for a certain y

> 0 and all f , g E M

51.e. functions which are not identically equal to +m.

Red interpolation functors

523

(b) v(f) = 0 H f = 0 ; (c)

f I 9 * v(f) 5 4 7 ) .

If a function v is specified on the subcone M C and satisfies conditions (a)-(c) with y = 1 and, besides, for any X > 0

(4 4Xf) = W f )

1

then v is referred t o as a monotone norm. Using v , we can define the subcone

M” := { f E M ; v ( f ) < m } . An example of a monotone quasi-norm is the function

llfIl4

:=

):

f(t)p

1IP

(0

’

When p := 0, we put

Since the functions in

M are nonincreasing, the latter definition obviously Lo := L o ( R + , $) [see

matches the definition of the norm in the space formula (1.2.10)]. Therefore,

dt Mp=-MnLp(n4,,),

(OIPIOO).

Definition 4.2.18. (a) A normed Abelian group @ is called a parameter of the E-method on +

the category A if for some monotone quasi-norm v we have

and, moreover

Selected questions in the theory of the real interpolation method

524

(b) A normed space 9 i s called a p a r a m e t e r of t h e € - m e t h o d o n t h e category

3

if for some monotone norm t h e equality in condition (a) is satisfied,

and

For the further analysis, it is useful t o note that for

9 we can take not a

group, but a subgroup in case(a) and a cone in case(b). 0

The following definition holds. Definition 4.2.19. If

x'

E

A and 9 is a parameter of the €-method, then the space E q ( 3 )

consists of the elements z E C ( 3 ) for which t h e quantity (4.2.34)

IIzIIEu(a,:=

inf{X

> 0; IIX-'DxE(.;

2 ; r?)llly

5 1)

is finite. 0

Here

Dx

:

M +M

stands for the dilation operator:

(Dxf)(t) = f ( W . A more natural definition is the one in which the quantity (4.2.35)

Il"llE;(a)

:=

IIE(-; t ; x'>llQ

is used instead o f (4.2.34). Unfortunately, the set of elements z for which it

is finite does not form an Abelian group. Under the additional assumption

that the operator

Dx

is bounded in \k for some X E ( O , l ) , we obtain an

Abelian group. However, i n this case also the correspondence 3 + IT;(-?) is generally not an interpolation functor on the category

2. The condition

under which E; is a functor can be easily derived from Proposition 4.2.20 formulated below and the following result:

Red interpolation functors (4.2.36)

EY.= E E ; ( L , , L o )

525

7

L 0 ( B + , $) is defined by formula (1.2.10). The

where the space Lo :=

relation (4.2.36) follows from definitions and the equality

E(t ; f ; L,, Lo) = f'(t) (see Proposition 1.9.4). Proposition 4.2.20.

If !# is a parameter o f the €-method on the category E* is a functor on this category.

d (respectively, 2),

then

Proof (Banach

Since, in view o f t h e definition of the E-functional

case).

[see (3.1.3)], in the case under consideration we have (4.2.37)

A-'E(At;

z;

2)= E ( t ; A-'z; d ) ,

formula (4.2.34) can be written in the form (4.2.38)

IIzIIEl(w) :=

inf { A > 0; llE(.; A-'z;

Further, in view o f the embedding A(Ml, M,) (4.2.39)

X(O,l]

E

Q

~)IIY.

5 1)

C !#,

.

Let us verify that the quantity (4.2.37) has the property of a norm. The validity of the condition 2

=0

* 1141 :=

ll"llEu(2) = 0

follows immediately from definition (4.2.38), and the converse, from inequality (4.2.46), which will be proved below. The positive homogeneity of also follows from the definition. We now use the inequality (4.2.40)

E ( - ; -, C zn. d )5

where a ,

:=

0

C An A,/C A,

< C A, < 00,

a,E(.;

11 . I(

5 ;d ) , An

It is assumed here that A, 2 0, and that the series Y ,- x, converges i n C(x'). If i n (4.2.40)

we take only A1 and

A2

[see (3.1.45)].

different from zero, this inequality and definition

(4.2.38) lead t o the triangle inequality for

11 . 11.

Let us now establish the completeness of the space E*(r?). Suppose that (Z,),~N

is a sequence in this space and

526

Selected questions in the theory of the real interpolation method

In view of inequality (4.2.46), which will be proved below, it follows that the

series C x, converges t o the sum C(2). It remains for us t o verify that the sum of this series belongs t o

+ 2"

Eq(J?). For this, for

E

An

:=

llZnl[

12

7

a given

E

> 0 we take

E JV

and apply inequality (4.2.40) with the chosen (A,)

and (z,,). Taking into

account the monotonicity of the norm in Q and definition (4.2.38), we then obtain for x :=

C xn

Consequently, z E E ~ ( J ? ) . Suppose now that x E A(x'). According to the definition of the Efunctional [see (3.1.3)],

Then for X := IIxllac~,we have

so that the identity (4.2.37) and condition (4.2.39) give

I / E ( - ;;

q*I

I1X(O,l]llV

< 00 .

Taking into account definition (4.2.38). we hence obtain

Further, suppose that x E Ee(x'); then (4.2.42)

IIxllccy,= inf { t t>O

+ E ( t ; I ; 2))

Real interpolation functors

527

(see the proof of Proposition 3.1.10).

Using the monotonicity of t h e E-functional, we choose to > 0 so t h a t

E(to - 0;

2

; 2)2 to 2 E(to

+ 0; z ; 2).

Considering inequality (4.2.42) we then have (4.2.43)

I E(t0 + 0; x ; d )+ E(t0 - 0; z ; x') .

I l ~ l l ~ ~ j ~ ,

Consequently, for any s E (O,1) we get X(O,l](4

II4lc(a, I 2E(sto; 2 ; 2).

Hence, in view of the identity (4.2.37) and the fact that t h e E-functional is nonincreasing (in (4.2.44)

t ) ,for y :=

we obtain

IIx(o,l]ll*

3

I)*

y Iltllc(a,I 2t' IIE(-; ; d )

We now suppose that to 5 11z)( :=

(0

< t' I t o ) .

IIzIIEu(a,.Then it follows from (4.2.42)

and (4.2.43) that

11~11c(a,5 % I 2 1141 . In the remaining case of t o

> 11x11,we choose E > 0 so that t o 2 (1+ E )

and put in inequality (4.2.44)

t'

llxll

+ E ) 1 1 ~ 1 1 . Then we have

:= (1

2

11~11c(a,I 7 (1 + E l 1141

*

Thus, we have established t h a t (4.2.46)

1121Iq2, I 2 mm(1, l / ~II"IIE~(~, ) *

Together with what has been proved above, the inequalities (4.2.41) and (4.2.46) establish that Eq(2)is an intermediate space for the Banach couple

2. It remains t o verify the interpolation property of t h e functor E g . It follows immediately from the more general statement (4.2.48), which will be proved below, and from embedding (4.1.4). In the case of the category

2, it can be easily verified that

E*(x') is a

normed Abelian group. Indeed, in this situation it is sufficient t o use inequality (3.1.44) instead of (4.2.40) [and, of course, now make use of definition (4.2.34)]. The embeddings

528

Selected questions in the theory of the real interpolation method

are proved following the same line of reasoning as in the Banach case. The completeness of

E * ( Z ) is also proved as in the Banach case, but now in-

equality (3.1.45) should be used instead of (4.2.40). Finally, the fact t h a t

EQ is a functor is established with the help of the same statement (4.2.48) (which was proved directly for the category 2). 0

Let us now verify that the family

( E Y )also belongs t o the class R of

functors of real interpolation. Moreover, the following theorem is valid. Theorem 4.2.21.

where the isomorphism constant does not exceed 2.6 If Q is a parameter o f t h e &-method in the category

2, for any Banach

couple we also have

Proof.According t o Theorem 4.1.8 and what has been said in Sec. 4.1.C, is sufficient t o verify that the functor

Ew

it

is B-invariant on arbitrary couples

and z E E q ( 2 ) . We shall evaluate E ( t ; T(z); 9 ) . If 2 = z o + z l , z i E X i , i = 0 , l and E > 0, then there exist elements yi E Y;.

3,?. Let T E B ( 2 , ? )

such that for y := IITlla(a,p, (4.2.47)

llyilly,

+

IT lleillx, E

7

= 071

.

Let now z1 be chosen so that for a given 7 > 0 we have

IIziIIxl I t

and

111 - zi11x0 = 1 1 ~ / I I x ~(1+ q ) E ( t ;5 ; 2).

'When x' is a Banach couple, the space Eq(x') is of quasi-Banach type (in the assumption that V is a parameter of the E-method on the category A). Indeed, equality (4.2.38) is valid here, from which it follows that 11 . IIEVcn, is positive homogeneous. In particular, E~(J?,) is a quasi-Banach lattice.

529

Red interpolation functors

In view of the arbitrariness of ~ , >q 0, this gives the inequality

for any 7’ > y. Taking into account the definition o f the norm in

Eo(2)[see (4.2.34)],

we thus obtain

Let us show that for certain Q’s, the parameter E

~ ( I ?can ~ )be evaluated.

For this purpose, we shall use Proposition 3.1.16 according t o which

E ( t ;f ;

Lo)=

sup (Ifl(s) - t s )

.

S>O

Recalling the definition of the Legendre-Young transformation [see formula

(3.1.16)], we get in view of (4.2.38) (4.2.49)

IlfIIE&)

= inf {A

> 0; Il(X-l IfDAIlo 5 11 .

In particular, this leads t o Corollarv 4.2.22.

If w : R+-+ ELW,

R+is convex, then L,&V

7

where the isomorphism constant does not exceed two. In the Banach case we have the equality.’

Proof.In view of inequality

(4.2.49) and Theorem 4.2.21,

7For the definition of the transform h

+h v ,

see formula (3.1.16)

530

Selected questions in the theory of the red interpolation method

EL%Z Ko where @ is defined by t h e norm

Since w is convex, the inequality

is equivalent t o the inequality

(4.2.51)

((A-*

lfl)A)v

5wv ,

[see the identities (3.1.17) in

this connection]. In view of these identities, the

left-hand side of (4.2.52) is equal t o X-lf.

Thus, (4.2.50) can be written in

the form

Since

f^

is the smallest concave majorant o f the function

If1

and w v is

concave as the lower bound of linear functions, the right-hand side is equal t o sup

%.Thus,

s>o

In particular, we put

Eclp := EL;

,

where w ( t ) := t-" ( a > 0 ) .

Then from Corollary 4.2.22 we obtain the relation Earn

Y(a)(.)srn

>

531

Red interpolation functors where 29 :=

& and "/a):=

(1+ ~ ) a - ~ / The ~ + isomorphism '. constant

here does not exceed two, and in the Banach case we have the equality.

A similar result is known t o hold for 0

< p < 00.

Namely, the following

theorem is valid. Theorem 4.2.23 (Peetre-Sparr).

Eap

(-)dq,

where

19 :=

a+l

and q :=

A.

0

As another corollary of Theorem 4.2.21, we shall consider one of the possible versions of the reiteration theorem for functors of the &-method. Corollary 4.2.24.

Eal) g Ea, where we put (4.2.52)

:=

KW(E~~,E~,)(Z,) .

Proof. It follows from

Theorems 3.3.24 and 4.2.21 that it is sufficient t o

verify t h e equality ~ ~ . f ~ ~ E + (= & IlfllE+(Em) ,)

'

Since .f = (IflV)"l [see (3.1.17)], taking into account the second identity in (3.1.17) and (4.2.49), the left-hand side can be written in the form (4.2.53)

IlfIIE+(Em) = inf

{'

>

1

; ((A-1 Ifl)")lla

5

'}

*

Since the function gA is convex, being the upper bound of linear functions,

(SAT= 9". and thus the right-hand sides of formulas (4.2.53) and (4.2.49) coincide t o within notations. In the same way, we also obtain the reiteration theorem in which three E-functors take part. It is simpler, however, t o make use of the following result which is t o some extent inverse t o Theorem 4.2.21. Theorem 4.2.25 (Asekritova).

IC,

2 E K U ( ~ , , L owhere ), the isomorphism constant does not exceed two.

532

Selected questions in the theory of the r e d interpolation method

Proof. Suppose that we put

:=

E I<*@). We estimate ( ( x ( I E e ( f )where , For this we note that i f f : R++ R+, then

element

K*(L,,Lo).

I

the following inequality is valid for t h e greatest convex minorant:

(t > 0) .

f ( t ) 5 (1 - +lf"(Et)

Here E E ( 0 , l ) is arbitrary. Choosing now E := 1/2 and putting

E

:=

E ( . ; I;J, we then have (4.2.54)

IIX-'DxEllo 5 112X-'Dx/zEll~.

Next let us make use of the fact t h a t

where K :=

K ( . ; I;J. For any nonincreasing function f

in view of Proposition 1.9.4 and definition (3.1.16), we have

I<(t; f ; (L,,

LO))

= inf (f(s)

+ts) = fv(t> .

s>o

From these two equalities it follows then that

In view of Proposition 3.1.18 and formula (3.1.17), we have

( ( D A K ) ~=)( D ~ x K r = DxK

.

Therefore, the previous equality together with (4.2.54) gives

IIX-'DxEJ(* 5 2 IIX--'DJ<1II@ *

= 2 IlW. ; X-'a: ; X)Il* =

2

=

Il"llK*(2,

*

In view of definition (4.2.34), this leads t o the inequality

:

R+4 R+,

533

Real interpolation functors

11"11E,(a, I 2 II"IIKu(a) Conversely, suppose that

2

E

*

E o ( 2 ) . Then in analogy with the previous

case

In view of definition (4.2.34)

it follows hence that

From this inequality and the reiteration theorem for the K-method (Theorem 3.1.24), we immediately arrive at Corollary 4.2.26.

Eo(Ea,, Eel) S Eo, where we have put

*

:=

Eo (Eoo(Lm,Lo), Eo1(J5m,Lo))

B y way of example, let

*

us consider two cases of calculating the parameter

0 in relation (4.2.52). Other similar facts can be obtained with the help of the results analyzed in Sec. 3.8. Corollary 4.2.27. Let w; : BZ+ + Then

where we have put

R+ be

decreasing functions (i = 0 , l ) and w E Conv.

Selected questions in the theory of the re d interpolation method

534

\Tr := w,"

Proof. Since w v (\Tr*)V

. w(wp/w,").

E Conv, the function Q is concave as well. Then Q =

[see (3.1.17)]. Therefore, according to Corollary 4.2.22, the relation

t o be proved is equivalent t o the identity

.

KLY_(KL"7,K v ) 2 K L Z L2

00

Its validity follows from Corollary 3.3.13.

In particular, when w i ( t ) := t-"', a;> 0,

0<6

< 1 , we get

where

CY

(i = 0 , l ) and w ( t ) := t',

is determined by the formula CY a0 a1 .- ( 1 - 6 ) +6-.

a+l

a1+l

ao+1

A similar conclusion can also be obtained for t h e spaces Eaipir0 < pi

< co.

However, a more natural result is obtained not for these spaces but for the subspaces E:ipi. Recall t h a t according t o definition (4.2.35) we obtain +

II~IIE,&(a,:= llE(*>; 2 ; WllL;,-.

7

so that in view of definition (4.2.34) we have

Theorem 4.2.28 (Peetre-Sparr). If a.

# a1 then 7

where

CY

E:m

)BP

2 E:pl

+6al.

:= ( 1 - 6)a0

Proof. For a given norrned Abelian group X

II

*

IIY

:=

and for a given r

(11 . Ilx)' .

