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
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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
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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
n and using the fact that (a,,) is convex, we obtain from (4.5.31) and (4.5.25) the inequality 7
*
an
I ~ ( nz v;;
:=
( x ,Y ) )I En(xv) .
Putting x := ry-lzv and taking into account (4.5.27), we obtain
In order to prove the second statement of the theorem, we note that in view of the definition of convex minorant and the relations (4.5.25) and (4.5.28), there exists an infinite sequence (nk)c N such that
Efnk ; zq ; ( X , Y ) ) = Enk(%?) Then for t := nk we obtain from (4.5.31)
so that the inequality
Selected questions in the theory of the real interpolation method
624
can be written for z := 7 - l ~ It~was . mentioned earlier that both constants are independent of
2
and (a,,).
Thus, t o complete the proof of the theorem, it remains t o verify the existence of an element z E
X for which K ( . ; x ; 2)E 7'. In view of the
first equality from (4.5.29), this will follow from the existence of an element z E
X for which
(4.5.32)
E ( t ; x ; (X, Y)) ta
for some a
> 0.
( t E at+)
In order t o find such an element z, we take using condition
(4.5.21) an element z, such t h a t
Choose a number q
> 2 (which will
c
be specified later) and set
00
x :=
q-mxm.
m=l
Then in view of conditions (4.5.19) and (4.5.20), we have n-1
C
q-mxm
EX
I + ~2
+ ... +
cxp-1 ,
~ 2 n - 1
m=l
whence it follows that
On the other hand, in view o f (4.5.26) and (4.5.27) we have
Using inequality (4.5.33), we obtain hence
625
Inverse problems of red interpolation Taking in this inequality q (4.5.35)
> 2 so large that
E p - l ( z ) 2 $ q-"
,
considering that E,,(z) decreases monotonically, and we obtain from this inequality and from (4.5.34) the relation
E"(X)x
7-L-p
,
where a := log,q.
It remains for us t o note t h a t the equivalence constants in these relations, and hence in the theorem t o be proved, depend only on the quantity (4.5.21). Remark 4.5.13. Let ( a n )satisfy the Az-condition, i.e.
6 := supQL"/a~" < 00
.
Then using the inequality
f74 I If(U I 2 M t / 2 ) (see Bergh and Lofstrom [l],Lemma 7.1.3), we obtain from (4.5.31) the stronger relation
where the equivalence constants depend only on the quantity (4.5.21) and 6.
B. Let us consider one more inverse problem. Namely, we shall determine the extent t o which the family of interpolation spaces
(z~,)~ defined the
c o u p l e d . As (4.5.36)
Ka(T) = K a ( 2 " ),
we see that the couple
r? is defined t o within
relative completion. On the
other hand, the relation (4.5.37)
(LI,L,)S,
(L1,BMO)sm
(0 < l9
< 1)
Selected questions in the theory of the real interpolation method
626
(see Sec. 3.9.B) shows that two different relatively complete couples may
generate coinciding families of the spaces under investigation. It should be noted, however, that the isomorphism constants in (4.5.37) tend t o +m as
6
---f
0 or 1. The result presented below shows that a relation similar t o
(4.5.37) is impossible when the isomorphism constants are uniformly bounded in 6 E ( 0 , l ) . Theorem 4.5.14. In order that the relation -..
(4.5.38)
2 Y&,
28,
(0 <: 19 < 1)
holds with uniformly bounded isomorphism constants, it is necessary and sufficient that (4.5.39)
( S ) O
E (+)O
. -..
( - f C )=~X8,. , Besides, as we pass from the couple zct o the couple (2?")O, the K-functionals of elements
Proof (suficiency).
In view of (4.5.36),
-
in E0(I?") do not change. However, for 0
< 29 < 1,
(2?")8, C"2C) . Therefore, we have
This leads t o the sufficiency o f condition (4.5.39). In order t o prove the necessity, we consider an element z E Ac(x'). Since 4
Ac(X) = A(*)
-+
+ X,J,
E
Pg,
L+
C(P) ,
it is worth while t o consider the I<-functional of z in the couple
?. Let
us
verify that (4.5.40)
K ( . ;z ;
2)x K ( . ; a:; 9 ) ,
where the equivalence constants are independent of z. For this we consider the function F :
R+x Ac(x') -+ R, defined
by the formula
Inverse problems of r e d interpolation
627
CIea rly,
c
2
K ( t ; z;;
2).
x = c xi
Since the right-hand side is not smaller than (4.5.41)
K ( . ;z ;
K ( t ;z ;
2),we have
2)I F ( - ;z ; 2).
Let us now prove the inverse o f this inequality. Since z E A'(-?), we have
Consequently, if q concave function
>
1 and
(ti)%-,
is the sequence constructed for t h e
+
K := K ( . ; z ; X )and the chosen q, then n, rn < 03
(see
Propositions 3.2.5). In view of the same Proposition 3.2.5, we obtain
Using Theorem 3.2.7 on A'-divisibility, we obtain from the l e f t inequality in (4.5.42) the existence o f a finite sequence (4.5.43)
z =
C
z2i+1
9
K ( .; z 2 i + 1 ;
c
( ~ ~ i + ~A'(x'>, )
such that
2)I y q K ( t z i + l ) m t z , + ,
for all 2i+ 1 E [-m,n] and y := 9. From this inequality and the definition of the function F , we have
The upper bound in (4.5.53) does not exceed
628
Selected questions in the theory of the real interpolation method
Consequently, for all t E
(4.5.44)
lR and any di E (0,1), we
F ( t , z ; 2)5 7q
c( i
t -)9’K(tzi+l) tz;+1
have
.
Using the arbitrariness in the choice of di, we define them for a given and t
E
>0
> 0 so that
for
& > 1 and
for
& 5 1. For such a choice o f di, we obtain from (4.5.44) and (4.5.42)
the inequality
~ ( t2 ); 5 7 q ( l +
E)
t C K(tzi+l)min(l, -) hi+l
5
Together with (4.5.41), this leads t o the equivalence
(4.5.45)
F ( . ; 2 ; 2) x K ( . ; 2 ; 2)
with constants independent o f I and
d.
Since we never used, while proving inequality (4.5.41). the fact that z E A C ( ~ ’ )we , can write
F ( . ; 2 ; P) 2 K ( . ; 2 ; P) . Taking into account (4.5.38) we obtain from this inequality and (4.5.45)
K ( . ; 2 ; P) 5 T F ( . ; z ; 2)5 y 1 ~ (;.5 ; d>x min(1, t) . Thus, the element
(4.5.46)
2
belongs t o A‘(?), and the equivalence
F ( . ; 2 ; 2 )x K ( . ; 2 ; 9 )
629
Inverse problems of red interpolation
holds. Since in view o f (4.5.38) the left-hand sides of (4.5.45) and (4.5.46) are equivalent, for x E
Ac(x') we have
K ( . ; x ; 2)= K ( . ; 2 ; ?) with constants independent of
x.
Arguing in the same way for the couple
A(2') = A'"(r?)Z A'(?) and for an element t belonging t o
?, we hence obtain
= A(?'")
,
A(-f'"),we have
.
K ( . ; 2 ; 2'")
2 ; +)
x;= KLiJ-f') ,
yi" = KLb,(?'"),
; a ( : %
Since -#
t h e norms of these spaces are equivalent on A(-fc) [and on
A(?'")]. Passing
t o the closure, we get
( X yLz (Y,')O . 0
Condition (4.5.38) is equivalent not only t o the relation
(2')' (?'")',
but also t o a similar relation where the operations,of closure and relative completion are transposed. This statement can be obtained by modifying the proof presented above. However, it is simpler to make use of Proposition 4.5.15. (2'")O
E
(20)".
Proof.It should be noted first (4.5.47)
o f all that for any couple
JLt(?) r yJ (i = 0 ,l )
? we have
.
Indeed, in view o f (3.5.18) and Theorem 3.6.1, the left-hand side is embedded into the right-hand side. Moreover, the two spaces coincide since in view of Theorems 3.4.9 and 3.5.5(c), we have
630
Selected questions in the theory of the real interpolation method
Let us now use Corollary 3.5.16(b) for the couple
?
:=
2‘. Together with
(4.5.47). this gives
(x;)Oz ~ ~ ~z ~( ~2 ~n c0(Zc) )( 2. ~ ) Since
C ( 2 “ )= C ( 2 ) the right-hand side is given by
~ ~ ~ n co(2) ( 2 = ~) ~ ~ = (x:)‘ ( .2 ~ )
C . Concluding t h e section, l e t us consider an inverse problem connected with the reiteration theorem o f Lions and Peetre. According t o this result, the “path” (r?sq),0 < I9 < 1, connecting Xo and X1 is “linear”. Indeed, t h e analogous path connecting X,,, and Xslq,0 < 19 # I91 < 1, coincides with the corresponding part o f the path between Xo and XI. Since the two segments having two points in common lie on the same straight line, the above analogy makes natural the following result which t o a certain extent is inverse t o the Lions-Peetre theorem. Theorem 4.5.16 (T. Wolf). Let the Banach spaces X;, 1
5 i 5 4,
be continuously embedded into a
separable linear topological space in such a way that (4.5.48)
A(X1,X4)
A(X2,X3) .
Let us further suppose that for some I9,cp E ( 0 , l ) and q , r E (O,+m] we have
Then putting
( := we obtain
(PI9
~ - Q + + Q
, 1c,
:=
I9 1-cp+I9cp
’
Inverse problems of red interpolation
Proof. It is sufficient
631
t o prove that
where t h e constant 7 (here and below) depend only on X i , 1 5 i
-
5
4.
Indeed, the first inequality is equivalent t o the embedding
(X17X4)'l
x 3
7
(see Bergh and Lofstrom [l], Theorem 3.5.11), while the second inequality is equivalent t o the embedding x 3 L+
(Xl,X4)',00
*
Then taking into account (4.5.49), we obtain from the Lions-Peetre iteration theorem [see Theorem 3.8.10(a)]
Xz = (Xl,X3)rpr
(Xl,X4)V',r = (xl,X4)~r.
The formula for X3 can be obtained similarly. Let us now prove inequality (4.5.50). Let z E A(Xl,X4).Then z E X3 and in view of the first inequality in (4.5.49) we have
Similarly, we obtain
Substituting the second inequality into the first one, we obtain (4.5.50). Let us prove inequality (4.5.51). For this we take a certain be specified below) and suppose that z E X3 and
(Y
> 0 (it will
t > 0. We must show
+
that there exist elements z1E X1 and z4 E X4 such that z = z1
ll3,-1llx1+ t 1l4lx4 IY ll4lx3t+ . We put
24
and
Selected questions in the theory of the red interpolation method
632
In view of conditions (4.5.49), there exist elements
5:
E X i such that z =
xi + x: and xi = xi + xl, where
II4llx2 + 21 IIx:llx4 5 Y U 9 11~11x3
7
Then
= x:
2
+ 5: + 2; and considering the definitions of u and v , we have
Let us repeat this line of reasoning, using 5; instead of x , and so on. For an arbitrary n E
N ,this gives the expansion
(4.5.52)
=
5
+ C1SjSn a: + C l S j S n 4
with the estimates
Let us now choose a so that the constant a y in (4.5.53) is less than unity and put x1
c
:=
z;
,
54
:=
ncN
c
5;
.
ndV
Then in view of (4.5.52) and inequality (4.5.53), both series converge in X1 and in
X4 respectively, and x = x1 + x4. Then it follows from the second
inequa Iity (4.5.53) that
+ t llx4llx4 5 PtJI
11~1IIx1
c
ndV
Thus, inequality (4.5.51) is proved.