Then from Theorem 4.2.23 and equality (4.2.55) we obtain

> 0, we put

Red interpolation functors

535

(4.2.56) Here we have put Qi d i := , qi := Qi 1

+

A , r; := Q i + Qi + 1

1

(i = 0 , l )

Since the K-functional o f the couple on the right-hand side in (4.2.55) is 4

equal t o L ~ ( X B o q o , ~ ~where 1 9 1 )w,; ( t ) = tri on the right-hand side is equal t o L ~ , L ; ( ~ ~ o q , ,On ~ ~the l qother l ) . hand, in view of Theorem 4.2.15, this 4

+

space coincides, up t o equivalence o f norm, with the space ( X ~ o m , X ~ l q l ) ~ ~ , where we have put

r := (1 - d)ro +dr1, Further, since do

# dl,

77 := drJr,

s := r p

.

we obtain, applying the Lions-Peetre reiteration

theorem 3.8.10,

Calculating the parameters appearing in the right-hand side and using Theorem 4.2.15 once again, we finally get

It remains t o note [see (4.2.55) that in t h e right-hand side we have E;,(z). Finally, it should be observed that in spite of a bilateral relation between the functors of the

E- and K-methods (see Theorems 4.2.21 and 4.2.25),

the theory of the E-method is more complicated. For example, the analog of the key fact o f the K-method, viz. K-divisibility (see Theorems 3.2.7 and 3.2.12) is apparently not valid for the €-method. Nevertheless, most of the results described in Chap. 3 have corresponding analogs for the E-method also. As a typical example, let us consider the analog of the density theorem

3.6.1. For its formulation, we assume that 9 is a parameter of the E-method on the category

2,for which the dilation operator D Xis bounded in 0 for

X E (0,l). Recall that in this case E i ( X ) is a normed Abelian group (which is complete if 0 possesses the Fatou property). We assume in addition that for any X > 0 and any sequence (fn) c 0 n M such that limn-roollf,,ll~ = 0, we have

Selected questions in the theory of the real interpolation method

536

(4.2.57)

lim

n-o3

llXfnlla

=0

.

Then the following theorem is valid. Theorem 4.2.29.

The set A ( 2 ) is dense in E ; ( - f ) for each a-couple Mo n M , is dense in the cone CI, n M and iff (4.2.58)

CI, n M

Proof (suficiency).

6 Mp

for p := 0 , m

iff the subcone

.

We take z E E G ( 3 ) and construct the corresponding

approximating sequence. For this purpose, for n E yn E

-f

ZT

we take an element

X such that

and then put

In this case z, E XI and z, = (yn - z) - (x - y-,,) E X o so t h a t (2%)c A(x'), and it remains t o verify t h e convergence of this sequence to z . For this purpose, we shall estimate t h e E-functional of the element z -

2,.

According t o the definition of the E-functional, for any z satisfying the inequality llzllx,

5 t , we

have +

E ( t ; z - 2,;

X )I 112 - Yn

+ Y-n

- 211x0

I

Here y 2 1 is the constant from the generalized triangle inequality for the a-space Xo. Taking the greatest lower bound over all z and taking into account the second inequality from (4.2.59) and the fact that the E-functional is decreasing, for

t 5 2-" we have E(t;z-sn;~)13y2E(t;z;~).

Similarly, choosing z := y+ and using (4.2.59), for 2-"

5t I 2" we have

537

Red interpolation functors

Finally, for t

> 2"

v(z)

we put

:= z

- z,

llzllxl

where

I t . Then

in view

of (4.2.59)

II4z)IIxi

I Y'

{IIgnIIxi

+ IIY-nIIXi + IIzIIxi} I3 7 9

*

Since z is arbitrary, it thus follows that

~ ( 3 7 2 tz; - 2,;

2)I inf{llz - z,

- w(z)~~x,, ; llzllxl

5 t}

=

+

=

E ( t ;2 ; X ) .

Combining these estimates and assuming that X

:= (3y2)-l and t h a t

+

E := E ( . ; z ; X ) , we obtain

E ( *; x - z n ; 2) 5 X-' { E . x(o,z-n]

+E

+ D A ( E ( ~ " ) x ( O , At - ~ ~ ~ )

. X(X-12n,m)l .

Using further the monotonicity of the @-norm, the generalized triangle inequality and the boundedness of the operator

DA

in @, and taking into

account (4.2.35) and (4.2.57), we obtain 112

- znllE;(q

-, 0

for n

-,

00

if this is true for the sequence

(IlE. x(o,~-~)IIo

+ lIE(2") . Xn + E . (1 - x n ) l l ~ ) n c ~

x,,

Here := X(o,x-lzn). Let us verify that each of the two terms tends t o zero. While estimating the first term, we shall assume that E is unbounded on R+.Indeed, if

E E M,,

then in view of condition (4.2.58), for p

function f in M

n a, which

is unbounded on

:=

00

R+.Then

there exists a

starting from a

certain no we have

IIE . x(o,z-~)llo I Ilf

*

x(o.2-n)Ilo

>

and it remains t o apply the following arguments substituting f for E. Thus, suppose that E choose a sequence

4 M,.

Using the hypothesis of the theorem, we

c Mo n M ,

( f k ) k e ~

for which

538

Selected questions in the theory of the red interpolation method

lim

k-w

llE-fkll@

Since f k is bounded and

=0

.

E is unbounded and nonincreasing, for each k E PV

there exists n := n(k) such t h a t

1

( E - fk)X(O,Z-") > E ' X(0,Z-n) . 2 Since the sequence n(k) -+ n+w lim

00

as

k + 00,

IIEX(o,S-n)ll@ 5 2

we have

IIE - f k l l 0

it remains t o verify that for any function

@. Further, l e t for a given

Since

fk

E

M0f l M,,

> 0, the number k

E Mo,there exists m :=

*

n M , we

fE

For this purpose, we take a sequence ( f k ) C

=0

have

converging t o

f in

:= K ( E ) be such t h a t

m(&)such that the support of f k and

(1- x m ) = X ( ~ - I ~ - , ~do ) not intersect. For this reason,

2 n(e) the point suppfk. Therefore, for any index n satisfying the condition 2" 2

) that for n Further, there exists a number n ( ~ such

2" 6 max {X-12m(c), 2"@)}, we have

Finally, for the indices n indicated above we have

Red interpolation functors

539

Thus, relation (4.2.60) is proved, and we have established that

llz - Z,,I(~;(Z)

+0

as n

+ 00.

Let us prove the necessity of the conditions of the theorem. Suppose t h a t

A(x') is dense in E ; ( X ) for any a-couple x'. Then L , n Lo is dense in E;(L,, LO).In view of Proposition 1.9.4, here E;(L,, Lo)n M = 0 r l M . Since, moreover, L, n M = M,, it follows that Mo nM, is dense in CP n M . Further, let us verify the necessity o f condition (4.2.58). For instance, we assume that

(4.2.61)

0 nM

c Mo

and consider a Banach couple

x'

such that

We can take X o := c[O,11 and X1 := c1 [0,1].Since in view o f embedding

(4.2.62) the function E ( . ; z ; x') belongs to M , for any

5,

it follows that

Ei(d)= EinMm(x'). Therefore, we can assume that 0 n M

c M,,

and hence

C

Mo n M ,

by

hypothesis. Since the definition of the parameter o f the method implies that -#

the inverse embedding also holds, the linear spaces EG(X) and

E&onMm(z)

coincide. Then, taking into account (4.2.36), we have by Theorem 4.2.21

[see the formula preceding equality (4.2.49)], it follows that

l l ~ ( -f; ;i , ) l l M o

:= i d { t

> 0 ; ~ j l " ( t= ) 01 = sup If(s)l . S>O

Moreover,

540

Selected questions in the theory of the real interpolation method

Consequently, we obtain

Combining this equality with (4.2.62) and Theorem 3.5.9, we obtain

E&,nM,(Z) G Ac(x') = Xf . Thus,

E $ ( Z )and X ;

coincide as linear spaces, and since they are conti-

nuously embedded into

E(a),we have in view of the theorem on closed

graph

By hypothesis,

A ( d ) equals X1 and is dense in E G ( 2 ) . Therefore, the

obtained relation contradicts (4.2.62), and the embedding (4.2.61) does not hold. Similarly, taking t h e transposed couple (4.2.62), we see that the condition

c M,

ZT

:=

( X I , X o ) , satisfying

is necessary.

U

Example 4.2.30. Let us take @ :=

1

L;, 0 < p < 00

1

(4.2.63)

w-"(t)

dt

and suppose that

< 00 .

0

Then A(@ is dense in

E$(x').

Indeed, it follows from (4.2.63) that

L; n M q! Mq

for q := 0,00

.

It can easily be verified that condition (4.2.63) is also necessary.

D. Finally, let us consider three other interpolation methods whose functors belong t o class

R. In contrast

t o the methods considered earlier, they

R e d interpolation functors

541

all can be expressed in terms o f the J-functors rather than A'-functors. We shall begin with the analysis of t h e Lions-Peetre method of averages. For +

this purpose, we consider a couple CJ of Banach lattices over a measurable space

R with

a a-finite measure p.

Definition 4.2.31.

The couple

6 is called a parameter

(0)

# A(&) c Ll(R)

of t h e method of average3 if

f

Let us now consider a subset of those elements z E

E(d)which

can be

represented in the form (4.2.64)

z =

J

u(w)$p(w)

n with a strongly measurable function u :

R + E(x') with range in A ( 2 )

Let us define a space J3(d)with the help op t h e norm (4.2.65)

IIzIIj6(y) := inf u

max

II~~~o,(x.)

.

i=O,1

Here the lower bound is taken over all representations (4.2.64), and @(X) denotes the space o f all strongly measurable functions f :

fl + X for which

the norm

Proposition 4.2.32.

If 6 is a parameter of the method of averages, then J6 is a functor.

k f . Let us show that the linear operator I defined by the right-hand side of (4.2.64)

A(6(d))continuously

(CJo(Xo),CJ1(X1))

maps into E(x'). Here

is a Banach couple since CJp;(Xi)-+

i ._ .- 0 , l . Indeed, it follows from the embedding

6(x') :=

E(6)(E(d)),

A(6) -+ Ll(R) that

542

Selected questions in the theory of the red interpolation method

The equality (4.2.66) now defines an isometry

J g ( d ) z A(a))/KerI. Hence it follows that

J$(d)is a

Banach space. Besides, in view of the

previous inequality we have

-

Thus, the embedding

J6(2)

C(2)

is also established. Let us now verify the validity of the embedding (4.2.68)

A(d) +J 3 ( 2 ) .

Since A(6) # {0}, there exists a subset Ro and

If for

x := xno E @,n xE~ ( d we )put := p(Ro)-'X

.(W)

it follows that

x=J n Since, in addition,

udp.

. .x ,

c R, such that 0 < ~ ( R o<) 00

R e d interpolation functors

5

lIxlIJ~(B)

IlxllA($)

543

*

llxllA(a)

7

and t h e embedding (4.2.68) is proved. Thus, J $ ( d ) is an intermediate space o f the couple

2,and it remains t o

establish the interpolation inequality. If T E L(d,?)and z is representable in the form (4.2.64). then

Tx = J T u d p and n

Let us now define the method of means as the family of the functors {Js}, where 6 runs through the parameters in Definition 4.2.31. The fact that this family indeed forms an interpolation method follows from the result established in Theorem 4.2.33.

J6 = J*, where Q := J$(&). +

Proof.Since J\u is minimal on the couple L 1 ,we have in view of the definition of Q

and it remains t o prove the inverse embedding. For this purpose, we take an element z in the open unit ball of the space

J&(X'). Then there exists a representation o f the type (4.2.64) such that (4.2.68)

1(~11o~(x~) 5 1 (i = 0 , l )

Let us take an arbitrary q

R,,

:=

.

> 1and suppose that

544

Selected questions in the theory of the r e d interpolation method

Without loss of generality, we can assume that p(R,,)

# 0.

If, in addition,

p(Rmn) = 00, using t h e a-finiteness o f t h e measure, we can represent Qmn in the form of a union of the disjoint sets with finite measure. Since in this case

the changes t h a t have t o be introduced into the proof under consideration are obvious, we shall confine ourselves t o the analysis o f the case when

p(Qm,)

< 00

Thus,

Z. Z}is a

for all m , n E

{Q,,;

m,n E

family of disjoint sets of finite positive

measure. Obviously,

Let us now define a function V : R --+ A ( 2 ) equal t o zero outside

U Rmn

and put

V ( w ) := Then z can also be represented in t h e form 5

=

J

v(w)dp(w)

n

and it can be easily seen from the definitions of

llV(~)lIx,I q'+' IIu(w)Ilx, Further, l e t the function w : R

--t

Qmn

s ( R + ) be defined by the formula

:= J ( q m ; V(W); X)X[qm,qm+l)

and equal t o zero outside

U Rmn.

V that

(i = 071) .

+

W(W)

and

for w E Rmn

From this definition and the preceding

inequality, it follows that

(4.2.69)

llw(w)llLi :=

J

$dt 5 q'+'lnq llu(w)Ilx,

(i = 0,1) .

R+ Since u E

A(6(2)),

it follows therefore that w belongs t o t h e space

A(6(El)). Then by analogy with (4.2.67), means that

t h e embedding A(6)

w is integrable on R. Thus, we have

L)

Ll(Q)

Red interpolation functors

545

where we put

zmn :=

J

u(w)dp(w).

Qmn

Since here

for the vector function

h in the integrand on the right-hand side we have

9 h 9

= - f(t)

.

Thus, in view of the definition o f the functor Jq and the choice o f into account

9,taking

(4.2.68)and (4.2.69)we get

llxllJq(T)5

IIJ(t;

+

h ( t ) ;x>llq 5

9 & llfllJ~(&) I

9

ll~llo,(x,)I q3

< - m a llwllo,(L;) L q3 i=o,1

*

:=0,1

Since q is arbitrary, it follows hence that 2 belongs t o the unit ball of

Jq(2).

Thus, we have established that

J$(2) L

Jq(2).

0

The parameter J3(z1) appearing i n Theorem

a;

L:;8, 9 E (0, l), i and Lofstrom [l],Theorem 3.12.11.

for the case

:=

4.2.33is calculated only

= 0 , l [see i n this connection Bergh

Let us now consider the (generalized) method of traces developed by Lions. For the definition o f the functors of this method, we shall use Definition

4.2.34.

7) is called a parameter

A couple 6 of Banach lattices over the space (B+, of the method of traces if

546

Selected questions in the theory of the red interpolation method

and if, further, the operator

is bounded in Oo while the operator

is bounded in

91.

If 8 is such a parameter, then we put

ll"lITi(l)

(4.2.70)

:= inf {Ilflloo(xo)

+ Iltf'(t)llo,(a))

where the lower bound is taken over the set of absolutely continuous functions

f

: JR, + C ( x ' ) such that

(4.2.71) lim f(t) = 2 , t++o

t:lim f ( t ) = 0

(convergence in

C(r?))

.

The fact that T6 is a functor and the family (T6)forms an interpolation method follows from the result contained in Theorem 4.2.35.

T6 2 JG.

Proof. Let

us first establish the embedding of the right-hand side into the left-hand side. For this purpose, we take x E Ja(r?). Then there exists a

representation

such that the following inequalities hold:

Let us consider the function

Red interpolation functors

547

Then we have the inequality 60

ds

I M ~ ) II I ~J ~I I ~ ( S ) I I ~y~ = B ~ ( I I ~ ( * ) I I ~. ~ ) ( ~ ) t

Since the operator Ho is bounded in

Q0,

we obtain

Besides, tf'(t) = -u(t) so that

Iltf'(t)ll@l(Xl) = II~ll@l(Xd. Combined with (4.2.72), this gives II"IITa(2) I 2 ( 1

+ llHoll> 11~11J&

-

Let usnow establish the inverse embedding. For this we choose z E

TG(-~)

and assume that f is such that the limiting relations (4.2.71) are satisfied and (4.2.73)

Ilfll@o(Xo)+ Iltf'(t>ll@l(Xl)5 2 11~11Ta(2).

Weput u ( t ) := -tf'(t). It follows from definition (4.2.70) that ~ ( tE) A(x'). Also, in view of (4.2.71). we have

1

(4.2.74)

dt

u(t) 7 = -

Rt

1 f'(t)dt

=z .

R+

Further, let

where the function and

'p

E C r ( l R + )has the support [2-',1],

is nonnegative

548

Selected questions in the theory of the r e d interpolation method

This equality together with (4.2.74) implies t h a t (4.2.75)

5

=

J

u(s)

ds s =

J mt

Rl

J

=

e(t>

dt

.

R 1

It follows from the assumption about the support of cp that 2t

I I ~ ( ~ L) (max I I ~ ~9)

J

IIu(S>llxldt= ( m m cp>Hi(IIullxl)(t).

t

Since the operator H1 is bounded in

Q1,

we obtain in view of (4.2.73) and

the choice of u

llcll*l(xd L 2(max 'P) IlHlll .

II"IIT&(R)

.

It remains to estimate t h e norm of ii in a o ( X 0 ) . Integrating by parts and taking into account the choice of u , we get

Hence it follows that m

m u Ilc(t)llxo I

ds

Id1 J Ilf(s)llxo y

= (max I d l ) ~ o ( l l f l l x o ) ( t )

t

a0 and (4.2.73),

Using the boundedness o f Ho in Il~llOo(X0)

I

we finally get

lcp'l . I l H O l l . I l f l l * o ( x o )

I2 max

19'1 I l H O l l

I

II"IlT~(d).

Together with equality (4.2.75), the obtained estimates of the norm of give

lI4lJ6(X) I 2(m=

IV'I

. llffoll + m a IcpI . llHIl>II+&(2) .

U

Red interpolation functors Thus, t h e embedding

549

T&(d) ~t J$(x') is established as well.

0

Finally, we shall describe one more interpolation method which, j u s t as

the &-method, is based on t h e theory of approximation.8 In order t o determine this method, we consider a certain intermediate space ip o f t h e couple +

(Ly,Lo)E A

which has a monotone norm, and put

llfll~

:= i n f { X

> - ; IIX-'Dxflla- I 1 1

Further, we define for an arbitrary couple

2

E

A t h e space B a ( 2 ) ,

assuming that for z E C(x')

where the lower bound is taken over all sequences (zn) satisfying t h e condition

nlymIIz -znIlxo = 0 . If in the above definition we substitute ip for 6,we obtain the definition of I(znI(xt5 2 "

7

72

E

7

BG(2). If the operator Dx is bounded in Abelian group coinciding with B a ( 2 ) . The fact t h a t Ba is an interpolation functor

for X E ( 0 , l ) . we obtain

the set an

and that the family

{Ba}

is an interpolation method follows from the result presented below without proof. In order t o formulate this result, we choose p

:= p ( 2 ) E (0,1] such

that the following inequalities are satisfied : 1JP

(i = 0 , l )

.

The existence o f such a p for each a-couple is guaranteed by the AokiRolevich theorem (see Bergh-Lofstrom [l],Lemma 3.10.2). Further, let Tp be an operator defined by the formula "he

so-called telescopic method for the proof of inverse theorems, as proposed by

S.N. Bernstein.

550

Selected questions in the theory of the r e d interpolation method

By T i 1 ( @we ) denote the set o f functions from f E Lo + Li,for which llfllT;1(4)

:=

l l ~ p f l l o< +a

*

Theorem 4.2.36 (Asekritova). (a) If

0 is a parameter of the E-method and 0

-

Lo or X

+

-

= X " , then

E&f) s B+(@)(Z). The same is true for

EQ and BT;I(&) also.

(b) We put \E := Ea(Li,L0). Then

= &(Z) .

B,y(Z)

In particular, we obtain Corollary 4.2.37.

If the operator Tp is bounded in

E4

a, then

BT;I(*) .

0

E. Concluding the section, we shall consider one more application of Theorem 4.2.2 t o the proof o f the theorem on the K-divisibility. Thus, we shall give a new proof of this fundamental fact. Thus, let x E (4.2.75)'

E(d)and (9,) c Conv be such that C

IS(.;I ; Z) 5 w

:=

C 9, .

We shall limit ourselves t o the (basic) case

cp,(l)

< 00

and

551

Real interpolation functors

(4.2.76)

w !$ ~5;

(i = 0,1)

Let us consider the functor F := F(pn)defined by the formula

Obviously,

F(2)consists exactly of those

elements z for which the K - d i -

visibility takes place relative t o the sequence (9,). Therefore, it is sufficient to establish that (4.2.78)

F

K L ~,,

where the isomorphism constant 7 is independent of (9,).Indeed, if the ~
according t o (4.2.77), the following representation holds z =

C z,

(convergence in

(X)),

with 4

K ( . ; 2 , ; X )I YPn

7

nEN

*

In order to establish (4.2.78), we must verify that F is a functor. Only the embedding

A-tF has to be proved. For this purpose, we take z E

A(z),such that condition

(4.2.75) is satisfied for it. Then

K ( t ; ; x') 5

IlzllA(2)

1 min(l?t)5 - I l z l l A ( d ) w ( t )

so that for 7 := 1

4)

II4IKLy,(2) 5 7 lIzlla(a) .

4)

7

552

Selected questions in the theory of the red interpolation method

Thus, the norm (4.2.77) does not exceed y ( ( ~ ( ( ~and ( d the ) , embedding is proved. Further, we shall make use of the fact (see Proposition 3.2.13) that the

zl

-.

are K-divisible with the constant couples L , and and definition (4.2.77) lead to

F(I?)= KL&(I?) for I?

(4.2.79)

:=

~ ( 2= ) 1. This fact

( Z I , ~ , ).

It follows from this equality and Definition 4.2.1 that F E

R. In view

of

conditions (4.2.76) and (4.2.79),

n ~ ' b#, ~ ( i ,( ) i =o,q

I

Therefore, according to Corollary 4.2.3(a), we have the isomorphism (4.2.78) with a constant independent of F [i.e. of (pn)].Thus, the K-divisibility is established. Remark 4.2.38. The restriction (4.2.76) can be eliminated from the proof. For this it should be noted that it suffices t o carry out the proof only for relatively complete

I?

couples [see (3.1.32)], and further item (b) from Corollary 4.2.3 has to be used. However, then an additional analysis has to be carried out to establish the independence of the constant of the couple I? and of the sequence (9,).

Stability of real method functors

553

4.3.Stability of Real Method Functors A. Th e theorem on the "commutativity" o f the complex and real functors, obtained by Lions, was the first of results similar t o those considered here. In an equivalent form, this result can be written as

where, as usual

1 .6 , := (1 - r]>s, 7761, - .-

+

Here

P,

-+2 Po

Pl

(O
C, is a functor o f complex interpolation (see Example 2.6.6).

T h e proof

is mainly based on peculiar properties of the comples method. Actually, it will be shown below that such a statement is true for an arbitrary functor

F

and a broad class o f functors of the IC-method.

Let us begin with the formulation of a less general b u t more frequently encountered result. Theorem

4.3.1(Brudnya'). ds

If the operator (Sf)(t)= J min(l,t/s)f(s) - is bounded in the couple S R+

$ of

Banach lattices, then for any functor

(4.3.2) F(I{ao, Ka1)

F we have

KF(8) *

Proof. Let us verify that without any restrictions imposed on the parameters of the K-method,

(4.3.3)

F(l<@o>

K@l)

Lf

KF($)

. 4

Since the K-functor is maximal on the couple

-

+

the embedding on this couple. For L,,

(4.3.4) F ( & o , & l )

Ijl

(Q :=

L,,

it is sufficient t o verify

embedding (4.3.3) has the form

F ( 6 ) ).

Since the left-hand side is obviously embedded i n t o the space Q and is an 4

interpolation space of the couple L,, the embedding (4.3.4)follows f r o m the

554

Selected questions in the theory of the r e d interpolation method

fact that

& is the maximal interpolation space contained

in 9 (see Remark

3.3.9). Similarly, using the minimality of the 3-method on the couple prove that for any parameters of this method, we have (4.3.5)

Jqo,

-

il,we

.

F(J@,,

Since, in addition, we have in view of (3.5.34)

J*,

c--t

K@ (i = 0 , l ) ,

the embeddings (4.3.3) and (4.3.5) taken together give (4.3'6)

J F ( 8 ) L,F ( J @ 0 7

J@l>

Lt F ( K @ 0 7

K@l)

L,K F ( 6 )

'

It remains for us to note that if S E L($),this operator is bounded in F ( 6 ) . Consequently, the relation JF(6, KF(a,holds (see Corollary 3.5.15). Together with (4.3.6) this result leads to (4.3.2). 0

Let us describe a considerably broader class of parameters of the

-

Ic-

method for which relation (4.3.2) holds. For its definition, we assume that (4.3.7)

cpi

CO(E,)

(2

= 0,l)

and that one of the following conditions is satisfied. (a)

The couple

9 is relatively complete.

(b) The parameters

@j

are nondegenerate, i.e.

%\(Lo, u L L ) # (see

,

(i = 0 , l )

Definition 3.5.4).

In this case, the following theorem is valid. Theorem 4.3.2. In order that the relation

Stability of r e d method functors

be valid for an arbitrary functor relation be satisfied for

x'

:=

555

F, it

is necessary and sufficient that this

El.

Proof. The necessity is obvious.

In order t o prove the sufficiency, we note

that in view of (4.3.3) we must only verify the embedding

Kp(d)(z)

(4.3.8)

~f

.

F(K@,,

To prove this embedding, we make use of the fact that if one of the conditions of the theorem is satisfied, the following isomorphisms are valid:

JQ,(d) (i = 0 , l )

K@,(x')

(4.3.9)

-.

where Q; := K@,(L,)and

K6(d)

(4.3.10) where

6

:=

J @ ( x ' ),

F ( 6 ) and 6

:= K6(z1).

Indeed, (4.3.9) follows from Corollary 3.5.16( b) under t h e condition (a) and from Theorem 3.5.9(a) under the condition (b).

account the fact that, since condition (4.3.7) is satisfied, Kcp,(-f)

i = 0,l.

-

Here we take into

Co(r?),

Relation (4.3.10) is proved similarly. We must only take into account that

6 so that

6

F ( 6 ) Lf C(6)

:=

K&(x')

L)

Lf

CO(E,),

Eo(x'), and verify the nondegeneracy of the parameter

for the case when both parameters cp, are nondegenerate. But if the

parameter

6 is degenerate, then A($)

~t

F(6) := 6 ~t LO, U LL ,

A(+) is degenerate. However, the nondegeneracy of the parameter @ is equivalent t o the existence of a function g E Conv such

so that the parameter

that (4.3.11)

g(+m) = +OO

,

g'(+O) = +m

,

556

Selected questions in the theory of the r e d interpolation method

[see (3.5.9)]. Therefore, there exist functions g; (4.3.11).

E @;, i = 0,1,

Then the function g := min(go,gl) belongs t o

satisfying

+

A(@)and satis-

fies condition (4.3.11). The contradiction obtained completes the proof o f relation (4.3.10). Thus, the relations (4.3.9) and (4.3.10) are satisfied. It follows thus that in view of the minimal property of the J-functor, it is sufficient t o verify the embedding (4.3.8) for the couple 31. However, for this couple (4.3.8) is satisfied by the hypothesis of the theorem. 0

B. First o f all,

let us consider a generalization o f the Lions theorem. For

this we shall need the definition of the Calderdn-Lozanovskii construction. Namely, let cp

:

lR:

--f

R+ be

a homogeneous (linear) function that is

nondecreasing in each argument, and l e t

$

:=

(a0,Q1) be a couple o f

Banach lattices over a measurable space (0,C, p ) . Definition 4.3.3.

y ( z ) consists o f the set o f measurable (classes of) functions f such that there exist functions f ; E a;, Ilfille, 5 1, i = 0,1, satisfying The space

Moreover, we put (4.3.13)

Ilfllqp(~)

:= inf X

It can easily be verified that

.

cp(5)is a Banach lattice (in the norm (4.3.13)).

It follows fro m the definition that

4

In particular, since cp($)

= C(d) for cp(z,y)

:= max(z,y) and cp(@) =

A($) for cp(x,y) := min(z,y), cp(6) is an intermediate space o f the couple $. Henceforth, we put cp’(z,y) := zl-’y’, 0 I 19 5 1, and

Stability of red method functors

a:-'@:

:=

cpo(6).

557

This particular case is directly related t o the complex

interpolation method. Namely, t h e following theorem is valid. Theorem 4.3.4 (Culdero'n).

If one of the spaces

@i

is separable, then

,

C'(3) 2 @;"-;

where the isomorphism constant does not exceed two. 0

In the case when @; are complex-valued Banach lattices, isometry takes place. Remark 4.3.5 ( S h e s t a h ) . Without assuming t h a t (4.3.14)

@i

is separable, we have

C19(6)2 cpa(6)' .

The above results and Theorem 4.3.2 lead t o Corollary 4.3.6.

If the operator S is bounded in the couple G,then

where @ :=

(a:-#@!)'.

Proof. Everything follows

from Theorem 4.3.1 with F = Cs and from the

identity (4.3.14). 0

Remark 4.3.6.' An application of Theorem 4.3.2 makes it possible t o extend the range of applicability of the identity (4.3.15). We leave it t o the reader t o verify that it is sufficient that cp satisfies the conditions in the theorem cited above and that one has the embedding

Selected questions in the theory of the real interpolation method

558

(4.3.16) S-'(@A-ff@;) ~t S-'(@o)l-sS-'(Q1)ff . L e t us illustrate by an example that condition (4.3.16) can also be satisfied for +

the couple @ on which the operator S is unbounded. Namely, let @; := L z , where

Ilntl-"*-'

,

O < t 5 e-'

,

,

e-'
.

w ; ( t ) :=

(1 Here a;> 0,

i = 0,l.

Simple calculations show that

Thus,

( s w ~ ) ' - " s w , ) ~x s(w;-"w,s) and hence embedding

,

(4.3.16)is satisfied for the couple chosen.

Remark 4.3.7. A similar result may be considered for the upper complex method C8 (see t h e definition o f this method in Bergh-Lofstrom [l], Sec. 4.1). Under the hypotheses o f Corollary 4.3.6, t h e following relation holds here:

C8(K*,,%)

KO

,

where @ :=

The proof is based on the relation (Shestakov)

C"6)

G

(@;-";)c

.

In particular, if the @; have the Fatou property, the right-hand side is equal to

+A-ff@;

since the Calder6n construction preserves this property.

Th e following calculations might be helpful for applications o f the above resuIts.

Stability of red method functors

559

Example 4.3.8. (a) If, as usual, p;'

:=

1-9

9

-+ - , Po PI

then

(see Proposition 1.3.8).

(b) Similarly, for the Marcinkiewicz-Lorentz spaces, we have

(c) If Mi :

R+4 R+are convex functions equal t o zero a t zero, we have

for the corresponding Orlicz spaces

Let us consider an example illustrating the application of the theorems of the previous subsection t o functors which are in a certain sense a "continuation" of the Calder6n-Lozanovskii construction for arbitrary Banach couples. However, the possibility of such an extension is limited by the following statement. Theorem 4.3.9 (Lozanovskii). There exists a couple

6 of Banach lattices for which a:-'+;

is not an in-

terpolation space. 0

Nevertheless, the following statement of fundamental nature is valid.

560

Selected questions in the theory of the real interpolation method

Theorem 4.3.10 (Ovchannakov). The functor cp is a (not exact) interpolation functor on the category

L of

Banach lattices possessing the Fatou property. 0

The proof is based on the interpolation functors cpu and

cpl

which will be

described below. It involves rather accurate results from the theory of absolutely summing operators and nuclear operators (in particular, the Grothendieck inequality). Here we shall consider an elementary proof of a somewhat more general result according t o which cpc is an interpolation functor. It leads t o the Ovchinnikov theorem since the Calderbn-LozanovskiT construction preserves the Fatou property and hence cpc = cp for the couples possessing this property.

-

Theorem 4.3.11.

The functor pc is an interpolation functor on the category B L of Banach lattices.

Proof.Suppose that

It is sufficient to show that (4.3.18)

llTfllvp"(q5 Y

7

where the constant is independent of f and

T. The

proof is based on the

following inequality:

(which can be easily verified), where the sum is taken over all possible functions

E

: (1,..., N

} -, {-1,1}.

L e t us begin the analysis with the following basic case. Let us suppose that +(t) := cp(1,t). Since @ is monotone and homogeneous, the function t -i+(t) is nondecreasing, while the function t -+ +(t)/t is nonincreasing.

Stability of red method functors

561

In view of Corollary 3.1.4, we can assume, without any loss of generality, that

+ E Conv. Let us first assume that + E Conv. Thus,

(4.3.20)

+(+O)

:=

F i +(t)= 0 ,

In view of the definition of

:= lim

+'(+00)

t++m

cp(6) and the second inequality

+(t) -0. t from (4.3.17),

there exist functions fi such that (4.3.21)

If11 ~ f l I lfol+( -) If01

7

Ilfill*,

L1

(2

= 0,1>.