( Y V l
'
Inverse problems of real interpolation Remark 4.5.17
633
(T. Wolf).
(a) The theorem is also valid for quasi-Banach spaces X i . In order t o prove
t h e theorem for this case, it is sufficient t o use in the last stage the Aoki-Rolevich theorem according to which for some p E (0,1] and any sequences (zn),we have
(b) A result similar t o Theorem 4.5.16 is valid for t h e complex method as well (in the Banach case). Remark 4.5.18 (Janson-Nilsson-Peetre).
Let us consider a family o f functors (Fa)o
If ( T n ) n cc~.C(x’, ?), and SUP n
IITnllxi,~,< 0;)
&%
7
llTnllx~-i,Y~-i= 0 >
then the following relation is valid:
$&
llTnllFa(~),Fa~~) =0 .
In particular, the functors (.)+ and
C, for 0 < 19 < 1, as well as
many other
functors, have the strictly interpolation property. As regards a family
(Fa)0<,
strictly interpolation functors, the que-
stion o f t h e validity o f an analog of the Wolf theorem can be raised.” Thus, for given couples
x’ and ? one assumes that for some 7, p E ( 0 , l )
Fp+JZ) z I?&(?)
(0 5
7-
5 1- p) .
One has t o find out whether these spaces have the form F,(,)(Xo, Y l )with
[O,1 - p] + ( 0 , l ) . It turns out that, under certain approximation conditions imposed on the spaces Xi and yi, the answer t o this question is affirmative.
a certain function a :
”It can easily be shown that this is wrong without the condition of strict interpolation.
Selected questions in the theory of the real interpolation method
634
4.6. Banach Geometry of Real-Method Spaces
A. Investigations o f Banach properties of interpolation spaces are steadily going on.
Therefore, we shall confine ourselves to an analysis of the i n
our opinion most brilliant results. The readers interested i n the subject are referred t o the literature cited in the comments on this section (see Sec.
4.7 .A). Let us consider the reflexivity o f spaces o f the K- m e t h o d . T h e first of the results considered here is contained i n
(Beauzami). For a space y8pto be reflexive (0
< 19 < 1, 1 < p <
and sufficient that the closed unit ball topology of the space
m) it is necessary
B ( A ( 2 ) ) be precompact in the weak
E( 2) .
0
Using the material o f the previous chapter, we shall outline below a new approach and a broad generalization of Beauzami's theorem.
In order t o
formulate the corresponding results, we shall need some definitions. Definition 4.6.2. The operator
T E L ( X , Y ) is termed weakly
is weakly precompact.
compact
if the set T ( B ( X ) )
The set o f all such operators will be denoted by
P ( X ,Y ) . 0
A natural transition of these concepts t o couples leads to Definition 4.6.3. The operator
T
E ,C(?,f)
is referred to as weakly compact if
E
Lwc(A(x'),C ( f ) ) . The set of all such operators will be denoted by C w c ( X , ?
Banach geometry of real-method spaces
635
Further, we shall require Definition 4.6.4.
A couple 2 is termed weakly reflexive if there is a reflexive space among i t s intermediate spaces. 0
Proposition 4.6.5.
If the couple 2 is weakly reflexive, then
cwC(x') ;
(a)
13 E
(b)
L ( Z )= L w c ( Z )
h f . (a) Let
X
be a reflexive intermediate space o f the couple
-
that the embedding operator
i
:
A(2)
-+
x . We shall show
C ( 2 ) is weakly compact.
A(2) X ~t C ( 2 ) is the product of two embedding i = i l i 2 . Here the operator il : A(x') -+ X is weakly compact. Indeed, il ( B ( A ( 2 ) ) )is a bounded set of the reflexive space X , and hence it is precompact in the weak topology (the Banach-Alaoglu Indeed, since
operators:
theorem).
Since t h e product o f two linear operators one of which is
weakly compact is also weakly compact, the required property of
i is
established. It remains t o note that l d l a ( ~=) i. (b) If
T
E
L ( 2 ) then T
can be represented in the form T = 1lmT and so
condition (a) can be used. 0
Example 4.6.5'. +
(a) The couple I , = (I:,&)
is weakly reflexive. Here
636
Selected questions in the theory of the red interpolation method Indeed (see Bergh and Lofstrom [l],Theorem 5.6.1)
(1'00)~,
1;
(b) The couple (L,,L,)
. is weakly reflexive for the same reason.
(c) It follows from Theorem 4.6.1 (its generalization will be proved below, see Corollary 4.6.18) that the condition
12 E Lwc(z) is not only necessary
but also sufficient for the couple J? t o be weakly reflexive. Hence it follows, in particular, that the couple
(Lp,Wpk),where p E [ l , ~ is] ,
weakly reflexive. Finally, we shall use Definition 4.6.6.
The functor
F
retains the property of weak compactness if the weak com-
pactness of the operator
T l q ~ ): F ( 2 ) -+ F(?) follows from the fact
that T E LWc(.f,?). 0
The relation between this definition and the question under consideration is revealed in
Proposition 4.6.7.
If the couple
x'is weakly reflexive and the functor F
retains weak compact-
ness, then the space F ( 2 ) is reflexive.
12 E L w c ( + f )the , operator
I121Fca,is weakly compact. But this operator coincides with the identity map of the space F ( 2 ) . Thus, the b f . Since
closed unit ball of this space is precompact in the weak topology. Since the convexity and closure in the strong topology leads to the closure in the weak topology (Mazur's theorem), this ball is compact in the weak topology as well. In view of the Banach-Alaogly theorem, this shows the reflexivity of
F(2). 0
Banach geometry of real-method spaces
637
The proposition proved above motivates the necessity of investigating conditions imposed on a K-functor under which it retains weak compactness. The following result provides an exhaustive answer t o this question. Theorem 4.6.8 (Aizenster'n-Brudnyr'). For the functor Ka to retain weak compactness, it is necessary and sufficient
that the space
K a ( L ) be reflexive.
Proof (necessity).
Since the couple 2,
is weakly reflexive, the reflexivity of
F ( f w )follows from Proposition 4.6.7. (Suficiency). We require Proposition 4.6.9. Let the functor F satisfy the following three conditions:20 (a) It is regular and relatively complete;
(b) F' is regular; (c) F" Z F . Then
F
Proof.
retains the property o f weak compactness. In view of the large number o f maps appearing in the proof, it is
expedient t o use the "diagonal search" technique which makes our operations graphical. We shall begin by establishing a (canonical) isometry of the spaces
F ( z ) * *and F ( 2 ) " . For this we use the operator
EX
: 5 ( X o ) *extending
linear functionals on A ( 2 ) t o X o . Here X is an intermediate space of the couple
x'(see Lemma 2.4.4).
Since F ( 2 ) and F ( 2 ) ' are regular, we have
Conjugating the first of these operators and putting 2oRecall that F ' ( 3 ) := F(J?)' and F"(f") := F(@". In particular, condition (b) implies that F(x')' is regular in the couple 2'.
Selected questions in the theory of the real interpolation method
638
we obtain the required isometry (4.6.1)
Ef
:
F(2)”.
F(Z)**
Everywhere below, we use the notation (4.6.2)
:
OG
y
:
G ( 2 ) - - t C o ( 2 ) , SG
:
A(Z)-tG(2),
P(2) q d ) , CS
adding, where necessary, the symbol
2
(for example, in (4.6.2) we write
a c ( f ) ,7 3 , etc.). Besides, we put
&I := omitting
:
CO(2)’ZCO(Z)* ,
in this notation if the couple is specified in the context.
Using this notation and taking into account (4.6.1), we can easily verify the commutativity of the diagram N
C(2”)
C0(2)**& E’
(4.6.3)
U
F
~ N
F ( 2 ) ” &F(Z”)
<
IUF
While verifying t h e commutativity, we must only take into account the following identities (see Proposition 2.4.6):
Co((x’)‘= A(2‘)
,
A(Z’)* = C((x’”) .
Lemma 4.6.10.
Im(6F) is dense in F ( 2 ) ” .
Proof. Let us verify the commutativity of the following diagram:
Banach geometry of red-method spaces
639
A( Z)**
C(x’l)* C *,O )X l(‘
I
Y*
Note that, according to our agreement, map, and
E
E
= €3, for the right lowermost
= €2 for the upper map. The same refers to t h e map OA on the
left- and right-hand sides of the diagram. To prove the commutativity, it i s sufficient to verify that for any two elements
2’’E
A(Z)** and
L’ E
A(,?‘)
we have -1
*
I‘
(E>Or(a)z”,2’) = (aA(2t)EBrY21z
I 7
)
’
After a transition of conjugate operators on the right-hand side and taking into account the equality
“;\(2)(4 =4 O ( R , )
7
following from the definition of O F , we obtain on the left-hand side of the previous relation (2”,(E22’)lco(R,))c(2,)
= (2”,Y2J‘)L-(rn,)
Considering t h e definition of the map
E2:(z*)= Z * ( A ( 3 1 )
(z*
E,
.
we have
E C0((X’)*),
whence we obtain for the right-hand side (oA~~,)(-~;~,~’’)IA 2‘) ( ~= ~ ) (~;itz”, , z‘)
= (z”, ~ 2 ~ 4 c* ( ~ ~ )
Thus, the commutativity of the diagram under consideration is proved. Putwe obtain from this relation and (4.6.3) the commutating 7 := ~af,-y>,, tivity of the following diagram:
Selected questions in the theory of the real interpolation method
640
N
F(Z)**,-+F(x'7") E (4.6.4)
6;t
6A
1-
A(x')** -A(d") r)
Since y is an isometric embedding, y* is a surjection, and hence the mapping 7 is also a surjection. In view of condition (a) of Proposition 4.6.9 the set A ( 3 ) is dense in
F ( 2 " ) . Therefore, the facts that 77
is a surjection
and that the diagram (4.6.4) is commutative ensure the density of
F(
I m F in
q**. Let now
T E ~ " ( x ' , ? )We . put
TF
:=
TIF(a): F(x') 4 F ( ? ) .
:=
T ( , ( q : A(x') --+C ( ? ) .