Here we assume that t h e right-hand side vanishes a t the points where a t least one of t h e functions fi is equal t o zero.

Let us consider the set

and verify that (4.3.22)

[If

- fXnc8)Ilccq + 0

for s + 00

.

For this purpose, we put

52p

:=

{w;-ss-l}

,

Ql"' := { w ;

->

.}

Then from the definition of the sum C ( 5 ) and inequality (4.3.21) we get

Ilf - fXnc.)Ilc(d) L IlfX,pllQo + IlfX,~.)llQl

s

This relation and the conditions (4.3.20) lead t o the statement (4.3.22).

Let us now use Proposition 3.2.5. According t o this proposition, for a given q > 1 there exists an increasing sequence (tk)E-2m c R+u {+m} such that (a) t--2m := 0,

tzn :=

+00

for m,n

=0

limk+-oo

tk

limk++m

tk=+m

< +00, f o r m := +m

,

f o r n :=

;

(4.3.23) +00

562

Selected questions in the theory of the real interpolation method

(b) for t E [t2k,tZk+2],we have

(c) finally, for all

t E (O,+m)

For the sequence (tk), we consider the sets

In view of this definition and inequality (4.3.24), we have

where

Xk

:=

xn,.

Therefore, for the functions

the following inequality holds:

(4.3.28)

fh 5 q lf;l

(i = 071) .

Let us now take in inequality (4.3.19)

Then this inequality gives

If we denote the left-hand side by g&(w), then taking into account embedding (4.3.17) and inequality (4.3.28) we have

Stability of real method functors

563

A similar inequality also holds for the function

Thus, we have proved that

Finally, let us consider the function fN

fxk

:= Ikl
and for a given w E fl define the number

In view of the definition of the function gk, we then have

Combining this relation with inequality (4.3.29) and taking into account the definition of the norm of cp($),

we get

Selected questions in the theory of the red interpolation method

564

Since in view of (4.3.23) and (4.3.20)

fN

+ f in

c($),T f N

+ T j in

c(\t).

Therefore, from inequality (4.3.30) and the definition of relative completion, we obtain inequality (4.3.18) withg q(q y := -.

+ 1)

q-1 To complete t h e proof, it remains t o consider the case when @ fZ Convo. We put @ o ( t ) := $(+O)

Then @ = @o

+@'(+m)t,

$1

:= @

- $0 .

+ G1 and it can be easily verified that

q 9 o ( b ( q

A

4)

q90(6),vI(6)).

Consequently, it is sufficient t o prove inequality (4.3.18) for q o ( $ )and cpl(6) separately. But for the former space, this inequality i s obvious, while for the second it has been already proved, since

(PI

E Convo.

0

Remark 4.3.12. -t

By analogy, 9 ' is also an interpolation functor on the category B L . --+

Let now

F be an arbitrary functor whose contraction t o B L coincides

with the functor y c (or yo). The examples of such functors will be considered below. Here we shall formulate Corollary 4.3.13.

If the operator S E L($)then

Proof. It is sufficient t o apply Theorem 4.3.1. 0

gFor q := 1 + 4,we obtain the minimum value 7 = 3

+2 4 .

Stability of r e d method functors

565

Finally, let us consider examples o f interpolation functors which coincide, t o within equivalence o f notms, with the functor cpc or cpo on the category d

B L . Such a functor was constructed for t h e first time in 1971 by Peetre for the power case. We shall consider a generlization for the case of an arbitrary cp. However, we shall be only interested in the case when the limiting relations

(4.3.23)are satisfied for the function +(t) := cp(1,t). Two of the

definitions presented below refer t o exactly this situation. Example 4.3.14(Peetre).

It should be recalled that the sequence

convergent in the Bsnach space

(xk)kcz

is termed unconditionally

x if for a certain y > 0 and all ( A , )

C

R

we have

Let us define the space

(x'),+,as the set of elements x

E C ( x ' ) for which

there exists a representation

(4.3.32) x =

C xk

(convergence in

~(2))

k

E A(x') and the sequences ( 5 unconditionally in the spaces X , and

such that x k

space is chosen as the least upper bound of the constants in corresponding inequalities o f the type

(4.3.31)over all representations (4.3.32).

It can easily be verified t h a t the correspondence r? +

(r?)w .

IS

an inter-

potation functor. Example

4.3.15(Gustavson-Peetre).

Replacing in the previous definition the absolute convergence by weak un-

conditional convergence, we arrive a t the definition o f the functor

(2, cp).

It should be recalled that the definition o f weak absolute convergence differs from the concept o f unconditional convergence in the respect that instead of

(4.3.31),the inequality

Selected questions in the theory of the r e d interpolation method

566 is used.

Example 4.3.16 (Ovchznnikov). Let 'p be such that @ E Convo. For the weighted couple put

20' := (wo, wl)we

(~(5 :=) 'p(wo, w'). Further, let the space of sequences 1; be defined

by the norm

11Zllr;

c

:=

1

1/P

IZkWkllP

{ k E z

.

We shall consider all functors G for which

22") &

G( I$, I: )

for all w' and let c p I denote the minimal among these functors. Its existence follows from Theorem 2.3.24 (Aronszajn-Gagliardo). Similarly, we consider

all functors H for which

H(1;"0,I;"')

A )"(:I

for all w and l et 'p,, denote the maximal among these functors. The relation of these functors t o the Calder6n-Lozanovskii construction and t o each other is described in Theorem 4.3.17. (a) (Ovchinnikov)

(c) (Junson)

567

Stability of red method functors Besides cpr)’ = GU, where

@ ( t ):= l/p(l/t).

C . Let us consider another type of stability theorem referred t o a subfamily of the family { E @ ( x ’ ) } For . this purpose, we define a linear approximation famiry as a couple

(B, {&}nEz) consisting

of a Banach space

B continuously embedded in a linear topological space 7, and family A := (An),€Zof subspaces of 7.

a monotone

Thus, (4.3.33)

R, C An+*,

12

EZ

.

We assume in addition t h a t

Let us define the best approximation of the element x E function e A : (4.3.35)

7 x Z --t R+U {+co},

7 as

the

defined by t h e formula

e i ( x ; B ) := i d {IIx - allB ; a E A,}

Here the right-hand side is taken equal t o +m if

2

(n E

$B

Z).

+ A,.

Obviously, the sequence (4.3.35) is nonincreasing and hence belongs t o o f bilateral nonnegative nonincreasing sequences s : 23 the subcone Add)

-t

lR+ u {+m}.10 c

Further, l e t us suppose that of the €-method on the category

is a subcone which is a parameter

6 (see Definition 4.2.18).

Definition 4.3.18.

The approximation space E,(A; B ) consists of elements z E 7 for which (4.33)

( I x l l E * ( A ; B ) :=

II(‘$(z;

0

“The sequence s = +co is excluded.

B))ne-ll@ <

*

Selected questions in the theory of the real interpolation method

568

does not contain the sequence identically equal t o +oo, we have

Since

&(A; B ) C B

(4.3.37)

+U

An .

n

U A,

Let us define in

a structure of normed Abeliand group, assuming

n

that

"

llxll

:= inf (2" ; z E A,}

A,

n

for I

#

0 and that the left-hand side is equal t o zero a t z := 0. Let us

now define in

B

+

U A,

the norm of the sum

n

(4'3.38)

Il"llS+uA,

._

inf

'-

{11I011B

+ IlzllluA,) .

z=zo+z~

In all cases of practical importance, the normed Abelian group

B +U A,

is

complete. In particular, the completeness can easily be verified for the case when for some no, (4.3.39)

A, = (0)

for n 5 no

.

Proposition 4.3.19.

If the sum

B + U A,

is complete and Qr possesses the Fatou property, then

& ( A , B ) is a Banach space.

Proof.We require Lemma 4.3.2Q. For the sequence

(2,)

sufficient that for any (4.3.40)

t o converge t o I in B

+ U A,,, it is necessary and

k E 23,

lim ef(z - z, ; B) = 0

n-m

.

Proof. If the convergence indicated above takes place, then in view of definition (4.3.38), for a given any n

E

> 0 there

2 N, and some k ( n ) , we have

exists a number N , such that for

Stability of red method functors

569

Hence it follows that k(n) 5 log,e. Therefore, for all

ekA (z - z, ; B) 5 e&)(z - z, ; B)< E Thus, (4.3.40) is satisfied for any

ej$,)(z

- z, ; B) < e/2 be valid. 115

for n

we have

.

k.

> 0 we first

Conversely, if (4.3.40) is satisfied, for a given e such that 2k(c) < a/2 and then

k 2 log,e

choose k ( e )

N, such that for n 2 N , the inequality Then

- znllB+UA, < - e&)(z - 2, ; B)+ 2J4') < &

2 N,

0

Let now

(2,)

c & ( A ; B) be fundamental, i.e.

Since the cone of 9 is continuously embedded in the cone S+ of the sequences (with the topology of pointwise convergence), we s : 25 + R+U {+a} have from (4.3.38)

lim

m,n+w

ef(zn - z, ; B ) = o

k E 25. According to the lemma, we therefore find that (z,) is a fundamental sequence in the sum B+(U A,). In view of the completeness of

for any

this Abeiian group, there exists in it an element z to which (z,) Using the lemma once again, we obtain

Iim eAk ( z - z , ; B ) = O

( ~ E Z ) .

n-63

It follows from this relation and the triangle inequality that

ekA (z - 2, ; B)5 lim e f ( z - z, ; B) n-63

+

converges.

570

Selected questions in the theory of the real interpolation method

Applying t o both sides of this inequality the @-norm and using the Fatou property, we get 112

lim

- Z ~ ~ ~ E + ( AL; B )

112,

-

2 m l I ~ + ( .~ ; ~ )

n-w

Since the right-hand side tends t o zero as m

-+

co,2,

-+

2

in & ( A ; B ) .

U

Let now &(A;

g ) :=

( E Q o ( AB; o ) , E a , ( d ;B,)) be a Banach couple

and F be a functor. Let us analyze the validity of the equality (4.3.41)

F [ E G ( d ;I?)]

2

EF(q(d;F ( Z ) ) .

It can be proved that for this the "splitting" condition is essential: (4.3.42)

F(@o(Bo),al(B1)) F ( 6 ) ( F ( g ) ).

Henceforth, we shall assume t h a t this condition is satisfied. In order t o formulate the relevant result, we require Definition 4.3.21.

A linear approximation family (B, A) will be said t o satisfy condition ( V P ) if there exists a family o f linear operators (Pn)nEz E L ( B ) such that (a)

Pn : B

(b)

PnIA,

-+

&+I,

12

E

z;

= IdA, ;

We can now formulate the main result. Theorem 4.3.22 (Brudnyz'). Let the splitting condition (4.3.42) be satisfied for

Em(A,g ) and l e t the con-

dition (VP) and further (4.3.39) be satisfied for the approximation families

(A, B;), i = 0 , l . If, moreover, t h e operator I? defined by the formula

Stability of r e d method functors

571

00

(4.3.43)

(rZ)n

:=

(n E

Izkl

z ),

k=n+l

is bounded in

6,then for any functor F the relation (4.3.41)

is valid.

b f . We shall require Lemma 4.3.23.

If the operator l7 is bounded in 0, then the norm in &(A, B) is equivalent t o the norm

where the greatest lower bound is taken over all sequences (xn E

that

11

- xnllB + 0 as n

A,) such

--t +00.*'

Proof.The triangle inequality and the positive homogeneity of the function (4.3.44) follows from the definition. The remaining property of norm follows from the inequality (4.3.45)

IllxIl* L 7 2 II~IE*(A;B)

71 IIzIIE+(A;B)

which will be proved below. Here x is an arbitrary element in the constant

7i

B +U A ,

and

> 0 do not depend on x.

Suppose that 2 E E o ( d , B ) . We choose elements

x, E A,, n E 23,

such that 112 - xn11B

Here e,(z)

:= e:(z;

5 2en(x)

(n E

z) *

B).

l? is bounded in 0, the fact that z belongs t o & ( A ; B ) implies that e , ( x ) + 0 as n --t 00. Therefore 112 - x , 1 1 ~ --t 0 as n --t 00. Since the operator

Further, from the obvious inequality

11xn - xn+lllB

5 2(en(x)

+ en+i(x))

it follows that "We assume that the right-hand side is equal to +oo if such a sequence (2,) does not exist.

Selected questions in the theory of the red interpolation method

572

II(IIzn+l

Here

(Tz),

- znllB)n€Zll@

:= z , + ~ ,n

I 2(1 + llTll@> llen(z)ncZ1l* .

E Z, is the shift operator. Since the operator I'

is bounded in 9 by hypothesis [see (4.3.43)], the operator T is bounded as well. Thus

llzIl* I 2(1+ Ilrllo) II~IIE~(A;B). Let us now suppose that 11z1)*<

00.

Then for some sequence (Z,,),~Z

in definition (4.3.44) we have (4.3.46)

ll(llzn+1 - z n l l ~ ) n c ~ I I l a2 1 1 ~ 1 1 '

*

Further, llz - ZnIIB

I

1(zk+1

- ZLllB

7

kzn

where in view of (4.3.46) and the boundedness of the operator

r

in O , the

series on the right-hand side converges. Since the left-hand side does not exceed e,(z),

we have

en(z) I

r [(I(~L+I

- ZkllE>&Z]n-l ,

12

E

.

Applying the O-norm to both sides and using the boundedness of I?, we obtain

Further, we shall use Lemma 4.3.24. Let the space & ( A ;

B) be

constructed according t o the approximation

family satisfying condition (VP) and (4.3.49)

and l e t the operator

bounded in O . Then there exist continuous linear operators

P such that

:

@(B)-+&(A; B ) , R

P is a surjection and

:

&(A; B ) + 9 ( B ) ,

r

be

Stability of real method functors

573

Let (Pn)nEz be the family of operators in condition (VP). We note

&f.

a t once that in view of (4.3.49),

(4.3.48)

Pn = 0

for n 5 no - 1 .

Suppose t h a t Qn :=

Pn+l- P,.

From the properties (a)-(.)

of Definition

4.3.21 we obtain

for any a E A,. Taking the infimum over a, we hence obtain 112

-Pnzll€J = (1

+

.

IIpnll)

SUP n

In view of t h e definition o f Qn and the monotonicity of en, we have llQnzllB

I 2(1+

SUP n

IIPnIl) e n - l ( t )

3

If we take 2 in &(,A; B ) , then applying t o both sides the @-norm, we obtain

(Q,,+~z)~ E ~@z( B ) Let . us now suppose that for z E &(,A; B )

Rz :=

(Qnz)nEz

Then it follows from the previous inequality that

I

I I R ~ I I ~ ( B ) 2(1+

SUP

IIPnII) I I ~ IIIZIIMA;B) I .

n

Thus, the operator

R is constructed.

Let us now define the operator for

(tn)

P

:

@(B)4 & ( A ; B),assuming that

E @(B) P[(zn)] :=

C

Pn+25n

*

n45

In view of property (c) in Definition 4.3.21, we have IIPn+2znllB

I (SUP IIPnll) 11znIlB

(nE

z, 7

Selected questions in the theory of the red interpolation method

574

and hence the convergence of the series contained on the right-hand side of the definition o f P follows from the fact that

(5,)

belongs t o iP(B),

from condition (4.3.21), and from t h e boundedness of the operator

iP. Let us prove the boundedness of the operator Yk

:=

Cng

Then Yk-Yk-i

Pn+2Xn.

P. For this

= P k + 2 X k and for

X

:=

r

in

we consider

En&

Pn+2Xn,

we have

j i T

1 1 2

- YkllB = 0 *

Taking into consideration the definition o f

11x11* [see (4.3.44)], we hence

obtain

llxll* 5

IIpnll>

II(IIxnlb>ll@

*

In view of Lemma 4.3.23, this leads t o the continuity of the operator P. It remains t o prove formula (4.3.47). Suppose that x E E a ( A , B ) . Then we have

( P R L = P K Q n z > n c ~ I=

C

Pn+A?nx

.

nE Z

But in view of property (b) in Definition 4.3.21,

P,Pn = P,, Therefore,

Pn+2Qn

for m

>n .