In particular,
TA
Let us use one of the criteria of weak compactness (see, for example, Dunford-
Schwartz 111, Theorem V1.4.2):
* S**(X**)
S E Lwc(X,Y)
(4.6.5) Here
ICY
L-)
.y(Y) .
is a canonical embedding o f Y into
P. Thus,
it is sufficient t o
establish that (4.6.6)
( T F ) * * ( F ( ~ )L-, * *tcF(p,(F(?)) ) .
Since it is proved in Lemma 4.6.10 that 6F(A(r?)**)is dense in F(r?)**, embedding (4.6.6) will follow from the inclusion (4.6.7)
(TF~F)**(z") E K.~(Q(F(F)) (5" E A(d)**) .
In order t o prove this inclusion, we shall use the obvious identity (4.6.8) Since
Tr = (~~(p)Z'~b,(y))** .
TA also
belongs t o C w c ( A ( X ' ) , C o ( ~ ) )because , Co(?) is closed in
C(?), while T(A(r?))
L-)
Co(?), we have according t o criterion (4.6.5)
Banach geometry of red-method spaces
Applying to both sides the mapping
and taking into account that the
E;
commutativity of (4.6.3) (for the couple
641
f ) leads t o the identity
EjuGq) = t p and that according t o Definition 2.4.10
we obtain (4.6.9)
.
In view of condition (c) of the proposition and the definition of the map
Theorem 2.4.12 we have
F(?)" n Kpo
(c0(?)) = Kpo(~(?)c) ,
and condition (a) implies that F(?)" = F ( p ) , (4.6.9) leads t o the embedding (4.6.10) Let us add
<~(TF~F)** E ~ p (F(?)) o . along the arrow in diagram (4.6.3) for the couple ?:
F(f)
-
F(?)**
SF(?)
The commutativity of this diagram follows from the definition of the canonical embeddings
K
[cf. (4.6.8)] so that the combined diagram is also
commutative. This leads t o the identity
Selected questions in the theory of the red interpolation method
642
from which, taking into account the identity ‘cpo :=
E ~ K ~ ~ ( we P ) obtain ,
the relation
Substituting this identity into the right-hand side of (4.6.10) and exploiting
that
5~
is an injection, we obtain from (4.6.10) the required embedding
(4.6.7). 0
Further we require some additional information from the theory of Banach lattices.
For t h e proofs of t h e results presented below, see, for example,
Kantorovich and Akilov, Sec.
3 of Chap. 4 and Chap. 10. For formulating
these results, l e t us recall some concepts.
A Banach lattice @ will be said t o be order continuous if from the monotone convergence t o zero of a sequence of non-negative functions (fn)
c@
follows that
lim llfnllo = 0 . Further, @ will be referred t o as monotonically complete if the fact t h a t a sequence function
f
(fn)
bounded in @ monotonically increases and converges t o a
implies that
f
E @.
Finally, we define an associated lattice @+ by putting
llfllo.
:= SUP
{ J fs+
; llsllo I I}
.
For a special case of this concept, see Definition 3.7.1 Lemma 4.6.11.
If @ is order continuous, then @+ = @*. 0
Lemma 4.6.12.
If @ is order continuous and monotonically complete, then @++ = @. 0
Banach geometry of real-method spaces Lemma
643
4.6.13.
If 9 is reflexive, then 9 and @+ are order continuous and t h e equality 9 = 9++is satisfied.
A well-known theorem o f Ogasaware states that the reflexivity of 9 is equivalent to the order continuity and monotonic completeness of 9 and 9*. Here the ordering in 9 ' is induced by the corresponding ordering in @ (i.e. the functional f* E 9 ' is non-negative if (f*,f)a 2 0 for all f 2 0). An application o f the two previous lemmas leads t o the required result.
b f .
0
Returning t o the proof of the theorem, we shall use the relation between the K-method and its discretization [see formula (3.3.30), in which it is convenient t o put q := 21.
4.6.14. Ke 2 K Fd G ) ' where F := Ka. Lemma
Proof. In view of Theorem 3.3.4,we have 4
Ka = Kqi,) = COrbF(tm)(.; L,)
.
A discrete analog o f this theorem, which is obviously also valid, gives
Finally, l e t us take into account the fact that the couple 4
the couple L,,
F(e,).
I', is a retract of
4
and therefore, the space F(1,) is a retract of the space
Hence, the application of the analog of Proposition 2.2.23 for
coorbits gives +
Corb,(i,,(.; L,)
+
Z
Corb,(i,,(.; l,) .
The three relations proved above lead t o the required result.
We can now establish the validity of the following important statement for the functor Kq.
644
Selected questions in the theory of the red interpolation method
Lemma 4.6.15.
The functor Ka is regular and nondegenerate.
Proof. In view of the discrete analog of nondegeneracy of the functor
Corollary 3.6.3, the regularity and
K&;-), where F
Ka, is equivalent t o
:=
the regularity and nondegeneracy of the parameter of F(f,)
L.Since the space F(1,) +
lattice F(?,)
in the couple
is reflexive by the condition of the theorem, the
is ordered continuous in view of Lemma 4.6.13. If z E
there obviously exists a nonincreasing sequence
(2,)
converging t o z such +
that the supports of all z, are finite. Then all I, belong t o A(l,) in view of the ordered continuity, and dense in
F ( L ) , and the regularity
111
- znllF(;,)
u1 3 # 0
+
F ( L ) , i.e.
Then the couple
is
show that
*
Suppose that conversely, for instance, the embedding F(r,) place.
c F.+ ( L )
0. Thus, A(l,)
is proved.
Let us now verify the nondegeneracy of
F(fm)\(z:
F(L),
(A(L),l:)
-
ZL takes
has a reflexive intermediate space,
and therefore it follows from Proposition 4.6.5 that the embedding operator +
i : A(l,) -+ 1: is weakly compact. Let us consider the subspace
x
:=
{(z,,)E ~ ( f , ); z, = o for n > 01
P : A ( L ) + X o f multiplication by the characteristic function of the set {n E 25;n 5 0). Then the operator ZP is also weakly and the operator
+
compact. However, by the definition of the norm i n A(2,) have for all
(2,)
and in 1; we
EX
IliPzlll~m=
SUP
1znl =
ll(~n)llq;m) .
n
Thus,
and hence the space X is reflexive. But X equality. We arrive at a contradiction. 0
2~
1, in view of the previous
Banach geometry of real-method spaces
645
We now have everything needed t o prove the theorem. We must verify the fulfilment of conditions (a)-(.)
o f Proposition 4.6.9 for the functor
F := Ka. Let us first verify the validity of conditions (a). The regularity of F has already been established in Lemma 4.6.15, while in view of the
-
discrete analog o f Theorem 3.6.7 and Lemma 4.6.14, the relative completeness follows from the relative completeness o f F(l,)
-+
in t h e couple .,I
But
a reflexive space is relatively complete in any enveloping Banach space (see
Corollary 2.4.13). Let us now verify the validity of condition (a) of Proposition 4.6.9, i.e.
the regularity of F’. In order t o do so, we use discrete analogs of Theorems 3.7.6 and 3.7.2. The first o f them is applicable t o the functor since, according t o Lemma 4.6.15, the parameter
K&rm,
F ( L ) is nondegenerate.
Applying this theorem and taking into account Lemma 4.6.14 we obtain the relation
[for the definition o f the discrete analog of the J-functor on the right-hand side, see (3.4.53)]. In this case A(G) is dense in
F ( L ) + , in view of Lemma 4.6.13 and
arguments similar t o those applied in the proof of the regularity of F(rm) in
F(rm)+ is a regular intermediate space of the couple 6 , and by t h e discrete analog o f Theorem 3.6.1, J&im) is a regular functor. Lemma 4.6.15. Thus,
In view of (4.6.11) the functor
F’ is then also regular.
It remains for us t o verify condition (c), i.e. the relation F”
%
F . For
this purpose, we use the second of the above-mentioned duality theorems (the discrete analog o f Theorem 3.7.2) in relation (4.6.11). This gives
In view of Lemma 4.6.13,
F(L)++= F(rm) so that the right-hand side is
equal t o K&-m, 2 F (see Lemma 4.6.14). Thus, all conditions of Proposition 4.6.9 are satisfied for 0
F
:=
Ka.
646
Selected questions in the theory of the red interpolation method
4.6.16. For the space K a ( 2 ) t o be reflexive for any weakly reflexive couple 2,it is necessary and sufficient that K a ( L ) be reflexive. Corollary
Proof.This follows from Theorem 4.6.8and Proposition 4.6.7. 0 +
It is not always easy t o verify the reflexivity o f the space K a ( L ) in applications. Let us note here an important case when the required property follows directly from the reflexivity of the parameter @. Theorem
4.6.17.
Let @ be a quasi-power parameter o f the K-method"
and @ be reflexive.
Then Ka retains the property of being weakly compact.
Proof. As in the proof of Theorem 4.6.8,it is sufficient t o verify conditions 4.6.9.The regularity o f the functor Ka is established as follows. Using the same arguments as those applied in Lemma 4.6.15, we prove that A(Z,) is dense in 0 from the reflexivity o f @. Since @ is a quasi-power parameter, Ka 2 Ja (Theorem 3.5.3 on equivalence). Thus, KQ is equivalent t o the J-functor with a regular parameter, and according t o Theorem 3.6.1it is therefore regular. Further, from the identity
(a)-(c) o f Proposition
( S f , g ) = (f,SS)
9
where the bilinear form is defined by
(f,d
:=
dt
J
f(t)g(llt)
t>
R+ it follows that the operator S is bounded in the associated lattice @+ as well. Since @+ is also reflexive (Lemma the duality theorem
3.7.2show that the functor
21This means that the operator S is ia
( S f ) ( t ) :=
4.6.11),the above reasoning and
J
mi+
t)
L(@). &call
ds f(s) - .
also that
Banach geometry of real-method spaces
647
K:, E (J*)' 2 K*+ is also regular [condition (b) of Proposition 4.6.91.
Here Ka E Ja and
Ka+ G J*+ (Theorem 3.5.3), so that applying twice the duality theorem 3.7.2, we obtain
Using now Lemma 4.6.13, we get
i.e. the condition (c) of the above-mentioned proposition is satisfied. To
-
complete t h e proof, it remains t o verify that the functor Ka is relatively complete. Since
Ka 2 Ja
Co we have for some y > 0
Since t h e K-functional is convex, it follows that Co = KcO(t,).Thus, from t h e previous identity and Theorem 3.3.15 we obtain
-
Ka = Ka",,o(t,)
Since
d
=: K6
.
Co(iM), we can apply Theorem 3.7.6 according t o which
( K a y E K& . Thus, the relative completeness of
K* is
a consequence of the relative
-#
completeness of
6 in the couple L,.