= Qn and the series on the right-hand side is given by

Since

as n

the right-hand side of the above equality is equal t o x. Thus, equality (4.3.47) is proved. Note also that the same formula shows that P 4 00,

is a surjective operator. 0

Let us now prove the theorem. In view o f Lemma 4.3.24, the constructed

operators P and R are retractive maps, and the couple E4(d, of t h e couple we have

6(g)

:=

g ) is a retract

(@o(B,-,,iPl(B1)).Therefore, for a given functor F

Stability of real method functors

575

F(6(g))5 F(&(d;g))2

F($(g)).

According t o the splitting condition (4.3.42), it follows then that

F(E6(.A,2)) 2 EF@, (.A; F @ ) ) . 0

Remark 4.3.25. Without the assumption concerning the fulfilment of the splitting condition, the above analysis leads t o t h e following description of the interpolation space on the left-hand side o f (4.3.41):

F(E$(d;S)) = {Z E B ; (Z- F'ne)nez E F ( $ ( g ) ) } * Let us specify the conditions under which the splitting condition (4.3.42) is satisfied for concrete interpolation functors.

Theorem 4.3.26. (a) (Lions-Peetre)

for p := ps

(0 < II) < 1).

(b) (Cwikel) If p # p ~the , space (lpo(Bo), lpl(Bl))19, does not coincide (as a set) with any space of the form {(z~); E S}. Here S is an arbitrary subset of the space o f sequences lm + I, .

Theorem 4.3.27. (a) (Culdero'n) If one of the spaces

@i

is separable, then

Selected questions in the theory of the real interpolation method

576

Here and below 9 9 :=

9.'-$9; ; 0 < 9 < 1.

(b) (Bukhvalov)

CyqB')) r s,(CyB'))

.

(c) (Bukhvalov) If ' P i = 9,k = 0,1,then

A similar relation holds for Vk as well.

Remark 4.3.28. (a) For complex-valued

a;,

equality of norms takes place. In the real-valued

case the isomorphism constant does not exceed two.

(b) We recall that 9i are Banach lattices of sequences. For arbitrary

Qi,

statement (b) of the theorem holds either if one o f the 9 i ' s is separable, or when both Oi's possess the Fatou property. Similarly, condition (c) holds when

( a t ) :=

9 has the Fatou property, or @-' satisfies the Az-condition

dW).

These facts lead t o the following result. Corollary 4.3.29.

Let the approximation family (B,, A ) and the parameters 9;satisfy the conditions of Theorem 4.3.22. Then the following statements are true. (a) If

9; :=

12,i = 0 , I and p (E&(d;@)Bp

Here

w8

:= wt-' w1.

:= p s ,

Ey(d;

0 < 9 < 1, then

.

577

Stability of r e d method functors (b) If one of the Oi's is separable, then

z)) &,(A;

08(E&(d; (c)

c8(3)).

C8(E,(d;2))2 E*,(A; C'(9')) .

(d) If 0; = 0 , i = 0,1,then CPI (E&(A;

z)) &(A

A similar statement is valid for 0

(P,

-

; rpr(5)) as well.

Selected questions in the theory of the r e d interpolation method

578

4.4. Calder6n Couples

A. In this section, we shall consider a number o f problems associated with a "basic problem" i n interpolation theory, viz. the description o f all interpolation spaces of a given couple (see Sec. 2.6), and analyze the possibility of solving this problem with the help o f the real method functors. In this connection, we recall the definition of 3-adequacy (see Definition 2.6.8).

It will be more convenient to use an equivalent definition (its equivalence t o the original definition follows immediately from Corollary 2.3.18). Let

F

be a family o f interpolation functors.

Definition 4.4.1.

The couple

3? is said to be F-adequate t o a couple ? if for

there exists a functor

(4.4.1)

any functor G

F E 3, such that

G ( 2 ) ~t F ( z ) ,

F(?)

In particular, if (4.4.1) is satisfied for

L+

?

G(?). +

:=

3,then the couple X

is referred

to as 3-adequate. 0

Henceforth, we shall use the notation

2

5 ? for the couples satisfying 3

the condition in this definition. Our aim is to study the properties o f the F-adequacy for the families

K

:=

(KQ) or 3

:=

(JQ)of the

real method

functors, and t o single out specific couples which are K-adequate. It was mentioned i n Sec. 2.6 that the first result i n this direction was obtained by Calder6n (see Theorem 2.6.9). According to this theorem and Corollary 2.6.10 formulated later, the couple

( L l ,L,) is K-adequate. Thus, we obtain

a complete description o f all interpolation spaces o f this couple. It should be

noted that in another independent paper appearing a t the same time B.S. Mityagin gave an elegant description of a large part of the set

Int(L1, Lm).

Henceforth, the Gadequate couples will be also referred t o as the Caldero'n

couples, or C-couples. There exist equivalent characterizations o f these couples, in which the following definitions are used.

579

Calderbn couples Definition 4.4.2.

I? is termed

The couple

intermediate spaces (4.4.2)

K - m o n o t o n e relative t o a couple

9

if for any

(X, Y ) E Int(2, ?),the following condition is satisfied:

K ( . ; y ; ?) 5 K ( . ; x ; 2),

xE X

+y E Y .

0

Definition 4.4.3.

The couple 2 possesses t h e C-pToperty relative t o a couple 9 if inequality (4.4.2) implies the existence o f an operator T E L(d,?), such that y = Tx. 0

Remark 4.4.4. Definitions 4.4.1-3 permit a “quantitative” formulation. For instance, assuming that the norm o f the embedding operators in (4.4.1) does not exceed a constant 7

> 0 independent o f the functors G, we arrive at the definition of

the ( F , 7 ) - a d e q u a c y . Similarly, we can define the (K,r)-monotonicityand

the ( C , 7 ) - p r o p e r t y . Using the theorem of the closed graph, we can easily verify that these definitions are equivalent t o t h e corresponding “qualitative” definitions. The relation between the properties contained in these definitions is described in Theorem 4.4.5 (Bmdnyr’-Kmgljak). The following statements are equivalent. (a) The couple

r? is Ladequate on the couple 9 .

(b) The couple

r? is X-monotone

(c) The couple

r? possesses the C-property relative t o the couple ?.

Proof. The

implication (c)

relative t o the couple

+ (b)

f.

is trivial, while the implication (b)

+

(a) was established in Corollary 4.1.15. It remains for us to verify that (c)

follows from (a). For this purpose, we assume that the inequality

580

Selected questions in the theory of the r e d interpolation method

K ( . ; y ; F) 5 K ( * ;I

(4.4.3)

2)

;

is satisfied and find the appropriate operator

T. Since the

adequacy takes

-). The first embedding implies that z E O r b , ( f , 2 ) ~t K@(r?) so that K ( . ; I; 2)E 9. But then it follows from inequality (4.4.3) that K ( - ;y ; ?) E 9 as well. place, the embeddings (4.4.1) are fulfilled for G := Orb,(z,

Using the second embedding from (4.4.1) in this situation, we obtain

y E Ka(P) + Orb,(Z,

P) .

In view of the definition of orbit, it follows hence that for some T E

L ( 2 ,P),

we have y = Tx. 0

Corollary 4.4.6.

If x'

5 f , then the couple F is relatively complete.

x:

Proof.We shall require Lemma 4.4.7. The couple

2 possesses the C-property relative t o ? iff for every x E C ( 2 )

we have (4.4.4)

Orb,(Z,?)

S

KLz(?)

.

-+

Here w, := K ( . ; I ; X ) .

Proof.It follows from inequality (4.4.2)

that the element y belongs to a unit

ball of the space appearing on the right-hand side in relation (4.4.4). Taking into account the definition of orbit functor [see (2.3.10)], the condition of Definition 4.4.3 can be written in the form of an embedding in set theory: (4.4.5)

Kp=(F) c Orb,(Z,F) .

Here both spaces are continuously embedded into C(?), and hence the embedding (4.4.5) is continuous. The inverse embedding follows from the minimal property of the orbit (see Theorem 2.3.15). This property implies that if z E

K L z ( z ) then ,

Cdderdn couples

581

But w, coincides with the K-functional of

x

so that the required embedding

takes place. U

Let us now prove the corollary. For this we take in (4.4.4) an element in

2

A(d)\{O}. Then according t o (2.3.15), Orbz($,

(4.4.6)

f )2 A(?) .

On the other hand, in this case

Therefore, taking into account Theorem 3.5.9(c), we have

KLyrn~(P)

Kate,,(P) = A'( P) .

Together with (4.4.6) and (4.4.4), this leads t o the relation

A(?) Z A'(?) = A(?') Thus,

.

P Z p.

0

The corollary readily proves examples o f couples that are not Gadequate (like (C, C')). Later it will be shown that there exist relatively complete but not K-adequate couples. Let us single out the subcategory of the category

6,for which the rela-

tion of the x-adequacy is transitive. For this we require Definition 4.4.8. Let

C

be a subcone of the cone Conv. Then the couple

x'

is termed

C-

abundant if for any function cp E C there exists an element x E C ( 2 ) such that

(4.4.7)

K ( *; I ; 2)M

'p

,

Selected questions in the theory of the r e d interpolation method

582

where the equivalence constants are independent of cp and

x.

Theorem 4.4.9. Relation

5 K:

is transitive on the category o f Convo-abundant couples."

Proof.Let i 5 f

and l e t the two couples be Convo-abundant. The cone

K: Convo consists of those 'p E Conv for which (4.4.8)

cp(+O)

:=

)jy0 cp(t) = 0 ,

p'(+m) := lim t-m

t

=0.

Therefore, the condition of the Convo-abundance indicates, in view of (4.4.8) and Corollary 3.1.14, that elements

x satisfying the equivalence (4.4.8) be-

long t o Co(J?). Henceforth, we shall use Proposition 4.4.10.

If condition (4.4.1) o f Definition 4.4.1 is satisfied for the family of functors G := O r b , ( i ; where z runs through the elements of ED($), then e),

2

5 ?. K:

0

We shall prove this fact later; continuing our line of reasoning, we claim that the couples 3 ,? and (4.4.9)

2are Convo-abundant and

z. z.

d 5 ?, ? 5 K:

K: +

We must show that X

5 K:

For this purpose, i n view of Proposition

4.4.10, it is sufficient t o establish that +

(4.4.10) K ( - ;Z ; 2) 5 K ( - ;S ; d

=+ z

E

) ,E ~C o ( d ) =+

Orb,(Z,Z) .

"Sec. 4.6 contains a convenient criterion of the Convo-abundance of 2. It implies, in particular, that for this it is sufficient that there exists an element z E C ( 2 ) whose K-functional is equivalent to a power function.

Calder6n couples

583

Since r E Co(z), in view of Corollary 3.1.14

K ( - ;z ;

f) E Convo.

Then

? implies that there exists an element

the Convo-abundance o f the couple

y E Co(?), for which

K ( . ;y ; Therefore,

?) M K ( . ; 2 ; 2 ) .

K ( . ; y ; ?) 5 y K ( . ; x ;

z),and according t o condition (4.4.9)

and Theorem 4.4.5, there exist operators

R E L ( z , ?), T E L(?, 2) such

that

But then z = T R x , and hence condition (4.4.10) is satisfied. It remains for us t o prove Proposition 4.4.10. For this purpose, we require

Lemma 4.4.11. Suppose t h a t an element x z = xo

+

51,

$ Co(z). Then for an arbitrary decomposition

x ; E Xi, the following relation holds:

Orb,(2 ;

a)

E Orb,,(z ;

0

)

+ Orb,, (2;

a)

.

Proof.Since x 4 E 0 ( f ) , the following three cases are possible:

(c)

20

E xo\x,O.

21

E Xl\X,".

We consider only the first of these cases (the remaining cases can be analyzed similarly). Since from the definition o f orbit and the relation Tx = Tzo

+

T x l , where T E L(x',?), it follows that

Orb,(2 ; -)

Orb,,(z ; -)

+ O r b z l ( 3; .) ,

it remains t o prove the inverse embedding. For this purpose, we consider the general Banach couple, see Def. 2.1.30,

Z

:=

(xo/xo n co(x'); xI/xln ~ ~ ( 2 ) )

Selected questions in the theory of the r e d interpolation method

584

(here

A(z') = (0,O) E Zo@ 2,)and the canonical projection P :

Then in view of the definition of

z

x'

4

f.

and condition (a),

Px=Px1#0. Since

A(Z) = {0}, we have -9

-9

Orbp,( Z ; .) = Orbp,,( Z ; -) = Prl. Therefore,

p+,= Orbp,(z;

a)

Orb,(x' ; *) .

Hence it follows that x1

E Prl(x')L-+ Orb,(x',x') .

Consequently, we have established

Orb,,(r?,x')L+ Orb,(x',x') .

-

Using the minimal property of Orb,, , we obtain the embedding

Orbz1(x'; .)

Orb,(x' ;

a)

.

This embedding and the condition zo E X: implies that

Orb,,(r? ; .) c+ Orb,($ ;

a)

I

Combining the last two ernbeddings, we obtain the required result. 0

Let us return t o the proof of the proposition. For this it should be noted first of all that if condition (4.4.1) is satisfied for all G := Orb,(-? ; .) with z E

C(rz'), then x' 5

K Orb,(x',x') then

9. Indeed, if L+

K+(x') ,

K ( . ; x ; d ) E 0 . Therefore, the second embedding from (4.4.1) gives

for G := Orb,(z; .) and w, := K ( . ; z;x')

Calder6n couples

(4.4.11) K L z ( p )L+ K a ( p )

-

585

Orbz(z, p) .

While proving Lemma 4.4.7,we established the equivalence o f this embedding and the C-property of the couple of Theorem

4.4.5it follows hence that

2 relative t o the couple P. In view x' 5 p. K

(4.4.1)be satisfied for all G := Orbz(J?; .) with 2 E Co(J?).We consider an element 2 @ Co(l?) and assume that for some y E C ( f ) , we have Let now condition

K ( . ; y ; f )5 K ( . ; 2 ; 2). If 2 = z0

+ zl,z;E X i , then

and by the theorem on the K-divisibility there exists a decomposition y =

yo

+ y1 such that

(4.4.12) K ( * ;y i ; But since zi E

?) 5 y K ( - ;2;;x') ,

X i it

(i = 0,l).

(4.4.12)that y, E q. But 4.4.6,Y c Z Y . Hence y; E i = 0,l.Let us now

follows from inequalities -#

according t o Corollary

4

x,

verify that

y; E Orbz,(l?,f) Indeed, let first

I;

(i = 0,l).

E Eo(x'). Then in analogy with (4.4.11)we obtain the

embedding

K L wm= ; since condition

(P)

L)

Orb=,(Z,?) ,

(4.4.1)is fulfilled with G := Orb,(X ; -) and

I

E Co(J?).

Since the left-hand side contains, in view o f (4.4.12)the element y,,

(4.4.13)

is proved for this case. Let us now assume that I ;

# Eo(J?).

Then as in Lemma

4.4.11,

Orbzi(2; .) t-' PViso that (4.4.13)is trivially satisfied in this case also. we get Combining (4.4.13)and Lemma 4.4.11,

Selected questions in the theory of the real interpolation method

586

y = yo

+ y1 E Orb,(2 ; F) .

Thus, we have established that for

2

@ Co(x’). Hence the embedding

K L w , I (P) ~ , c Orb,(x’ ; P) is satisfied for all

2

E C ( 9 ) . As before, it follows hence that

x’ 5 p. K

0 Let us consider the existence o f maximal and minimal elements (relative t o the K-adequacy) in the category

6. The problem o f minimal couples is

solved i n Theorem 4.4.12. The couple

XI is K-adequate to any relatively complete c0up1e.l~

Proof. In view

of Theorem 4.4.5, it is sufficient t o prove that i f ? is a rela-

Xl possesses the C-property relative t o Y . Let I

tively complete couple, then

us establish the following more accurate fact: (4.4.14)

K ( . ;y ;

P) 5 K ( - ;f ; XI) + 3 IT E L7(X1,F) ,

Here we can for y take any number of the form 6(?) is arbitrary (2’ depends on

E).

Recall also that

6(?)

y = Tf ,

+ E , where

E

>0

is the K-divisibility

constant (see Theorem 3.2.7). Let us now prove (4.4.14). Without any loss o f generality, we may assume that f 2 0. We take a fixed q

f=C

fn

7

(fn

> 1and :=

write

fX[qn,q”+1))

ndZ

Since

~ ( tf ;; i1)= ( ~ f ) ( t ):=

J

ds

f(s)min(l,t/s) , S

Rt

we have 131n view of Corollary 4.4.6, the statement is not valid without the condition of relative completeness.

Cdder6n couples

K ( .; f

587

;

=

C

~

(

ifn;

*

0

nE ZG

Then by the theorem on the K-divisibility, for a given

E

> 0 there exists an

expansion

y=

C

yn

(convergence in ( Y ) )

nEZ

such that (4.4.15)

K ( . ; y n ; f )5 ( 6 ( f ) + ~ ) K ( -f n; ;

z,).

In view of t h e definition of f n , gntl

(4.4.16)

K ( t ;f n ;

1

Zi) =

f(S)min(l,t/s)

ds y I qn(t) ,

'I"

where ¶"t 1

gn(t) := d n ( l , t / q n )

1

ds

f(S)

7-

9"

Let us consider the operator

T defined by the formula

It follows from this definition and the choice of the sequence (y,) that

T f = x yn=Y. nEZG

Therefore, in order t o prove (4.4.14). it remains t o estimate the norm of In view of the definition of

T and (4.4.15)

and (4.4.16), we have

T.

588

Selected questions in the theory of the red interpolation method

L (@)

+E)

c

,p+l

min(1,tlq")

ncZ

L

q(@)

+E)

J

J

ds

lh(s>l ; I

rl"

lh(s>l m i n ( l , t l s )

ds

.

nt,

Since on the right-hand side we have the K-functional of h in t h e couple

z,,

we obtain

K ( . ; Th; f ) 5 q ( h ( f ) + e ) K ( - h; ; Let now h E

Li,i E ( 0 , l ) .

z,).

Then it follows from the inequality proved above

that

IIThllKLi m (9) < - Mf.) + €1 llhllKLim (El) .

(= y t ) on t h e left-hand side and the norm in

Since we have the norm in y,"

(JqC (= L:), IlThllYi I (I(@)

+

E)

IlhllLr

(i = 0,1)

*

By the definition of operator norm, it follows that

Remark 4.4.13.

If f is a couple of Banach lattices, it follows from the above proof that the Tf 2 0 if f 2 0). Indeed, we have only t o note that if y 2 0, the element yn can also be regarded

operator T can be taken t o be positive (i.e.

as non-negative (otherwise, we should have replaced them by the elements

gn

:=

FlYnlIYnl

y). But if yn

2 0 (n E Z), the fact that the operator T

positive ollows from formula (4.4.17). Corollary 4.4.14 (Sedaev-Semenov). For any

E

> 0, the couple El

I 4 h particular, it

+ property.'^

possesses the (C, 1

is K-adequate.

is

Cdder6n couples

Proof. By definition E

589 (see Remark 4.4.4), we have t o establish that for any

zl)

> 0 the inequality K ( . ; g ; Zl) 5 K ( . ; f ; implies that there exists an T E L l + e ( z l )for which g = Tf.But this follows from statement

operator

(4.4.14) and the equality 6(&) = 1 (see Proposition 3.2.13).

zW

It would be natural t o expect that the couple is a maximal element. This is actually so; t o prove this, we require the following well-known fact. Theorem 4.4.15 (Hahn-Banach-Kantorovach). Let

Xo be a linear subspace o f the vector space X and Y be a linear (par-

tially) ordered space. We assume t h a t p : X -+

Y

is a sublinear operator15

Y is a linear operator such that Toxo 5 p ( x 0 ) for all xo E X o . Then there exists an extension T : X --f Y of the operator To such that Tx 5 p ( x ) for all x E X .

and To

:

Xo

--f

0

Let us prove t h e fact that the couple

zwis maximal. The following the-

orem is valid. Theorem 4.4.16 (Peetre). For any couple

+

X , we

have

.r

h

Proof. Let t h e elements z E E ( d ) and g E E(z,) (4.4.18)

K ( . ; g ; Em)5 K ( -; x ; r?)

be such that

.

In view of Theorem 4.4.5, it is sufficient t o establish that for some T E

Ll(r?, Zw) we

have

g=Tx For this we first of all note that in view o f inequality (4.4.18) and Proposition

3.1.17, it follows that "That is, p ( z 1

+

22)

5 p(zi) + ~ ( z zand ) p ( h ) = IAIp(z).

Selected questions in the theory of the red interpolation method

590

(4.4.19)

g

5 K(.; z;

2).

X the space C ( x ' ) , for Y the space C(s,), for p the function x K ( . ; x ; x'),and for To the linear operator given by the formula To(Xx) := Xg on the one-dimensional space fi.In Let us now take in Theorem 4.4.15 for ---f

view of (4.4.19), all the conditions of the theorem under consideration are fulfilled. Consequently, there exists a linear operator T : C(x')

---f

C(J?,)

such that (4.4.20)

T x = g and T y 5 K ( - ;y ; x') ,

y E C(x') .

Substituting into this inequality -y for y, we get

lTYl5 K ( - ;y ; x') ,

y€

W).

Since in view of Corollary 3.1.11

IF(* ; Y; x')IlLb,

= IIYIIX:

5

IlYllZ

,

we obtain from the previous inequality IITYIILb,

Thus,

i Ilvllx,

T E Ll(x',e,)

7

= 0,1

.

and g = T z .

Corollary 4.4.17.

The couple

2, possesses the (C, l)-property.16

0

Let us now verify that under certain conditions, the couple K$(x') := (Ka,,(z),K@,(x'))inherits the property o f K-adequacy from the couple of its parameters. This phenomenon was observed for the first time in the following particular case, which is important for applications. Theorem 4.4.18 (Cwikel). For any couple

60

x', t h e couple (x'doqo,x'~lql)

# 91.

0

I6In particular, it is Gadequate.

is Gadequate for 9, E ( O , l ) ,

Calderdn couples

591

We shall postpone the proof of this (and a more general fact) t o the next item, and consider now only a result providing an exhaustive answer t o the question concerning the inheritance of the property of K-adequacy. Theorem 4.4.19.

Let

*;

E Co(z,),

z = 0,1, and

2,f

Proof. In view o f Theorem 4.4.5,

be aribtrary couples. Then

K&(d)

we must establish the corresponding C-

property. We establish the following less accurate fact. Let a couple K d ( i o 3 )possess the (C,y)-property relative t o a couple

K$(Zl). Then t h e couple K d ( 2 ) has the (C,y')-property relative t o the couple K,jj(?). The constant y' here is any constant greater than y6(X). Proving this statement, we shall assume, without loss of generality, t h a t t h e couple

? is relatively complete (since K$(?)

= K $ ( F C ) )According . to

Corollary 3.5.16(b), it follows from the relative completeness of this couple and the condition

Qi

E Eo(i,)

with a certain J-space. (4.4.21)

E(K$(?))

t h a t each o f the spaces Kq,(?) coincides

Thus, L)

Co(f) .

Let now the condition (4.4.22)

K ( . ; 9 ;K&))

*

5 K ( . ; f ;K&))

3 IT E & ( K & ( L ) , K g ( & ) ) ,

=+ g = Tf

be satisfied. Further, l e t the following inequality hold: (4.4.23)

K ( * ;y ; Kg (? )) 5 K ( . ; 2 ; K s ( 2 ) ) .

Let us verify that there exists an operator transforming z into y. For this purpose, we note first of all that in view of (4.4.21) y E Eo(?) so that

K ( . ; y ; p) E Convo. We now take advantage of the fact that t h e operator

592

Selected questions in the theory of the real interpolation method

Sf = K ( . ; f ;

zl) has an “almost”

Namely, for any e

> 0 there

inverse operator (see Remark 3.5.14).

exists an operator

: Convo + C ( ~ I )such ,

that

(4.4.24)

h 5 S r h 5 (1

+ &)h

( h E Convo) .

Applying this inequality t o the function K ( . ; y ;

?), we

find the function

g E C ( i l ) for which (4.4.25)

K ( * ;y ; ?) 5 K ( . ; 9 ; Z1) 5 (1+&)K(.;y ; f ) .

In view of Theorem 4.4.12, the right-hand side inequality in (4.4.25) implies that there exists an operator

Ti E Lcp(zl,f), where p

:=

(6(2)+~)(1+&),

such that (4.4.26)

y = T1g

.

Let us now estimate the K-functional of the function Sg = K ( . ; g ; &) in the couple K$(i,) = ( $ 0 , (4.4.27)

K ( t , s g ; K&))

$1).

According t o Theorem 2.2.2,

= inf{Ilgolleo

+ t 11g1IIe1) ,

where the lower bound is taken over all gi E Conv for which go

Sg. Then 0

5 gi 5 Sg E Co(z,)

inequalities are satisfied for the functions h; := rg;:

Thus, the right-hand side of (4.4.27) is not smaller than

Hence it follows that

+ g1 =

and, in view of (4.4.24), the following

Cdder6n couples

593

Let us estimate the right-hand side o f this inequality with the help of an

, $ ( i l ) ) , such t h a t operator T2E L C y ( l + e ) z( K $ ( i m ) K (4.4.28)

g = TzK(.;

X ;

2).

Finally, it follows from Theorem 4.4.16 that there exists an operator T3 E

&(z;i,),such t h a t f

(4.4.29)

K ( *; X ; X ) = T ~ .x

Since Ka, are functors, T3 E

L1 ( K $ ( . f ); K&(z,)).

L e t now T := TlT2T3.

Then from (4.4.26), (4.4.28) and (4.4.29) we obtain y=Tx, and the norm of T as an operator from

K&(z) into K$(?) does not exceed

11T111 lT211 lT311, i.e. is not greater than

(1

+ - y . [6(x')+ €1 . E)3

It is not always easy t o verify the conditions of the theorem. The following result can be used conveniently in applications. Theorem 4.4.20 (Dmztrzev- Ovchinnikov). Suppose that t h e operator S is i n and

C(6)n L(\t).

Then for any couples

2

?

Proof. Obviously, this

result follows from t h e previous theorem. However,

we shall prefer a proof based on Theorem t o the case

4.3.1.We shall confine ourselves

6 = $ and x' = ?, leaving t o the reader t h e analysis of a more

general case. Since according t o Theorem 2.3.15 each interpolation space of the couple

K $ ( d ) is generated by a certain functor F , it is sufficient to

prove that for some parameter Q,

Selected questions in the theory of the r e d interpolation method

594

F ( K & ( - f ) ,)2 K\u (K&)

.

By Theorem 4.3.1 the left-hand side can be written in the form Further, since the couple is K-adequate by hypothesis, sented in the form couple

KFcs,(d).

F(@) can

be repre-

Ka(6). Finally, since the operator S is bounded in

6 we have in view of Lemma 3.3.14 (with Q

=

h;c*(rn,(d)K\u (%C-f,)

the

:= S)

.

0

Let us consider another result of this type generalizing Theorem 4.4.20. Theorem 4.4.21 (Nalsson). Let

x' and ? be arbitrary couples and 6 and 6 be the couples of exact

interpolation spaces relative t o couple

J$(?)

+

L,

and

L',

respectively. Suppose t h a t the

is regular and relatively complete. Then if

3 5 6,then K

0

Finally, l e t us consider the inheritance of the property of the K-adequacy upon transition t o dual couples. In order t o formulate the required result, we shall use Definition 4.4.22. The couple

x'

satisfies the weak upprozimation condition if for any z E

Co(x') there exist a constant 7 > 0 depending on z and a sequence of operators T , ( n > 0 ) such that (T,) c L C , ( 2 and ) T X ,

+x

in

~(d>

and, besides, T,x E A(x'), n E 0

PV.

595

CaJder6n couples

Obviously, a couple satisfying the approximation condition (see Definition 2.4.22) also satisfies the above condition. It will follow from Lemma 4.4.24 given below that the converse statement is not true. Theorem 4.4.23.

If 2 then

5 ? and if the couple ?' satisfies the weak approximation 3 ?' 5 X'.

condition,

Ac

Proof.Let us verify t h a t if 2

_<

? and if each functor Orby,(?';

.), where

3 {Df; F E J.F}of all dual func5 2'.For this we note that in view of Proposition 4.4.10, it is

y' E CO(?'), belongs t o the class V := +

tors, then Yk

sufficient t o verify t h a t for each o f t h e functors Orby@;

, ) a

y' E Co(?'),

there exists a functor Ka such that (4.4.30)

Orby@'; ?')

By hypothesis Orby,(?';

-+

a)

, Ka(2:')L+ Orb,)(?; 2').

Ka(?)

=

DG

for some functor G. Since

2

5 ?, 3

there exists a functor Jq such that

G ( 2 )L+ J*(Z), &(?)

~t

G(?) .

Passing t o dual functors and considering that, according t o Theorem 3.7.2,

DJq 2 Ka with 0 := W , we obtain from these embeddings

DG(?')

L-)

K.@')

,

K a ( 2 ' ) -+ D G ( 2 ' ) .

Thus, we have proved (4.4.30) and the f a c t that

?' 5 2'. It

remains

K: t o show t h a t

Orby,(?; .) E V for y' E Co(?'). For this purpose, we shall

require Lemma 4.4.24. The couple

2 satisfies t h e weak approximation condition iff for any interpo-

lation space X

c Eo(d)of this couple we have

Selected questions in the theory of the red interpolation method

596 (4.4.31)

X

L--)

(Xo)>"

Proof.Let us verify that (4.4.31) For this we take z E X

~t

follows from the conditions of the lemma.

Co(d)and assume that

(T,)

is the sequence in

Definition 4.4.21. Since X is an interpolation space,

Furthermore, T,x

E A ( 2 ) and

111

- T n x l l c ( ~+ ) 0.

Consequently, the set theoretical embedding X

~t

Therefore, z E (XO)". (XO)" holds in this case,

which, in view of the theorem on closed graph, leads t o (4.4.31). Conversely, let embedding (4.4.31) hold for any interpolation space X -+

C o ( d ) . For a given element z E C o ( d ) ,we consider the interpolation space X := O r b , ( d , d ) . Then it follows from the embedding (4.4.31) that for this space there exists a sequence ( z , ) , ~ N c A(d)such that

From t h e definition of the norm in Orb,(d; exists a sequence of operators (T,) SUPn

+

d ) it follows

c L ( X ) , such

now t h a t there

that x, = Tnx and

IlTnll~L Y.

0

Let us return t o the proof of the theorem. Since the weak approximation condition is satisfied for the couple

Y := Orbg,(?';

9')

L-,

?', we have by Lemma 4.4.24 (Yo)>"

for y' E Co(?'). In view of Theorem 2.4.34 we then have (4.4.32)

Orby(?;

3') = (Corbyt(2;

?'))I

Indeed, for this theorem t o be valid, it is only required that the closed unit ball of the space on the right-hand side be *-weakly closed in the space

A(x')*. Byt Y is generated by the orbit o f a single element and, as follows

Calder6n couples

597

from the arguments following the formulation of Theorem 2.4.39, the condition of *-weakly closure is satisfied in this case. Then equality (4.4.32) shows that the functor Orbv,(?';

= Orby(?'; .) belongs t o the set

a)

2, of

dual functors. 0

Corollary 4.4.25.

x'

Let t h e couples couple

and

?

?' satisfy the weak

be regular and relatively complete and let the approximation condition. Then if

x' 5 ?, it K:

follows that

?' 5 I?'.

x

Proof.

According t o Theorem 4.4.23, we have only t o verify t h a t the ine-

quality

x' 5 ? follows from the conditions

formulated above. Let G be

&7 an arbitrary functor. Then the condition of &adequacy allows us t o find a functor Ka such t h a t

G(2)

Ka(x'),

However, since the couples

Ka(?)

G(P) .

~ - t

+

2 and Y are regular and relatively complete, we

have in view of Corollary 3.5.16(b)

K@(x')z J*(Z),

K@)

= J*(?)

,

where Q is a certain parameter o f t h e 3-method. Consequently,

d 5 ?. &7

U

B. Let

us consider the K-adequacy of some concrete couples. We begin

with t h e proof of a fundamental fact which makes it possible t o obtain a large number of specific results. For this purpose, we shall require a few definitions. Definition 4.4.26. Let 9 and Q be Banach lattices. We say that 9 i s decomposible relative t o Q if for each function f E 9,for a sequence of disjoint measu-

598

Selected questions in the theory of the real interpolation method and for a sequence of disjoint measurable f ~ n c t i o n s ' ~

rable sets

(gn)nGN c Q it follows from the inequalities

that g :=

C

g, E Q and

with a constant y

> 0 which depends only on

@ and Q.