Now we make use of t h e fact t h a t (see
the proof of Lemma 4.6.13) that the relexivity of the lattice @ implies the monotone completeness and order continuity. It follows from these two properties that @ possesses the Fatou property" (see, for example, Kantorovich and Akilov [l],Chap. 3). Since C ( i , ) Fatou property as well. Let now unit ball
B(4)and
(5,)
also has this property,
d exhibits the
-.
be a sequence contained in the closed
converging t o I in C ( L , ) . Since convergence in norm
in a Banach lattice leads t o the convergence in measure [see (3.3.6)],
(I,)
converges in measure t o z. Then the Fatou property ensures that z belongs 22This means that the unit ball of 0 is closed relative to convergence in measure.
Selected questions in the theory of the red interpolation method
648
to
B(6).Thus,
we have established that
B ( 6 ) is closed in E ( J ? ~ )i.e. , 6
is relatively complete. 0
Corollary 4.6.18.
Ke(r?) with a quasi-power reflexive parameter 9 be reflexive, it is necessary and sufficient that 113 E L w c ( - f ) .
In order that a space
Proof. The
necessity follows from Proposition 4.6.5, while the sufficiency
follows from Theorem 4.6.17 and Proposition 4.6.7. 0
In particular, we can take
9 := Lp”, 1 < p < 00 and obtain Theorem
4.6.1. More generally, we can put
:=
L;, 1< p < 00,
where w E Conv
is a quasi-power function (i.e. Sw w w ) .
The well-known Davis-Figiel-Johnson-Pekzinski factorization theorem is an important corollary o f Theorem 4.6.1. Namely, the following theorem is valid. Theorem 4.6.19.
T
L w c ( X , Y )there exist a reflexive space R(T) and linear operators U E L ( X , R ( T ) )and V E L ( R ( T ) , Y )such that If
E
T=VU.
Proof. Let us consider the couple 2
IIYIII~T Then ImT
-P
where 7 := follows that
:= inf {IlTllx,~ 11
:= ( I m T , Y ) . Recall that
~ 1 ;1 Y~ = T s ) .
Y and the closed unit ball B(1mT)coincideswith y T ( B ( r ) ) , llTll-’. From this and from the weak compactness of T it
12 E LwC(z). Then the space
R(T )
:=
(ImT,Y)1/2.2
is reflexive, and putting
V :=
22,
U
-
:= i l T ,
Banach geometry of red-method spaces where il : I m T
~t
operator induced by
649
R ( T ) ,22 : R ( T ) ~t Y , and
T,we obtain the required
f
:
X
-P
I m T is an
result.
0 Remark 4.6.2Q.
It should be noted t h a t in the original proof o f Theorem 4.6.1, a construction in the proof of Theorem 4.6.19, proposed by the four authors, was used in an essential way. The above proof is based on another approach to this theorem. Finally, let us consider one more interesting result concerning the scope of problems under consideration. For its formulation, we consider a function ‘p
E Convo satisfying the additional conditions
(4.6.12)
lim ~ ( -~lim1 ‘p(t>= oo t-0
t
t-oo
.
Let us further consider the “abstract” Marcinkiewicz space
M&?)
:=
K L L ( f )*
The name of this space is due t o the fact that for the couple x’ := (L1,L,) we obtain the Marcinkiewicz functional space MV defined by the norm23
Another example is provided by the couple (Lp,Wpk). Here we obtain the Lipschitz space A:*Q defined by the norm
In particular, for cp(t) := t 8 , 9 E (0,l) we obtain the space Bp”” where
X := k9. t
23Recall that f’*(t)
:=
f’(s)ds where f’ is the decreasing rearrangement off 0
(see Proposition 3.1.18).
650
Selected questions in the theory of the real interpolation method
4.6.21 (Aizenstein). E Convo satisfy the conditions (4.6.12) and
Theorem
Let 'p
2 be a weakly reflexive
couple. Then there exists an isometry
obtained with t h e help of the map
K-~Q**.
M,(I?) into the second conjugate while
Here
Q :
K
is the canonical map o f
M z ( 2 ) ~t M , ( z ) .
0
For the spaces defined by norms
(4.6.13) and (4.6.14), Theorem 4.6.21
is obviously applicable. Thus we obtain, for example,
(m,)**N M , Similarly, for p :=
.
1,oo we have
Here m, denotes the closure i n
M , of the set o f
bounded functions and
Xi@
is the closure of the set of infinitely differentiable functions i n A;".
B. Another elegant result which will be considered here is devoted t o the possibility o f complemented embedding of the spaces
Zp
i n t o the space
Xap. Similar problems are very popular i n the geometrical theory o f Banach spaces.24 In order t o formulate the results, we assume that t h e parameters
29 and p satisfy the conditions 0<6<1,
l
Then the following theorem is valid.
24By way of an example, we recall Banach's well-known conjecture concerning the presence of a subspace isomorphic to I,, 1 5 p < 00, or cg in each infinite-dimensional Banach space. For a counterpart of this conjecture put forth by Tsirelson, see Lindenstrauss and Tzafriri [l].
Banach geometry of red-method spaces Theorem 4.6.22
651
(M.Levy).
X is a subspace of the space d$,,which is not closed i n C(d),then for any E > 0 there exists a subspace Y, o f the space X which is not closed in C ( 2 ) and is (l+E)-isomorphic t o I , and (1+E)-complemented + in X8,.
(a) If
(b) The same also holds for X couple
x'
+
X6,under the only
:=
is nondegenerate [i.e.
condition that the
A ( d ) is not closed in C(x')].
+ €)-isomorphism signifies that the inequalities
Here (1
hold for x E
Y,.
Here
phism. Similarly, (1
P
: KL;(x') --t
T
:
Y, + 1,
is a map carrying out the isomor-
+ €)-complemented
means that there is a projection
Y , with a norm that does not exceed 1+ E .
A more general statement will be proved below. Here we shall consider two impressive examples of application of the Levy theorem. Theorem 4.6.23 (Kadec-Pelczinski).
If X is an infinite-dimensioanl subspace o f the space L,(O, l), p E (2, +m), which is not isomorphic t o the Hilbert space, then it contains a subspace isomorphic t o 1, and complementary in
Proof. Let
us consider the couple
on ( 0 , l ) . Then L,
C(L,,LZ)
c Lz and Lz
(L-,
L,(O, 1) L2)
consisting of functions defined
hence
.
On the other hand
L,
2
(L,,LZ)$,
for 9 :=
-2 .
P
Here X is not closed in the sum. i.e. in Lz, since otherwise it would be isomorphic t o the Hilbert space. Consequently, statement (a) of Theorem
652
Selected questions in the theory of the red interpolation method
4.6.22 is applicable. 0
The following result providing a positive answer t o the question posed by Lions can also be derived from Levy's theorem, although in a more complicated way. Theorem 4.6.24 (Jamon-Nilsson-Peetre).
If the couple
x'
4
is nondegenerate, the relation
= X,l,l 4
xsopo
implies that f l 0 = dl and po = pl.
Proof. Let
us first consider the case when
Thus, for a nondegenerate couple
x'
I90
=
291
and 1 5 p
m.
we have
x,, x,,. 4
4
Then in view of statement (b) of Theorem 4.6.22, there exists a subspace X 4
of the space X,,,
which is not closed in C ( x ' ) and is isomorphic t o .,Z
now apply statement (a) o f Theorem 4.6.22, regarding 28,.
Let us
X as a subspace of
Then there exists a subspace Y of the space X ,which is isomorphic to
I,. Thus we have found
a subspace 1, isomorphic t o
I,.
In view of Banach's
theorem on t h e minimal linear dimensionality of the spaces
I,,
15 r
< m,
it follows therefore that p = q. We arrive a t a contradiction. Let us now reduce the general case t o the one considered above. When
&, # 01,we obtain from the
x
:=
+
relation
x,,, E x,,,, 4
3.8.10on reiteration that the space x',,. with 19 := X B , for does not depend on r (2 X ) . Thus, X,, 15 p If, however, r9 := do = d1 but 1 L: po < pl 5 00, then again does not according t o statement (b) of the same Theorem 3.8.10-f,T
and Theorem
4
< 00 < q < 00.
and 1 5 r
depend on r for 1 5 r 0
< 00.
4
Banach geometry of red-method spaces
653
Let us now prove a generalized Levy theorem. We shall use here the main idea of Levy's line o f reasoning and some facts from Chap. 3. Theorem 4.6.25. Let the weight w
E Convo be given and
let 1 5 p
< 00.
Then t h e following
statements are valid.
If X is an infinite-dimensional (closed!) subspace of the space KL;(x') E > 0 there exists a subspace Y, of X which is (1 6)-isomorphic t o I, and (1+E)-complemented in which is not closed in C ( x ' ) , then for any
+
K&. The same holds for X couple
x'
:= K p ( x ' ) under the assumption that the
is nondegenerate.
Proof. Let us begin with a more difficult case (b).
To this end we must first
find a sequence (en)nENfor which
and second prove the existence o f a projection
+
norm does not exceed (1
E).
P
: K L ; ( ~ '+ )
2 whose
Here
In order t o prove these statements, we require Prooosition 4.6.26. Let the couple
2 be nondegenerate and assume that the fundamental func-
tion cp of the functor (4.6.16)
K* [see (3.8.2)] satisfies the conditions
cp(t>lim cp(t)= lim -0 t +o t-oo t
Then the restrictions o f the norms
11
-
Ilc(m,
and
11 . I I K e ( ~ )
to
A(x') are
not equivalent.
Proof.Let us suppose that the statement of the proposition is incorrect, i.e. the norms of the spaces
K a ( 2 ) and E ( d ) are equivalent on A(x'). Since
in view of the embedding (3.8.9)
654
Selected questions in the theory of the red interpolation method
-
&(2)
K&),
and in view of condition (4.6.16) and Corollary 3.1.14
K L L ( 2 )Lt C0(2), we have (4.6.17)
Co((;r?)% Kg((;r?)= K L s ( z 0 ). +
So we may suppose that X is a regular couple. Let us define the function w by the formula (4.6.18)
w ( t ) :=
sup K(t,a:;x') Il~llc(n,
.
Then it follows from (4.6.17) that (4.6.19)
SUP t>O
-< 00 , dt)
w(t)
so that w also satisfies the conditions (4.6.16). Let us verify that this leads to
the closure of A ( 2 ) in C(@,
whence we arrive a t a contradiction with
the assumption of the nondegeneracy of
2.In view of (4.6.17)
and (4.6.19)
we have (4.6.20)
C(x')
K ~ w , ( 2,)
so that
K L (AC(Z), ~ C ( 2) )
2 KL%(KLzn(i,t), KL
% ) ( .~ )
Here we took into account the fact that A. = KLzn(l,t) [see Corollary 3.1.111. Since, further C = KLm.x(~,f), an application of Corollary 3.3.13 to the previm ous equality gives
K p ( 2 ) % KLh,(2), where we have put tw(t-1)
,
t
51 ,
w(t)
,
t >1 ;
g ( t ) :=
Banach geometry of red-method spaces
Let us take
2
655
E X1. Then, on t h e one hand,
K ( t , z ; 2)I t 1141x1
7
and, on t h e other hand, in view o f the embedding X1 (4.6.20)] we have for some constant y
K(t,a: ; 2)I Y 4 t )
ll4lXl
L)
K p ( 3 ) [see
>0
.