Definition 4.4.27. Let

6 and 6 be two couples of Banach lattices.

posable relative t o

6 if @;

We say that d as decomq;,i = 0,1.

is decomposable relative t o

U

Note that in this definition t h e measurable spaces on which the functions in C(6) and C($) are defined are in general different. Example 4.4.28. The couple

LAC) is decomposable relative t o the couple L d f i ) if (and only

if) pi _< qi, i = 0 , l . In particular, the couple L A f ) is decomposable. Other examples will also be considered later. Now we shall discuss the main result. Theorem 4.4.29 (Cwikel).

d and 6 of Banach lattices be relatively complete and 6 be decomposable relative t o 6. Then d L: 6. Let t h e couples

K;

Proof.We shall require a few auxiliary results.

The first of them is of interest

as such. In order t o maintain the continuity of presentation, we will give i t s

proof later. 0

17This means that (suppg,) n (suppgm) = 0 for n

# m.

Cdder6n couples

599

Lemma 4.4.30.

f E C(6)and for each t > 0 there exist measurable subsets A t ( f )such that

For any function

&

:=

(4.4.33)

t 5 s + At C A,

Moreover, for any

.

t > 0,

Here we can take for 7 , for example, the number 11. 0

Let now the function cp E Conv and a number q

(t,) c R+ be a

> 1 be chosen and

let

sequence of points chosen in agreement with Proposition

A,(g) ( t E B + )be t h e measurable sets in Lemma 4.4.30. For an arbitrary n E 2 3 we put 3.2.5. Let further g

(4.4.35)

E C(6) and At

A, := At,,(g)

:=

,

if in the sequence (ti) there is a point with index 2n. Otherwise, we put

A,

:=

272

> 0.

0 for

2n

<

0 and A , equal t o the entire measurable space for

Under these conditions and in this notation, the following lemma

holds. Lemma 4.4.31. Let the inequality (4.4.36)

K ( - ;g ;

6 ) I cp

be satisfied. Then t h e following equality holds: (4.4.37)

=

gXA,+i\A” n€Z

and moreover, for a certain absolute constant

71,

the inequalities

Selected questions in the theory of the real interpolation method

600 are

valid.

Proof.The identity (4.4.37)

can be derived in analogy with the statement of

Lemma 3.2.10 since in view of inequality (4.4.34), we can take the elements instead of the elements z O ( t 2 iused ) there. Similarly, inequality (4.4.38) can be derived from Lemma 3.2.9 in analogy with inequality (3.2.31). We

gXA;

must only use, instead of inequality (3.1.30) employed there for the elements

so(t2,),inequality (4.4.34) for their analogs g x A ; . 0

Let us now suppose that the function f belongs to (4.4.39)

'p

:=

C(6)and that

K ( * ;f ; 6), g > 47

(e.g., q := 45). Then the following lemma is true. Lemma 4.4.32.

If the points (ti) are constructed from cp and q is as indicated in formula (4.4.39), and if further the sets An are defined by formula (4.4.35) with g := f , then the following inequalities hold:

Proof. Let

us first consider the case when both the points t2n+4and tZn-2 belong t o the sequence ( t i ) . In view of the concavity of the K-functional it is sufficient to prove inequality (4.4.40) only for t := t Z n + l . However, with such a choice of the argument, the left-hand side of (4.4.40) is not smaller than

In view of (4.4.34) and (4.4.35) we have

Cdder6n couples

601

Here we use the properties of t h e sequence ( t i ) indicated in Proposition 3.2.5 [as applied t o 'p and q from (4.4.39)]. In view of the choice of y we obtain

-

1

7

I ~ ~ ( t 2 n - 2f; ; Q ) I - ~ ( t 2 n + l ; f ; 6)I 4 ~ ( t 2 n + 1 ;f ; 6) . Q

Together with (4.4.42) this inequality estimates from below expression (4.4.41). Thus, we obtain in this case the required estimate:

-

K(t2n+l

; fXAn+p\An-i ; a) 2

1

5 K ( h n + l ; f ; 6,.

In the remaining case, the estimation is even simpler since if, for example, t2n+4

#

(ti) then the term K(t2n+l;f ( l -

4

x i , + z ) ; Q ) in

the expression

(4.4.41) vanishes. 0

Let us now prove the theorem. In view of Theorem 4.4.5, it is sufficient t o establish that if (4.4.43)

K ( . ; g;

6)5 K ( * ;f ; 6),

then there exists an operator (4.4.44)

g = Tf

T E L(6,G)such that

.

Let us now find the required operator in the form of a sum of three addends

T,. In order t o define

Tl,

we construct the sequences ( t i ) from cp and q in

(4.4.39) and for an arbitrary function h E

C(6) and n E 25 we put

602

Selected questions in the theory of the red interpolation method

In particular, for the function

f,

determined from

f

in this way, there exists,

in view of the Hahn-Banach theorem, a linear functional L, E C(6)* such that

and, moreover,

In view o f Lemma 4.4.32 we then have

(4.4.47)

Ln(fn) 2

f I((tzn+1; f ; 5)

*

Let us now define the operator Tl with the help of the formula (4.4.48)

Tlh :=

Then for n E

C

Ln(hn) -

nE3z Ln(fN)

322 we

gXAn+l\An

*

have

and therefore

Taking into account the relative completeness of

5, we further

have

Let us now apply the inequalities (4.4.45), (4.4.47) and (4.4.38), and the identity (4.4.46) to estimate the right-hand side of this expression. Taking into account (4.4.39) and the relative completeness of 0, the right-hand side does not exceed

Calder6n couples

Thus, for all n E (4.4.51)

603

325, we have

IITlhnllluo I271 llhnlloo '

Since for the values o f n chosen, the supports of the functions Tlhl [see (4.4.9)] do not intersect pairwise, using the condition of decomposability of

@ relative t o Q (see Definition 4.4.27) we obtain the inequality

IlTlhllluO I27172 llhlloo where 72 =

^fi(@0,00).

9

Similarly, we can prove that TI is bounded as an

operator from Q1 in Ql. Thus, (4.4.52)

TI E L(6;$) .

Further, let us define the operators T2 and T3 by the same formula (4.4.48), but now summing over all n o f the form

3k

+ 1 and 3k + 2 respectively

(k E 23). Then in analogy with (4.4.52), we can prove that T2 and T3 6 into the couple $. If now T := TI + T2 + T3,then T E L(6,$) and, in view of the definition of are bounded as operators from the couple and (4.4.37),

Thus, (4.4.44) has been proved, and it remains t o prove Lemma 4.4.30.

So, f E C(6) and the couple 6 is relatively complete. According t o formula (3.9.10). for a given E > 0 there exist measurable sets Bt ( t E BZ+) such that

(4.4.53)

IlfXBtll40

Further, l e t

(ti)

-k t Ilf(1- XBt)l1@1 5 2(1

+ & ) K ( t f; ; 6).

be a sequence of points constructed for cp :=

and an arbitrary q

> 1. We shall

25. Then we put A, :=

U ilk(.)

begin with the case when

i

K ( - ;f ; 6)

runs through

604

Selected questions in the theory of the red interpolation method

where k ( s ) E

Z

t2k(S)-1

is defined by the inequality

5 s < t2k(s)+l

*

With such a definition o f this set, A, increases with s so t h a t (4.4.30) is satisfied. It remains t o estimate (4.4.33) for the K-functional of f. According t o t h e definition o f A, and (4.4.53) we have

Taking into account Proposition 3.2.5, we find that t h e right-hand side does not exceed

+

2 q y1 q-1

I

E)

K ( s ; f ; 6).

Similar calculations lead t o the inequality

Taken together, the inequalities proved above lead t o (4.4.34). (ti)

It remains t o consider the cases when the set of indices of the sequence is such that i 2 -m for some m < 00, and/or i 5 n for some n < 00.

Let us consider, for example, the case when m

< 00.

Then two situations

are possible: rn is even or m is odd. For m = 2 k , we chose At := B,-,, for

t < t - 2 k . In view o f Proposition 3.2.5, the fact that m < 00 implies that

Calder6n couples

for t

605

5 t - 2 k . It follows from this

inequality and (4.4.53) that (4.4.34) holds

for such t ’ s . On the other hand, if m := 2k

+ 1, we choose At

:=

0 for t 5 t - 2 k - 1 .

Then for such t ’ s we have (4.4.54)

=0 *

IIfXAtllso

Further, since

6 is relatively complete, for the same t ’ s we have S

Let usmake use of Proposition 3.2.5 once again. In this case, it leads t o the inequality

Combined with the previous inequality, this gives

I

From this expression and (4.4.54) it follows that inequality (4.4.34) is satisfied in this case also. Remark 4.4.33. Theorem 4.4.29 was obtained by Cwikel in a somewhat more general situation, which will be described now. First o f all, we shall generalize Definition 4.4.27 by introducing the concept of q-decomposability of the couple lative t o the couple q :=

00.

re-

6.The concept o f decomposability coincides with it for

Here we consider not only Banach lattices but also more general

function spaces consisting of measurable (classes of) functions. The only conditions imposed on them is t h a t for any measurable set A the operator of multiplication by X A acts in such spaces and has a norm which does not

606

Selected questions in the theory of the red interpolation method

exceed unity. Then the condition of q-decomposability of

0 in terms o f 9 is

obtained by replacing in Definition 4.4.26 the inequalities lJgnl(,p5 IlfxAnIl*,

n E RV,by the inequality

4

Further, we assume that the couples 0 and

$

are relatively complete, and

that the statements o f Lemma 4.4.30 are fulfilled for them.ls Then from the inequality

it follows that there exists an operator T E L(6,$) such that

g=Tf. The proof is obtained from a slight modification of the proof considered earlier. We have only t o take into account the fact that in view of the definition of the sequence ( t i ) , the inequality

holds for some constant 7 independent of

f

and g .

Let us consider several important corollaries o f Theorem 4.4.29. For this we shall use Definition 4.4.34 (Shimogaki). The Banach lattices 9 satisfies the upper (accordingly, lower) p e s t i m a t e

if there exists a constant M E ( 0 , ~ )such that for any finite sequence of disjoint functions

(f,,)C 0,we

have

'"Cwikel refers to them as the Holmsiedi couples since it follows from the Holmstedt l ) this property. formula [see (3.9.7)] that the couple (-?goPo, ~ , j l p has

Calder6n couples

607

or, respectively,

The role o f this definition in the situation under consideration is explained in Corollary 4.4.35.

Let

6 and $ be relatively complete couples o f Banach lattices such that

satisfies t h e lower pi-estimate and Q; satisfies the upper qi-estimate. Then

if the inequalities (4.4.55)

pi

are satisfied,

5 qi

(2

= 0,l)

6 is K-adequate t o the couple $.

Proof. Using the

inequalities in Definition 4.4.34 and Holder’s inequality,

we find that if (4.4.55) is satisfied, then 0; is decomposable relative t o Qi,

i =0,l. 0

In order t o display some concrete results o f importance in the applications, we shall use the following well-known fact.

Let M :

R+---t R+be a convex function

equal t o zero at zero and t o

infinity at infinity. Further, let LM(O,1) be the corresponding O r l i u space.

We put

Selected questions in the theory of the r e d interpolation method

608

Theorem 4.4.36 (Shimogaki). (a) If q

< &,u,

(b) If p

> P M , then

then the space

LG(O, 1) satisfies the upper q-estimate

this space satisfies the lower pestimate.

Corollary 4.4.37. Let LG(O, l), L*(O, 1) be two couples of Orlicz spaces and let PM,

< aN,

(2

= 071)

.

Then the couple Lfi(0,l) is Gadequate relative to the couple Lfl(0,l) 0

Another important corollary refers t o the couples LA@) := (L,,(wo),

Lpl(wl)). Since the space LA@) obviously satisfies the upper and lower pestimates, we obtain t h e following important fact. Theorem 4.4.38 (Dmitriev).

If inequality (4.4.55) is satisfied, then the couple LA@) possesses the Cproperty relative t o the couple

Ldc).

0

It should be noted that condition (4.4.55) is exactly the necessary condition for this result t o be valid. Namely, the following theorem holds. Theorem 4.4.39 (Ovchinnikov). If a t least one of inequalities (4.4.55) is not satisfied, then the couple LAIR+) does not possess the C-property relative t o the couple LdBZ+). 0

In particular, it follows from Dmitriev’s theorem that

LA@) is a Calder6n

couple. In this case, however, a considerably deeper result generalizing Calder6n’s classical result is valid.

Calder6n couples

609

Theorem 4.4.40 (Span.).

The couple

LA@)

possesses the C-property.

0

Finally, we consider a generalization of Theorem 4.4.18 by Cwikel. For

this purpose, we take two couples of quasi-power parameters of the Kmethod of the form L?. Since in this case, by the Lions-Peetre equivalence theorem (see Corollary 3.5.15) KL; g

J p , we

may use the notation

( s ) ~ , ~ .

Corollary 4.4.41.

If t h e inequalities (4.4.55) are satisfied, then for any two couples 2 and +

+

the couple (Xw,m,XwI,pl) is K-adequate t o the couple

+

?,

+

(Yvo,qo,Y+,l,ql).

Proof. Since in the case under consideration L$ 5 L$ and the action of K the operator S is bounded in these couples, it is sufficient t o apply Theorem 4.4.20. 0

We shall limit ourselves t o only one example illustrating this theorem (some other Calder6n couples are described in Supplement 4.7.2). Let L i p a ,

0 < cr 5 1, denote t h e space of functions f that satisfy t h e condition

According t o Proposition 3.1.19, L i p a Z (C,Lip l)am. Therefore, t h e previous corollary allows us t o conclude that ( L i p a , L i p p ) is a Calder6n couple for o

< a , @< 1.

610

Selected questions in the theory of the real interpolation method

4.5. Inverse Problems of Real Interpolation

A. Let us first consider an individual inverse problem where, for a given

E- (or K - ) functional, we must

function p possessing the properties o f the find an element z E C ( x ' ) for which

E ( . ; z ; x') coincides w i t h

or is equi-

valent to p. The first result of this kind is a classical

(Bernshtein).

Theorem 4.5.1

Let

(X,),,,

be an increasing sequence of subspaces of a Banach space

X

such th at (4.5.1)

Further, l e t (a,)c

(4.5.2)

and m

dimX, = n

=

R+be an arbitrary

lim a, = 0

n-m

X . nonincreasing sequence such that

.

Then there exists an element z E

X ,such

that

0

L e t us verify that none of the conditions of the theorem can be weakened. For instance, the impossibility to exclude the assumption about the finite dimensionality o f

X, is demonstrated

in

Example 4.5.2.

If Theorem 4.5.1 is valid for every sequence of infinite-dimensional spaces

(X,), then X

is reflexive.

Indeed, let us choose for

llfll

X1 the subspace K e r f , where f E X' and X, when n 2 2. Further, we p u t a1 := 1

= 1, and the space X for

and a,

:= 0 for n

2 E X.such that

2 2.

Then, by hypothesis, there exists an element

Inverse problems of real interpolation

611

Since according t o the Hahn-Banach theorem

on account of the equality

Since

f

llfll

= 1, we obtain

is arbitrary, it follows therefore that any nonzero continuous linear

functional attains its upper bound on a unit sphere of the space X ding t o the classical James theorem, a space

X

. Accor-

having such a property is

reflexive. 0

Let us verfiy that the second condition from (4.5.1) is also necessary. Example 4.5.3. Let X

:=

C [0,1]

and let

X, coincide with

the linear envelope of the set coincides with the closure of

{l,~,~4,...,~(n-1)z}.In this case, the set

the linear envelope of the set { t ( " - l ) z } n E ~. Since

< 00, CF=p=, ("-1)2 1

ac-

cording t o t h e classical Miintz theorem the second condition in (4.5.1) is not satisfied. Therefore, lim

E,(z) > 0 for the

elements z

E C[O,l]\(u).