Thus,
K p ( I ? ) ,it follows from the previous inequality that
Since KL&(-J?)
llh'(.,a:
SUP
+
; X)IlL&,I Y;u
.
Il4lXlS1 Taking into account the definition of h, we can derive t h e inequality (4.6.21)
sup K ( t , z ; ll4lXl 51
t 2 1.
2)I y T w (w (t)) ,
This inequality gives an estimate of the upper bound (4.6.18), i.e. w(t). For this purpose, we take
z from the open unit ball of the space C ( 2 ) . Then
there exists a representation
5
= 20
+ z1such that
+ 11~1IlX1< 1 .
11~0llXci
This leads immediately t o
K ( t ; a:; 2)< l + K ( t ; 21; 2) ( t E R,). Therefore, for
t 2 1, we have from (4.6.18) and (4.6.21)
w(t)
5 1+
sup K ( t ; 2 ; 2)5 1 ll~llX1I1
+ y;uw(w(t)) .
656
Selected questions in the theory of the red interpolation method
Consequently, for
t 2 1 t h e inequality
is satisfied. If w ( t ) tended t o infinity as
t + 00, then from the conditions
(4.6.16) for w we would obtain that the left-hand side of this inequality is zero. The contradiction obtained shows that (4.6.22)
lirn w ( t ) < 00
t-+m
Let us use the same line of reasoning for the transposed couple
TT
:=
(Xl, Xo). Since the quantity WT for this couple, which is defined by (4.6.18), is related t o w via the equality WT(t)
=tw(l/t) ,
t
E E2+
,
we obtain in analogy with (4.6.22)
l i m w T ( t ) < 00 .
t++w
In view of the previous relation, this gives
From this inequality and from (4.6.22) it follows that
w(t) 5 y min(1,t) ,
t E B+ .
Thus, t h e embedding
is satisfied.
Since the inverse embedding is obvious, taking into account
(4.6.20) we find that
Ac(Yo) = A"(X')2 C(X'')
.
Banach geometry of red-method spaces
657
In view of the statement of Proposition 2.2.17, it follows hence that
A(x')
Co(r?). Thus, A ( f ) is closed in C ( f ) , which contradicts the assumption. 0
Remark 4.6.27.
It can easily be shown that Proposition 4.6.26 has the following converse. + If the norms 11 . Jlc(2)and 11 IIK,cn,are not equivalent on A(X) for
-
any nondegenerate couple
x',then the conditions (4.6.16)
are satisfied for
the fundamental function cp of the functor Ka. Passing t o the proof of the theorem, l e t us first establish that the fun-
Ka satisfies the conditions (4.6.16) for E Convo). Indeed, according t o formula (3.2.8)
damental function cp o f the functor :=
LF (1 5 p < 00,
w
we have
{&)'t
=
1
w(s)p
-+J ds s
O0
1
-}
ds W(S)' s
1l P
.
After the multiplication by t P , for t large, the first integral on the right-hand side will not be less than
Hence it follows that
Similarly, for t small the second integral is not less than 1
1
ds
+oo
whence it follows that
lim cp(t) = 0 . t-0
fort+O
Selected questions in the theory of the r e d interpolation method
658
It now follows from Proposition 4.6.26 that the norms of C ( 2 ) and KL; are not equivalent on
A ( 2 ) . The
same holds if we replace the functor under
consideration by i t s discretization (3.3.30). Here we choose q (3.3.30) so that for a given S
> 0 we
> 1 in formula
have
Here we have put
where L;)(qn) := w(q")lnq. In view of (4.6.23), we can henceforth use in the analysis the norm (4.6.24). Let us now construct the required sequence of elements (en)",== C
A(x') possessing property (4.6.15). To this end we put No := 0 and choose el arbitrarily with the single restriction
Using now this equality, we choose a natural number given E
> 0 we
Nl
so large t h a t for a
have
c
( K ( q " ;e l ; @)P
4
'(4")
2
InI<[No.Nl)
.
Let us suppose that we have already found natural numbers No
...,ek-l
... < N k - l
and el,
(4.6.25)
lle;llw = 1
< N1 <
in A ( 2 ) such that
and that, moreover,
Let us now find an element ek E A ( 2 ) and a number Nk satisfy the conditions (4.6.25) and (4.6.26) for i := the norm
>
Nk-1
which
k. For this we consider
Banach geometry of real-method spaces
659
Since the K-functional is a norm for a fixed numerical argument which is equivalent t o the norm
11 IIz.z), the quantity )I . Ilk-1
this norm. Then in the set
is also equivalent t o
A(X) there exists an element for which the norm
(4.6.24) is equal t o unity, while the norm (4.6.27) is less than ( ~ / 2 ~ + ' ) ' l P . Let us choose this element for e k . Then condition (4.6.25) is satisfied for
>
it. Choosing the number Nk
Nk-1
sufficiently large, we can make the
inequality
<
be satisfied. Then from this inequality and t h e inequality
Ilekllk-1
it follows that condition (4.6.26) is also satisfied for
for i :=
Thus, the sequence ( e k ) c
ek
A ( 2 ) is constructed. Let
& ( +)"'
k.
us verify that this
sequence satisfies statement (4.6.15) o f the theorem. For this purpose, we by putting introduce a norm in the space of bilateral sequences (zn)nEz
Obviously, if the sequence
(2,)
is stationary and 2, := z, n E
Z,then
l l ~ l l w= III(zn>lll . Let now z be representable in the form
C
z =
akek
-.
(convergence in K ~ ( x. ) )
k 6 N
Let us first define for a given Ni-1
and put
Further, we set
I IkI < Ni
k E Z a number i
:= i ( k ) via the condition
660
Selected questions in the theory of the real interpolation method
In view of these definitions, for any
k E 22 we
have
Hence we obtain the inequality
condition (4.6.26) leads to
III(xk>III I
c
1IP
biIP
.
(iiN
Similarly, from conditions (4.6.26) and (4.6.27) we have
Banach geometry of red-method spaces
661
using the three estimates obtained i n inequalities (4.6.28) and (4.6.29), we arrive a t the required inequality (4.6.15).
It remains for us t o construct the projection on the subspace X
{c
akek
;
lakl'
:=
< co} o f the space K p ( 2 ) and verify that its norm does For this we use the fact that J(q-" ; . ; 2')is a norm
+ O(E).
not exceed 1
C0(T)*, which is dual t o the norm of K(q"; . ; 2)on C o ( 2 ) (see Proposition 3.1.21). Therefore, for each In1 E [ N ; - , , N ; )there exists an element fn E A(2'), such that, first, on A(?)
(4.6.30)
N
(fn, e;)
= J(q-" ; fn ; -?')K(q" ; e; ; 2)
and, second,
From these two relations and from (4.6.25) we obtain that for p' := we have
P -
P-1
Similarly, but using (4.6.26). we obtain (4.6.32)
1-
c
lnldh'.-i
E
(fn,ei)
<5.
,N)
Let u s now define the operator T by the formula
T X :=
C ie N
(
C
fn,z)ei.
lnlE[N-i J%)
Further, let z E K p ( z ) ,T h e n j a view of statement (4.6.15) of the theorem and the estimates obtained above, we obtain
662
Selected questions in the theory of the real interpolation method
Thus, the operator T is bounded in exceed 1
+ E.
K L g ( 2 and ) has a norm that does not
It follows from the definition o f this operator and statement
(4.6.15) that ImT coincides with
Let us now verify that the operator 21 ' 2 has an inverse. For this we must establish that for
If
E
2
E
is chosen so small that 7a'lP
< I, it follows
from this inequality t h a t
(Tlx)-' (= En,== (12 - Tla)")exists. In order t o establish inequality (4.6.33) we write z = CiEmaiei. Then in view of
the inverse operator
(4.6.15) and (4.6.32) we have
llTz - 2 1 1 w
I
c
IIidV
[ c (
fn,ei)
l~lW.-iJ%)
1 IW
- 1 aiei
+
Banach geometry of red-method spaces
663
According t o the left inequality in (4.6.15). the first term does not exceed IIzllw. The second term does not exceed
which in turn, in view of (4.6.26) and (4.6.15). does not exceed
Thus, inequality (4.6.33) is proved. Since (W-l=
c
(112 -
W" 7
ndV
it also follows from (4.6.33) that II(Tlz)-'II2 = 1 operator
+ O(&').
Then the
P := (TJz)-'Tis obviously a projection on X,and its norm does + O(&".
not exceed 1
(a) Let us indicate the simple modifications which have t o be made of t h e analysis of this case. Since by hypothesis
is closed in
K L ? ( ~ ' )the ,
X
is not closed in C ( x ' ) but
norms of these two spaces are not equivalent on
X . Therefore, we can use the arguments following the proof of Proposition 4.6.26, replacing A(x') by X . The sequence (en) constructed i n this way will then define a subspace o f the space X , which is (1 + &)-isomorphic t o 1, and (1+&)-complementary i n K ~ p ( 3 ) . 0
Let us consider a useful supplement t o Theorem 4.6.26, concerning the case p :=
00.
Theorem 4.6.28. (a) Statement (b) o f Theorem 4.6.25 holds for p :=
the function w (4.6.34)
E Convo is nondegenerare, i.e.
lim w ( t ) = ! i w(t> --* ,
t-00
and with substitution ~0 for I,.
t
00
if we assume that
664
Selected questions in the theory of the real interpolation method
(b) Statement (a) o f Theorem 4.6.25 is valid for p
:=
00
if instead of
(4.6.34) we assume that for all elements x E X , the following conditions are satisfied:
lim
(4.6.35)
K ( t ; x ; 2)
= lim
and with substitution
K ( t ; x ; 2)
=o,
t-ca
t-0
Q
for 1,
(a) Let us indicate t h e main modifications that should be made in the proof
of statement (b) of Theorem 4.6.25 for obtaining the required result. First, we modify the definition of
11 . [Iw
[see (4.6.24)] by putting
Henceforth, w will appear everywhere instead of G. Second, instead of inequality (4.6.26), we have t o use the inequality
The possibility of the choice of Nl that
el
> No
:= 0 follows from the fact
E A ( 2 ) (so that K ( t ; e l ; 2)NN rnin(1,t)) and from the non-
degeneracy o f w [see (4.6.34)]. Hence
and therefore it is possible t o choose the required N l . If all el ,...,e k - 1 and numbers Nl element
ek
< ... <
Nk-1
E A ( 2 ) is chosen by the conditions
are constructed, the
Banach geometry of real-method spaces
665
Here we put [cf. (4.6.27)]
The number
is chosen by the condition
N k
The possibility o f such a choice is associated with the nondegeneracy of w and is established in analogy with the corresponding statement for N l .
Further, we put
+
Then for x =
akek
(convergence in K L & ( ~we ) )have, in the notation
used in the proof o f Theorem 4.6.25(a) and by the same arguments, 111(xk)lll
= Il('k>llloo
7
&
3.
5
~ ~ ~ ( z k l ) & Zlull ~ ~ ~
A further proof o f inequality (4.6.15) for p
:=
00
is obtained as in
Theorem 4.6.25.
A certain difference appears in the proof o f t h e fact that the subspaces
are complemented in the space
T
:
K p ( 2 ) . For p <
00
the operator
K L ; ( ~-+) X has the form
while in the case under consideration every index single element
fi
instead of the family (fn)lnlE[~,-l,~,).
chosen as follows. Since
i
is associated with a
This element is
666
Selected questions in the theozy of the real interpolation method
for E
> 1 there exists an index n; E [Ni-,,N;)
such that
NOWfi E A(2') is chosen by the condition
Further, the operator T is defined by the formula
TX :=
C (fi,x)
e,
All further estimates related t o the operator T are established as in Theorem 4.6.25.
(b) In this case, the line of reasoning is similar t o the previous one, but instead of the nondegeneracy of w the conditions (4.6.35) should be used. U
Comments and supplements
667
4.7. Comments and Supplements
A. References Sec. 4.1. The first result on nonlinear interpolation is contained in the
paper Orlicz [2], in which the invariance of Orlicz spaces under the action of operators in t h e space Lip(L1,L,)
is proved.26 The generalization of
this result t o a wide class of symmetric spaces can be found in the work of Lorentz and Shimogaki [ l ] . The case of the Lipschitz operators acting from the couple
( L l , L , ) into the couple (Ll,, L,) was considered later by
Shimogaki [2]; for a generalization, see Maligranda [l]. In this connection, let us call attention t o Corollary 2.5.25 according t o which any interpolation
space which is regular on the couple
r? is a x'.
Lipschitz-invariant space if the
approximation condition is satisfied for
The first version of Definition 4.1.1 (for the case ? = x') was proposed by Gagliardo [5]. The difference between his definition and the one given in this book lies in that E := 0 in inequality (1.1.2). If we denote the space obtained by
Bo(d,?), it can be shown that B(r?, ?) = B0(d, ?) iff ? is
relatively complete. The general Definition 4.1.1 was proposed by Peetre [9] whose paper contains Proposition 4.1.3 and Examples 4.1.5(a) and (b).The concept of quasi-additive operator in Example 4.1.5(c) is due t o Sagher [l] who established for such operators an abstract version of the interpolation theorem in the paper by Bennet, Devore and Sharpley [l]. The problem of the description of &invariant was formulated by Gagliardo
+
subspaces of the couple X
[5].In this book we present a
(nonconstructive)
solution t o this problem for relatively complete couples. The constructive solution of a more general problem (Theorem 4.1.11) is contained in Brudnyi' and Krugljak [3]. The same paper presents Theorem 4.1.8 and Corollaries 4.1.4 and 4.1.15. The latter corollary provides an affirmative answer t o a question posed by S.G. Kre'in in the survey Dmitriev, Kre'in and Ovchinnikov [ l ] . I t follows, by t h e way, from Theorem 4.1.8 that there exist couples x' for which the spaces
C&(d) of the complex
26Regarding the space Lip(?,
method are not B-invariant. The
f),see Definition 2.5.20.
668
Selected questions in the theory of the red interpolation method
first concrete example o f such a couple and t h e corresponding operator was constructed by Cwikel
[2]. He interpreted his result as a negative solution of
the Lions problem (see Lions
[6])on the
Lipschitz-invariance of the spaces
of the complex method. It should be noted, however, t h a t the operator constructed by Cwikel is not a Lipschitz operator. The Lipschitz-invariance
of the complex-method spaces was established in AizensteTn and BrudnyT [2]
2.5.23(a) and the computability of the functor Cs). Sec. 4.2. All results of items A and B of this section were obtained by the authors and were partially published in the papers BrudnyTand Krugljak [2,3]. (a corollary o f Theorem
The method of traces was introduced for the power case by Lions. In a series of papers (see Lions [l-3,5])he analyzed its properties and outlined a number of interesting applications. Later, in the fundamental paper Lions and Peetre
[2]the methods o f constants and means were introduced for the power case, and the relation between these methods was established. The fact that all
the three methods are reduced t o the method ( - ) a p was established later by
[7].The L-method for t h e power case was introduced and analyzed by Peetre [lo]. Theorem 4.2.15 is taken over from his work. The functors E& were introduced and studied in Peetre and Sparr [l], from which we borrowed the important Theorem 4.2.23. A generalization of the method o f constants t o arbitrary functional parameters was proposed by Peetre [7],while its relation t o the K-method (Theorem 4.2.11)was established in the work o f V.I. Dmitriev [4],where Peetre
another approach was used. A generalized method of averages was also proposed by Peetre [7], while its relation t o the 3-method (Theorem
4.2.23)
was established by the authors. A somewhat weaker result is contained in Brudnyi' and Krugljak (J*O,*,
1'
[3].The equality
K*t,*f
7
which amplifies the corresponding result obtained by V.I. Dmitriev
[l]), can
[2,3](see
4.2.23and the duality 3.7. The general L-method was introduced in BrudnyT and Krugljak [3],containing Theorem 4.2.14on the relation between
also Bukhvalov
be obtained from Theorem
theorems presented in Sec.
the
L-and the K-method. The &-method was introduced by Yu.A.
who is the author or Theorem
Brudny'i,
4.2.21. Theorem 4.2.25 generalizing this
Comments and supplements theorem was obtained in Asekritova (31 (see also Nilsson [l]).
669 Theorem
4.2.29is also taken from the same paper by Asekritova. The f3-method of interpolation (named after S.N. Bernshteyn) is considered in a less general form in Brudny’i and Krugljak [l], where a relation with the E-method is established. A generalization of this result (Theorem 4.2.36) is contained in Asekritova’s paper [4].A generalized method o f traces was considered in V.I. Dmitriev [l]and Kalyabin [l]. Theorem 4.2.35was obtained in Brudnyy and Krugljak (31. Sec.
4.3. This section begins with the theorem on the “commutativity”
of the real and complex methods [see (4.3.1)]formulated in the work by Lions
[4](see also Grisvard [l]). Theorem 4.3.1was proved by Yu.A. BrudnyY in 1978. Its proof and generalization (Theorem 4.3.2)are given in BrudnyY and Krugljak [3].The same result is anounced in Dmitriev and Ovchinnikov [l]. The operation X;-’X: was introduced by Calder6n [2].Example 4.3.8 is also borrowed from this paper. The question o f the interpolation property of the Calder6n construction was posed by Yu.A. Brudny’i a t t h e end of the sixties. A negative answer t o this question (Theorem 4.3.9)was obtained in + Lozanovskii [3].The more general construction of cp(X) was introduced and analyzed in the series of papers by G.Ya. Lozanovskii (see, in particular, the articles by Lozanovskii [1,2]).The important Theorem 4.3.10was proved by Ovchinnikov [l]. The original proof obtained by this author is based on the relation between the CaIder6n-Lozanovskiioperation and the functors cp, and ( P I introduced by him (see Example 4.3.16and statement (a) o f Theorem
4.3.17). A similar result for the Orlicz spaces was independently obtained by Gustavson and Peetre [l]. This work employs the idea formulated in earlier work by Peetre [13](see Example 4.3.14)and the relation between the Calder6n-Lozanovskii operation with the functors from Example 4.3.15, constructed by the authors mentioned above. A generalization o f the result obtained by Ovchinnikov (Theorem 4.3.11)and a considerable simplification of the proof was proposed in the work by Shestakov [2]. The elementary proof o f this theorem presented here is taken from Krugljak (31.The relation between the approaches of Ovchinnikov and Gustavson-Peetre (statements
(b) and (c) of Theorem 4.3.17)was clarified by Janson [2].On approaches
670
Selected questions in the theory of the red interpolation method
t o Theorem 4.3.11and further generalizations, see the papers Gustavson (21, Ovchinnikov
[3]and
Peetre
[21].
A relation between the Calder6n construction and the complex method of interpolation is established in the papers Calder6n
[2](Theorem 4.3.4)and
Shestakov
[I][statement (4.3.14)].Corollary 4.3.6is amplified
Karadzev
[l] obtained
in a different way. Statement
by a result in
(4.3.16)is apparently
new.
4.3.22was obtained by Yu.A. BrudnyY and was published in the paper Brudny’iand Shalashov [3](for t h e case of complex interpolation funcTheorem
tors). The same paper describes an application o f the complex interpolation
[2].Other results of this kind [l]. The proofs of Theorems 4.3.26and 4.3.27on “splitting” are given in the papers Lions and Peetre [2], Cwikel [5],Calder6n [2]and Bukhvalov [4].
of the Lipschitz spaces introduced in Calder6n are contained in Bukhvalov
[2,3]and
Janson
See. 4.4. Calder6n’s theorem (Theorem 2.6.9)27and Mityagin’s theorem (it is formulated in 3.7.2) proved the description o f a set Int(L1,L,) in quite different terms. In particular, Mityagin
[l] obtained, as a corollary of
his result, t h e statement t h a t separable symmetric spaces belong t o t h e set
Int(L1, L,)
(nonseparable symmetric spaces may not possess t h e interpola-
[l]). This result does not follow directly from the Calder6n theorem (see in this connection the paper Mekler [l]). In connection with the Mityagin theorem a previous paper by Orlicz [l]. published as early as i n 1934 (!), is worth noting. This paper proves that the Orlicz spaces LN(0,l) are invariant under t h e action of linear integral operators bounded on the couple (Ll,L,). A generalization of this result t o symmetric spaces
tion property; see Russu
is contained in Lorentz
[l]. The
development of this subject described in
4.4 is essentially connected with t h e work of Calder6n (see, however, Zobin [l] written in line with the approach started in the work of Mityagin; Zobin’s result is described in 3.7.1). In particular, under the influence of
Sec.
Calder6n’s work, the concepts of the C-couple, C-property, K-monotonicity and K-adequacy (see Peetre [11,12]) were introduced. Theorem
4.4.5on the
27Accordingt o Sparr [2], Theorem 2.6.9 was obtained independently also by M. Cotlar (unpublished).