However, the statement of Theorem 4.5.1 is not satisfied in this case for nonincreasing sequences (a,)which do not tend t o zero either. Indeed, let us choose a, := 1, n E

function

RV. If the theorem

is valid, then there exists a

f E C [0,1], for which En(!) = 1 ,

nEN

.

Without loss of generality we can assume that

E,(f)= Ilfllc.

Then it follows

from the previous equality that

so that the element closest t o

f

in the subspace X,, coincides with zero.

In view of the classical Chebyshev-Haar theorem, this element is unique

Selected questions in the theory of the r e d interpolation method

612 in

X n and has the property that the difference between this element and

the function attains its maximum absolute value with a subsequent sign

+

reversal at least at the ( n 1)-st point. This means that in the case under consideration for each n there exist points tp’ < t p ) < ... < tn+* (n) in the segment [0,1] such that

f&’) Here

E

E

,

= E(--ly

f 5 k 5 n + 1.

{-I, I}.

Then there exists a point to E [0,1] in whose any neighbourhood the function f assumes the value of +1 as well as the value of -1 infinitely many times. Consequently, it is a point of discontinuity o f f , which contradicts t o the fact that

f

belongs t o C [0,1].

U

It should be noted that in both examples it is not difficult t o find an element (4.5.4)

2

such that

E,(z) x a,

( n + m)

Therefore it can be askes whether it is possible to generalize Theorem 4.5.1, replacing equality (4.5.3) assumptions (4.5.1).

by equivalence, but considerably weakening the

However, there exist examples which show t h a t a re-

jection of the requirement o f the finite dimensionality of X , makes Theorem 4.5.1 untrue even if equality (4.5.3) is replaced by an equivalence (4.5.4). Below, we give a general result i n which we take for

X , nonlinear manifolds

(for example, the set of continuous rational fractions o f a degree not exceeding n ) , or infinite-dimensional subspaces. In this case, however, the class

for which relation (4.5.4) is valid is considerably narrower of sequences (an) than i n Theorem 4.5.1. The initial proof was based on a direct construction of the required element

5.

Here we shall prefer another approach based on

a generalization of Theorem 4.5.7 which will be proved later.

In order t o

formulate this theorem, we shall recall the definition o f quasi-power function (see Example 3.5.2).

Inverse problems of real interpolation

613

Definition 4.5.4.

A function cp E Conv belongs t o the subcone P of quasi-power f u n c t i o n s if for some y > 0

.

Sp 5 yp

(4.5.5) 0

Remark 4.5.5.

If

then S =

S-

+ S+.

Putting

the subcones P- and (SEi.>(t)

where

E

E {-,

1- := (0,1] and I+ := [l,+w),we define

P+,replacing (4.5.5)

57i.(t) 7

tE

by the inequality

L,

+}. Obviously,

Q=P+nQ-. Remark 4.5.6. Let us consider t h e mapping J : Conv + Conv defined by the formula

1 (JP)(t) := ti.(,) . The quantity

J

is an involution of the cone Conv such that

JS+ = S- J

, JS- = S+ J .

In particular, we have (4.5.7)

J S = SJ .

Hence it follows that J is also an involution of the cone

P

ant that

JP* =

PT. Let us now formulate the main result providing a criterion of Convoabundance of the couple

2. According t o Definition 4.4.8,

the couple X

possesses this property if for any function cp E Convo there exists an element z E

X such that

Selected questions in the theory of the r e d interpolation method

614

K ( . ; 2 ; 2) cp

(4.5.8)

with the equivalence constant independent o f z and cp. This criterion is contained in Theorem 4.5.7 (Krugljak). For a couple

2 to

be Convo-abundant, it is necessary and sufficient that

there exists a nonzero element zo E

.

K ( . ; zo; I?) E P

(4.5.9)

Z(d)for which

Proof.We require Lemma 4.5.8.

If cp E

P,then for any

number r

> 1there exists a number X

:= X(T)

>1

such that (4.5.10)

~ ( tL )c p ( X t > L

X

;~ (

Proof.If cp E P,inequality (4.5.5)

t >( t E

x+)

is satisfied. Since in view of the concavity

of cp

cp(t)mt L cp for any

t E R+(recall

that

m,(t) := min(l,t/a)) we obtain from (4.5.5)

s [cp(t)m*lI7cp . Hence for any X

> 1we

have

Taking now X such that 1nX = y r , we get

rcp(t> I cp(W

7

which proves the left-hand inequality in (4.5.10). Let us apply the inequality just proved t o the function which, according t o (4.5.7). belongs t o

P. Then we arrive

$ ( t ) := tcp(l/t) a t the inequality

Inverse problems of r e d interpolation

615

which is equivalent t o the right-hand inequality in (4.5.10). 0

Remark 4.5.9.

It can easily be verified that condition (4.5.10) is necessary for cp t o belong to the cone

P.

Let us return t o the proof. Since the necessity is obvious, we must prove the sufficiency of condition (4.5.9). Thus, let this condition be satisfied and let a function cp E Convo be given. Our aim is t o construct an element

z E C(x') for which condition (4.5.8) is satisfied. For this purpose, we first

find for each (4.5.11)

t E R+an element vt such that

mt

5 K ( . ; wt ; x') 5 ymt ,

where 7 is independent o f t and For this we take

T

2.

:= 5 and assume that X := X(r)

> 1 is the number

mentioned in Lemma 4.5.8 applied t o the function cp := (4.5.9).

Further, l e t the element fo(t) E X o ( t E

+

K ( . ;zo; X ) in

X+)be defined

by the

inequality

+

llfo(t)llxo t IIzo - fo(t)llx,

I2K(t;

20;

2) *

We define the element ut E A(d)by the formula 211

Since X

:= & ( A t )

- f.o(t/X) .

> 1and the K-functional does not

I

2{K(Xt;

20;

increase, we have

d ) + K ( t / X ;z o ;

Similarly, we have that the function

Z)}I 2 ( 1 + X ) K ( t ;

2;

K ( t ; 2 ; x') is increasing t

K ( s ; u t ; 2)Is 1120 - fo(Xt)Ilx,

+

3 1120

- 5o(t/X)l(x,

5

2).

616

Selected questions in the theory of the real interpolation method

2s

5

{

2( 1

b

K(Xt ; G o ; 2) K(t/X ; 20 ; 2) At t/x

+

+ X)s/t K ( t ; 20 ; 2)

Taken together, the estimates obtained give (4.5.12)

K ( s ; ut ; 2)5 2(1

+ A) min(1,s / t ) K ( t ; zo ; 2).

In order t o estimate the K-functional o f ut from below, we use the representation ut = (zo - i o ( t / X ) )- ( G o - ? o p t ))

.

This leads t o the inequality

t K ( t ; ut;2)2 K ( t ; so; 2)- K ( t ; go(+

2)-

Since the last two terms do not exceed

t 2qx;

Go;

2)+ -x2 K ( X t ; G o ; 2)

in absolute value, which by inequality (4.5.10) does not exceed 4/r = 4/5, we obtain from t h e previous inequality

4 K ( t ; ut;2)2 (1 - -) K ( t ; 2 0 ; 2). 5 In view of the concavity o f the K-functional, we hence obtain the required estimate from below:

+

(4.5.13) K ( s ; ut ; X)2

1

5 min(1,s / t ) K ( t ;z o ; 2)

Further, we put Dt :=

5 K ( t ; G o ; Z)-'ut .

617

Inverse problems of real interpolation Then the estimates

(4.5.12)and (4.5.13)lead t o the required inequality

(4.5.11). Let now 'p E Convo be given. We choose a number q

> 1 and construct

a sequence (ti) in conformity with Proposition 3.2.5.According t o items (c)

and (d) of this proposition, t h e set of indices

2n

i varies between 2m - 1 and

+ 1, where m,n E RV u {+m} (since cp E Convo). 3.2.5,

Proposition

where

In view of t h e same

+ is defined by the formula

The following inequality also holds:

(4.5.15)

v(t)5 qq(tzi+l)mt2,+i(t)>

where t E [tz;,t 2 i + 2 ) and i

# -m, n. If however i

:= -m (:= n ) , inequality

(4.5.15)is fulfilled for t E (0,Lz,,,) (for t E (tZn,+m)respectively). Let us now choose an element z := z('p) satisfying relation (4.5.8). Namely, put 2

:=

Ci vtzi+lv(t2i+l) *

Then in view of inequalities

(4.5.11)and (4.5.14),we have

(4.5.16) K ( * ;2 ; 2)5 7

q+l cp. C (P(tzi+i)mt2,,,IY q-1

Moreover, we have from

(4.5.11)

K(tzi+l;2 ; 2)2

( ~ ( t 2 i + 1 )-

Y

C (P(tZk+l)mtlL+1(tzi+l) . k#i

Taking into account t h e relations (3.2.13),the sum on the right-hand side is estimated from below by the quantity Y v ( t z i + l ) ( q + l - 1). Putting q :=

q-1

47 + 1 therefore we obtain

(4.5.17) K(tzi+i ; z ;

1

2)1 5 ~ ( t z i + i )

618

Selected questions in the theory of the real interpolation method

Let now t E

R+be fixed. Then three cases are possible: there exists i such t belongs t o [tz;,tz;+z)or t E (0, t-2m)or, finally, t E [ t 2 n , +w). In the last two cases, we put i := --m or := n respectively. With such a choice of t h e number i, we have in view o f (4.5.17) and (4.5.15)

that

K ( t ; 2 ; 2)L

~ ( t 2 i + l ;z ; 2 ) m t z , + l ( t )

L

Finally, we have obtained the required inequality:

U

Let us consider a version of Theorem 4.5.7 which is useful in the applications. For this we define t h e subcone Conv- (Conv+) of the cone Conv with the help of the limiting relation lim p(t) = 0

(respectively,

Convo = Conv-

n Conv+ .

t-+O

lim

t++m

d t ) = 0) -

t

0bviously

Modifying the above proof, we obtain the following result. Theorem 4.5.10. f

The couple X is Conv,-abundant on I,, where I- := (0,1], I+ := [l,+w) and

E

E

{-, +}, iff there exists a nonzero element K ( . ; z,;

2,

-.

E C ( X ) ,such that

2)E P, .

0

Let us now return t o the inverse problem in approximation theory, which was discussed in the beginning of this section. Thus, we consider a family

of subsets ( X n ) n c ~ u (oof )the Banach space X and assume that

Inverse problems of red interpolation (4.5.18)

X , := (0)

619

X .

and ==

We define the best approsimation o f x by the formula

and analyze the existence, for a given monotone convex sequence (a,)tending t o zero, of an element z whose sequence o f best approximation tends t o zero “almost” as (a,). Obviously, without additional assumptions concerning t h e approximation family (X,), there is no such element. We assume that (4.5.19)

+

Xm X, C Xm+, (m,nE

and, moreover, that for any X E (4.5.20)

AX,, c X,,

(n E

N

U (0))

R

nV)

Finally, we assume that t h e sequence of sets

X,, does not “glue together”

as n + 00. In other words, (4.5.21)

y := inf [dist(Xn+ln S ( X ) , X n ]> 0 . n

Here the separation between the sets is defined by the formula

dist(A,B) := sup

inf

XEA

yEB

/Iz

- yllx

.

Under the assumptions formulated above, the following theorem is valid. Theorem 4.5.12 (Brudnyi). Let (a,,) c

lR+ be an arbitrary convex nonincreasing sequence tending to

zero. Then there exists an element x E X for which

and, moreover,

620

Selected questions in the theory of the r e d interpolation method

Here the constant y1 depends only on the quantity (4.5.21)

Proof. Let us consider t h e set 00

Y :=

IJ

x,.

n=O

In view of conditions (4.5.19) and (4.5.20),

Y is a

linear subspace of X. Let

us define on it the function

(4.5.22)

J1zlly :=

inf{n :

L E

X,} .

By (4.5.18) and (4.5.21) (0) = XO# X1 so that

Further, according t o the definition and condition (4.5.19), we have (4.5.23)

11x1

+ 4 1 Y I11ZlllY + 115211Y ,

and condition (4.5.20) leads to

Thus, Y is a quasi-normed space in the sense of Definition 3.1.33. Let us verify the completeness of Y . Indeed, in view of the definition of quasi-norm [see (4.5.22)], any sequence fundamental in

Y stabilizes, and hence has

a

limit. Thus, ( X , Y ) forms an Z-couple (see Definition 3.1.33). Let us now show t h a t Theorem 4.5.7 is applicable t o this couple. For this we shall outline the properties of the K-functional used in t h e proof. While seeking elements

wt and proving inequality (4.5.11), t h e concavity and the monotonicity of the I<-functional (these properties are preserved) were used as well as the positive homogeneity. In view of (4.5.24), in the case under consideration the relation

K ( t ; ys ; 2)= y F q y - 9 ; 5 ; 2)

Inverse problems of real interpolation

621

is satisfied instead of the homogeneity. Therefore, putting

where ut is constructed as in Theorem 4.5.7, and y defined by equation

5K(yt ; 50 ; 2)

y :=

>

dt>

we obtain +

K(y-'s ; Wt ; X ) = y-'K(s ; U7t ; 2)x y-'rn,t(s) = y-lrnt(y-ls)

.

In view of t h e arbitrariness of s and the choice of y,we obtain

K ( . ; Wt ; 2)x cp(t)rnt uniformly in 2

t.

Further, we define

c

:=

2

:= ~ ( 9by)the formula

wtzi+l .

Using t h e previous inequality and the inequality

C

K(*;

zi;

( x , y )I) C

K(.; 2 ; ;( x , y ) 7)

which is valid in view o f (4.5.23), we complete t h e proof in the same way as in Theorem 4.5.7. Thus, Theorem 4.5.7 is valid for the couple

( X ,Y ) .

Let us now calculate the E-functional o f this couple. Since 0 E X,, we have

xnCX1+xnCXn+1

*

It follows from this expression and (4.5.22) that

E ( t ; 2 ; ( X , Y ) ) := inf {It. = inf (1 . Hence we obtain the formula

- Yllx ; IlYllY

5t)

- yllx; Y E X[t]I .

=

+

. K ( y t ,20 ; x

Selected questions in the theory of the r e d interpolation method

622 (4.5.25)

E ( t ; 2 ; ( X , Y ) ) = Ep](z) ( t E R+) .

It is important for the further analysis to mention the following properties of En(z) which follow from conditions (4.5.18)-(4.5.20): (4.5.26)

En+m(z

+ Y) I

(4.5.27)

&(AX)

= 1x1En(2)

(4.5.28)

lim En(s) = 0

n-co

En(2)

+ E ~ ( Y )( n ,m E N U (0)) ; (A E R,12 E lV U (0)) ;

.

Let us now use the relation between the K - and E-functionals of the I-couple ( X , Y ) , which in this case are expressed by the first formula from (3.1.18) and formula (3.1.46). Thus, (4.5.29)

K ( . ; 2 ; (X, Y ) )= EV (. ; 2 ; ( X ,Y ) ) ; E'(. ; 2 ; (X, Y ) )= K " ( . ; 2 ; ( X ,Y ) ) .

Recall that f'denotes the maximal convex minorant

If1

and that

+

f " ( t ) = inf { f ( s ) s t } , S>O

fA(t)=

.

sup {f(s) - S t ) s>o

Assume that the conditions of Theorem 4.5.7 be satisfied for the couple ( X , Y ) (this will be proved later). Then for any function 'p E Convo there exists an element zv such that (4.5.30)

K A(. ; 2 ; ( X ,Y ) )x

'pA

.

Let now (an)be the convex sequence in the hypothesis of the theorem, which tends to zero, and l e t a : R++ R+ be the convex function defined by this sequence. We put $0

Then

'p

:= a v .

E Conv, and for any

E

> 0 we have

623

Inverse problems of red interpolation

+ t s ) 5 a, + nt <

cp(t) = inf ( a ( s )

E

s>o

if n := n ( ~is) chosen so that a, shown that

< ~ / and 2 t < ~ / 2 n Thus, . we have

lim cp(t) = 0 . t-0

Similarly, we have for t + +oo

cp(t)/t I ao/t + O

.

Thus, we have establihsed that cp E Convo. Applying now (4.5.29), (4.5.30) and relation (3.1.17), we obtain (4.5.31)

E f t ; zv ; (X, Y ) )x

( c z V ) ~= a7t)

with the equivalence constants independent of t. Taking t
<