Comments and supplements
671
equivalence of these properties was proved by Brudny’i and Krugljak Corollary
4.4.6for
? = x’
[2,3].
was established by using another approach by
4.4.9and 4.4.19 were obtained by Krugljak. Theorem 4.4.12 was proved independently by V.I. Dmitriev [5]and Cwikel and Peetre [l] (“modulo” theorems on K-divisibility). This theorem was preceded by an important Corollary 4.4.14 (see Sedaev-Semenov [l]). Theorem 4.4.16 was proved in the paper Peetre [12],where the more general case of the couple ,fm(G) was considered. The important Theorem 4.4.18was obtained by Cwikel in 1976, but its proof was published much later (see Cwikel (31). A similar proof of a more general result (Theorem 4.4.20)was published earlier in the paper by Dmitriev and Ovchinnikov [l]. The proof of Theorem 4.4.20presented in this book was proposed by Yu.A. Brudny’i (see Brudny’i and Krugljak (31). A generalization of this result (Theorem 4.4.21)is contained in Nilsson (21.The duality theorem 4.4.23was obtained Cwikel [4].Theorems
by the authors in an attempt to grasp the following statement presented in Nilsson
[2].
x’ is relatively complete, while the couples ? and ?‘ are regular, t h e condition x’ 5 ? leads t o the condition ? 5 2’. The If the couple
K:
K:
proof is based on the concept o f the coorbit of the element
x’ E C(x’l)
introduced by this author (see item 3 o f the cited work). In the definition of this concept, the operation o f (abstract) completion is used, and therefore, in contrast t o the statement o f Nilsson, the constructed object is not an interpolation functor.
Thus, the validity o f the above statement remains
disputable. The fundamental theorem
4.4.29 and
i t s generalization given in Remark
4.4.33 are due t o Cwikel. It should be noted, however, that in his paper Cwikel [4],the author just postulates the property of a couple, which is established in Lemma 4.4.30 (such objects are referred t o in this work as the Holmstedt couples).
The statement o f the lemma mentioned above [2].The proof of The-
was anounced in later work by Cwikel and Nilsson
4.4.36 (Shimogaki 111 and Boyd [l]) can also be found in the book by Lindenstrauss and Tzafriri [2]). Theorem 4.4.40 (with a somewhat more
orem
Selected questions in the theory of the real interpolation method
672
accurate estimate of constants) was initially obtained in Sparr [l] with the help of a deeper generalization of the Calder6n method. This result was preceded by the theorems on the K-monotonicity of the couple couple
( Lp,L,), 1 < p <
00
(Lp(w~),LP(w1)). 1 5p <
( L 1 ,L p ) ,the
(Lorentz and Shimogaki [2]) and the couple
00
(Sedaev
[l]). Another proof of this theo-
rem was proposed independently in a paper by Cwikel [l].We should also mention the paper Arazy and Cwikel [l], where it is shown that Sparr's theorem can be obtained from the above-mentioned result obtain by Lorentz and Shimogaki. The paper Cwikel and Nilsson [l]contains the following generalization of the result obtained in this paper: +
I n t K ( x o , ~ d l pnl > IntK(-Gop0,x1) = IntK(-&om,-f~lpl). Here 0
< 80 561 < 1, 1 5 po,pl < 00
and I n t K ( z )denotes the set of all
spaces of the form Kcp(2). Theorem
4.4.38was proved for q; <
00
in V.I. Dmitriev
[5], where a
principally new approach was proposed. The fact that this result cannot be improved (Theorem
4.4.39)was established in Ovchinnikov's book [7]. The
methods used there have some other interesting applications. In particular, we can mention the following new interpolation theorem. For i t s formulation, + + .-, ~ ( X a o p o , X a l pand l ) assume that Z , p E { ( z o , ~ )0; < z, < we put X Z := -#
l} and p',q'E { ( z o , z ~ )1 ; 5 xi
5 00).
Then the following theorem is valid
T E L(2z,s,?,,,f)be given. Then T + maps the space Xa(~),r continuously into the space Yo(+, 0 < 9 < 1. Here, as usual, K ( 8 ) := (1- 9)ao + 29/31, but the parameters r and s satisfy the (see Ovchinnikov [7]). Let the operator
-#
following relation:
_1 - _1- - ( 1 s
r
-8)(; 1
- - 1) + + I 9 Po
Obviously, the result is not new for pi
1 (-1 - -)+ . 41
5
Pl
qi (z = 0 , l ) since it follows from
the interpolation theorem of Lions and Peetre.
See. 4.5. The classical inverse theorem 4.5.1was proved by Bernshte'in in
[l] for the case o f polynomial approximation i n the space C[O,l].The
method proposed there can easily be extended t o the general situation (see Timan [l]). The same book contains the theorems of Chebyshev-Haar and
Comments and supplements
673
Miintz used in Example 4.5.3.James's theorem applied in Example
4.5.2is
contained, for example, in the book Diestel [l]. Both examples were proposed
4.5.12. The inverse problem, which emerged in the analysis o f this theorem (Theorem 4.5.7),was solved completely for the K-functional by N.Ya. Krugljak. Theorem 4.5.14 by Yu.A. BrudnyY who is also the author o f Theorem
(or a result similar t o it) is apparently contained in a manuscript written by
B. Jawerth. This unpublished work is mentioned in t h e report by Peetre (231. Theorem 4.5.16on four spaces is borrowed from the paper Wolff [l], while a generalization o f this result is contained in Janson, Nilsson and Peetre
[l].
Sec. 4.6. The subject concerning Banach properties of the real-method
spaces was extensively developed by B. Beauzamy and his school (see, for example, the book by Beauzamy
[l] and the dissertation by Levy [1,2], from which Theorems 4.6.1and 4.6.22were taken). Theorem 4.6.8,providing the complete proof of the problem o f interpolation of weakly compact operators by the functors of the real method, was obtained by M.K-H. AizensteYn and Yu.A. BrudnyY. It was preceded by the works of Aizenste'in and BrudnyY [l] (Corollary
4.6.18).AizensteYn [l] (Theorem 4.6.21)and Aizenste'in [2](pre-
serving the property of weak compactness by the functors z8pfor 0 and 1
<9 <1
m). For the proof o f the properties of Banach lattices used
4.6.11-4.6.13)see, in addition t o the source indicated there, the book by Lindenstrauss and Tzafriri [l]. A remarkable factorization theorem 4.6.19was proved by Davis, Figiel, Johnson and Pelczynski [l] in the text (Lemmas
using a different approach. This approach has played a significant role in the works of Beauzamy and Levy mentioned above. In the dissertation of Levy cited earlier, a number of interesting applications of the result obtained in this way were considered. In particular, the proof of Theorem 4.6.23,initially obtained in the paper by Kadec and Peluynski [l] using other methods, was taken from this dissertation. Theorem
4.6.24 was proved in the paper of
Janson, Nilsson and Peetre [l]. A generalization of the results obtained by Levy (Theorems
4.6.25 and 4.6.28) was proposed by N.Ya. Krugljak. The
case of the quasi-power weight was considered earlier in diploma work of Tsyatson
111.
674
Selected questions in the theory of the r e d interpolation method
A number of papers are denoted t o the Banach properties of the spaces of the complex method. W e shall mention here only the papers Cwikel and Reisner
[l]and Pisier [l].
Comments and supplements
675
B. SuDDlements 4.7.1. Real Method for Finite Sets of Banach Spaces Some problems i n functional analysis necessitate the generalization o f the real interpolation method t o the category families of Banach spaces (n belongs t o
>
l)." In this
6nof admissible ( n + 1)case, x' := (XO, ..., X,)
6, if Xi are Banach spaces continuously embedded into a fixed
separable topological space. Thus,
g 1 coincides with
couples considered i n this book. the space o f operators
the category o f Banach
Many concepts (like sum, intersection,
L(d,f ) ,K- and
J-functionals and corresponding
interpolation methods) can obviously be transferred from the category of couples t o the general situation.
However, an attempt t o prove the main
facts o f the real method theory for the category
6, with
n
> 1 involves
considerable difficulties of combinatorial and geometric nature associated with the peculiarities o f the arrangement o f spaces X i in the sum
C(-?).
Moreover, the main results of the real method theory are, as will be shown later, incorrect for this category. In order t o describe the situation under consideration, we denote by := { ( t o , t l , ..., in); ti 2 0) consisting of those of ( t O , t l , ...,t,) for which t o = 1. Let us define on the set fi; the K-functional of the element z E C ( 2 ) via the formula
fii the subset t
:=
~ (; z t; 2)= inf
n l
C t; ~~z;llx, ; 2 = C z; . i=O
{ i=O n
Obviously, the function K ( . ; x ; X ) belongs to Conv, consisting of concave non-negative functions defined on E2;. We denote by majorant of the function f :
f
:= inf{g;
nt;
If1 I g ,
f^ the
least concave
+ R.Thus,
g E Conv,}
.
Corollary 3.2.4 describing the set Ex (Conv) of extreme rays of the cone Conv (= Convl) states that it is generated by the set of functions of the "There exist cases when n = 00 (see Zobin's theorem and the papers of Coifman, Cwikel, Rochberg, Sagher and Weiss [l-31).
676
Selected questions in the theory of the red interpolation method
i ~ where , A runs through convex subsets R+. Unfortunately, this will > 1. As far as we know, there is no description of the set Ex (Conv,) for n > 1, but the available examples indicate the complexity
form
not be so for n
of i t s structure. Probably, the basic property of the K-functional (viz. the K-divisibility) is therefore not observed in this situation. Namely, for the -+
couple
for
L1 := (L:", L;*',...,LTV1),where
i 5 i 5 n and ti [5]).
:=
1 for i = 0 one has the following theorem
(Asekritova
There exist functions g1,gZ
E Conv, and an element z E
E(d)such
that
K ( . ; z ; 2)5 91
+ 92 .
At the same time, for any constant y
> 0 and any decomposition z = zo+zl,
a t least one of the inequalities +
K ( . ; z i ; X ) I y g i , i=1,2 is not satisfied. The couples for which the statement of this theorem does not hold will be referred t o as K-divisible. Apparently, there are only a few such objects -+
in the category B , for n
> 1.
K-divisible (n+l)-family
x'is introduced in Asekritova [ 5 ] . Namely, we put
for
t
Then
In this connection, the concept of a weakly
E
2 possesses the property of weak K-divisibility if for a certain constant
y > 0, any z E C ( 2 ) and any gl,g2E Conv,, such that ; z ; x')
I 91 + 92
7
there exist elements z1and z2 for which
Comments and supplements
K ( . ; zi;
677
2)5 ysng; ,
In the work by Asekritova
i = 1,2
.
[5],the conditions which are necessary for I<-
+
divisibility and sufficient for weak K-divisibility of a relatively complete ( n
1)-family x' are formulated. Using these conditions it is possible to establish, for example, the following important fact.
A set of abstract Marcinkiewicz spaces
M;(d)
:=
( M w ( d ) ,...,Mwm(x')) (x' E &)
is K-divisible if one of the following conditions is satisfied: (a)
x'
is K-divisible;
(b)
x'
is weakly K-divisible, and t h e operator Sn is bounded in the space
L Z , i = O , l , ...,m . Here for w E Conv, we put
In particular, if
x'
is a couple, then
M ; ( d ) is K-divisible.
Let us consider
some more examples from the cited work. (a) If 29i
+
X is
+
a couple, then the (n
< 1 and 15 qi 5 00,
(b) If 0 is
+
+ 1)-family ( X , Y ..., ~ ~x,~,,,,,), ~ , where 0 <
i s weakly K-divisible.
a Banach lattice, then the ( n
+ 1)-family
...,O ( w n ) ) is
(@(wO),
weakly K-divisible. Here
(c) A set of Nikolskii spaces with a dominating mixed difference
(Hio,H,X',...,H,x") is K-divisible. Here Xi E R : \ { O } ,
15 p 5
00,
and
Selected questions in the theory of the r e d interpolation method
678
A k ( h e )we denote the k-th difference in the direction o f the vector e with the step h, while {el, ..., en} denotes the canonic basis in Rn. It is assumed th at 0 < A; < k; (i = 1, ..., n ) . As usual, by
For K-divisible couples, the reiteration theorem obviously holds:
where
x'
E
g,, 6 E gm,and we put
K&)
:=
( K o o ( z ) ..., , K@*(i)),
A similar theorem is valid in the case of weakly K-divisible families
2 under
the additional assumption about the boundedness o f the operator S, in lattices Oi. These are the fact related t o the reiteration theorem for the ( n
+ 1)-
families, which are available so fat. It should be noted t h a t in a series of papers by Fernandez
[l-31 an attempt was made to extend t h e reiteration
theorem t o an analog of the real method with power parameters (in the category
6~with N
:= 2" - l), proposed by this author. Unfortunately,
this result proved t o be incorrect (see Asekritova [2]).
In this connection,
the results in the papers cited above, which are based on this reiteration theorems (in particular, the analog of the Marcinkiewicz theorem), are also incorrect . In conclusion, let us review the problems under consideration. The first example of interpolation of ( n
+ 1)-families
(in the Hilbert case) is given
in the work by Foias and Lions [l]. Another result concerning spaces L, is described in Kerzman [l]. A generalization of the method of averages and the method of constants by Lions and Peetre for the category
Gn is
considered in Yoshikawa [l]. He established, in particular, t h a t unlike for
the case of couples, these methods are essentially different. Th e same paper presentes an application of the results obtained t o the interpolation o f spaces generated by the corresponding operators of n-parametric semi-groups o f operators acting i n a Banach space.
In Sparr [l]a generalization o f the
679
Comments and supplements real method with power parameters for the category
finis considered.
This
author constructed an example showing t h a t the Lions-Peetre equivalence theorem (see Theorem 3.5.3) does not hold any longer in this situation. Further, he singles out t h e (n
+ 1)-families for which an analog of the Lions-
Peetre “fundamental lemma” is valid (see, for example, Bergh and Lofstrorn
[l], Lemma 3.3.2, condition
(F)). He shows that condition (F)is satisfied
+ 1)-families
(lE(X), ..., li:(X)), where 1 5 p; 5 00, X is a Banach space, and the vectors A’ E R”are such t h a t 1R” is their affine
by the (n
envelope. Here
On the basis of this fact, Sparr [l obtained ] interesting results on the interpolation and embedding of the ( n
+ 1)-families of the Sobolev and Nikolskii
spaces with a dominating mixed derivative and, respectively, with a mixed difference. It should be noted that it follows from the results obtained later by Asekritova and presented above that if an ( n
+ 1)-family is K-divisible,
then the condition (F) holds for it. In view o f what has been said earlier, this provides a new approach t o the results obtained by Sparr. Finally, we should mention a new theorem on interpolation o f operators acting in (n+l)-families of “mixed” Sobolev and Nikolskii spaces, which was obtained in Bakhvalov, Further, l e t us consider an interesting application Berezin and Bessonov [l]. of the concept of K-divisibility t o the Peetre intersection problem, see Peetre [17]. In the general case, it consists in specifying conditions imposed on the spaces Xo,XI, X2 and on the parameter 0 under which holds
This relation is not true for arbitrary X; even in the case of power parameters (see Triebel
[l]). Some sufficient conditions for the validity of this equality
in the power case were established by Peetre [17]. Asekritova [5] showed that if the triple
( X o ,XI, X2) is relatively complete and K-divisible, then
relation (4.7.1) holds for any parameter 0. It should be noted, however, that the condition of the K-divisibility o f the triple (Xo,Xl,X2) is far from
Selected questions in the theory of the red interpolation method
680
being necessary. For example, it was shown recently by Maligranda [3] that
(4.7.1) is valid for any triple ( X o ,X1,X,) consisting of Banach lattices. Finally, l e t us consider an interesting result obtained by N.M. Zobin, in which the inverse problem is solved t o a certain extent, viz. in a given set
M of Banach spaces, the minimal (in number of spaces) set I? for which M = Int(2) is found. The following situation is considered in the paper Zobin [l].Let P,, : R" R+, Qm : R" R+ be two symmetric norms.29 Then the mixed norm P,,(Q,) is defined in a natural way on the --f
space
R" @ R".The set
--f
of all Banach spaces with such norms will be
denoted by Symm(n, m). Below we shall describe an (infinite) set of
X for
which
Symm(n,m) = Int(I?)
(4.7.2)
.
It should be emphasized t h a t here we speak about exact interpolation spaces. Since
R1 @ R"
we can, in view of the can be identified with Rm,
fundamental result by B.S. Mityagin, for n := 1, take
x'
:=
(ly,lE).
Thus, the theorem which will be presented below may be regarded as a generalization of the main result o f the work of Mityagin [l]for the finitedimensional situation. In order t o formulate the theorem, let us agree to
R"is canonically embedded in Rmso that xr for a given a E R"
assume that Let further
I(z) := card{i E N ;
= 0 for z
> n.
> z;}
be the number o f points a t which the function z* :
N
--f
El+ increases,
and C(z) be the number o f intervals on which this function is constant. We Put
Z ( z ) := max{I(z),C(z)} . Further, in analogy with the continuous case, we define finite-dimensional Lorentz spaces A:
and finite-dimensional Marcinkiewicz spaces p:
(u E
Rn\{0}) by putting 2gThe norm u : R' 4 R+is called symmetric if v(z) = v(z*)for all z E RE. Here I* is a nonincreasing transposition of the sequence
(IZ~~)~=~,...,~.
Comments and supplements
681
/ k
k
\
As before, we define the mixed norms l ; ( p r ) and
lk(Xr). Let
2 be the
infinite set of finite-dimensional Banach spaces defined by these norms with vectors
Q
for which
Z(A)5 n
(4.7.3)
.
Theorem (Zobin [l]). (a) Formula (4.7.2) is valid for
(b)
x'
x'
as above.
is minimal in the following sense. If a space Y E Symm(n, m ) belongs
t o I n t ( i ) ,there exists a sequence o f norms in t h e set
{I1 . I I x
;X E
x'},
which converges t o the norm o f Y .
Remar k. (a) Condition (4.7.3) is obviously satisfied for any vector
Q
E
pLm\{O}
for
n 2 m. (b) It follows from statement (a) o f t h e theorem that if Y , XI,..., X , are the spaces in
2 and Y # X i , 1 5 i 5 T , then Y 9 Int(X,, ...,X,).
Concluding the section, it should be noted that for n
>
1 the simplest
results concerning duality, valid for couples, are invalid in the category
s,,.
For example, it is shown in Dore, Guidetti and Venni [l] that the equality
C(x')' = A(2')
Selected questions in the theory of the real interpolation method
682
is generally incorrect in this case.
4.7.2. Calder6n Couples Let us consider a few more results concerning couples i n which interpolation is described (or not described) by the real method. First of all, an alternative description of the set
Int(L1, L,)
obtained by
B.S. Mityagin is
worth while mentioning.
Theorem
Let
y
(Mityagin [l]),
be a set of operators o f the form
f
+~
is a transformation preserving measure and
X E Int(Ll,L,) H
E
:
( of y), where y : R
R
+
+ lR
{-1, l}. Then
Y c &(X) .
0
This result leads t o the natural question about the description of a minimal subset of the set o f operators C ( x ' ) . The invariance relative to this set is sufficient for the space to belong to are given i n Zobin and Zobina
Int(2). Some results in this direction
[l-31.
Let us begin with the description o f K-monotone couples of Banach lattices.
Recall that the fundamental results obtained by
A. Calderbn, B.S.
Mityagin, G. Sparr, V.I. Dmitriev and M. Cwikel have already been mentioned i n Sec. 4.4 and in the comments to this section (see Dec. 4.7.A).
The following profound result underlines the special role played by couples of weighted L,-spaces. Theorem (Cwikel-Nibson
Let
[.??I).
6 be a couple of Banach lattices with order continuous norm on a space
with a separable measure, and assume that the couple (@o(wo),@l(wl)) be uniformly K-monotone for any weights w;. Then
PI E [ i , + o l . 0
6
L,- for some
Comments and supplements
683
It follows, in particular, that if a Banach lattice X differs from L, (1 5 p < m), then for a certain weight the couple (X(wo),X(wl)) is not a Calder6n couple (see Cwikel-Nilsson [2]). More extensive results are obtained for couples of symmetric spaces (see Sec. 2.6 on their definition). For example, Peetre [12] established that e,(G) possesses the C-property relative t o the couple of Marcinkiewicz spacesm (Mw,Mv1)if (pi E P(i = O, l).31 In view o f Theorem 4.4.16, it obviously follows that the couple Md := (M,,,Mv1) is K-monotone with the indicated weights. It is shown in Cwikel and Nilsson [3] that t h e couples (L,,M,) and (Ll,M,) are K monotone for all ‘p E Convo so that the condition ‘pi E P- is not necessary for the K-monotonicity of the couple o f Marcinkiewicz spaces. On the other hand, it should be noted that Peetre’s result described above cannot be improved. Namely, in the paper by Cwikel and Nilsson cited above it is shown + 5 Md for any weights wi,then ‘pi E P- (i = 0 , l ) . At t h a t if L,(G)
K the same time, there exists a couple of Marcinkiewicz spaces, which is not a Calder6n couple. Namely, it was shown by Ovchinnikov [2] that the couple
+
(L1 n L,, L1 L,) (on R+), which obviously coincides with (M,,, M,]) for cpo(t) := min(l,t), cpl(t) := max(l,t), is such a couple. V.I. Ovchinnikov’s result also gives a negative answer t o the following question posed by J. Peetre.
Is every couple o f symmetric spaces having the Fatou property K-monotone? Similar results were also obtained for couples o f Lorentz spaces A+ (see Peetre [12]). A more complicated result concerning the K-monotonicity of the couple (L,,A,)
(with cp E Convo) was established in the paper by
Cwikel [3]. Recall that IlfllA,
:=
1
=+
f’(s)d9(s) .
Very little is known about the K-monotonicity o f couples o f the Orlicz spaces 30Here it is convenient to use a somewhat different definition of M+,. Namely, t
llfllMv
:=
SUP
dt1-l
s f’(s)ds.
t>o
311n other words, for any y we have 0
ds
5 ycp(t) (t > 0).
Selected questions in the theory of the r e d interpolation method
684
L d . We shall mention only one result from Ovchinnikov [7], according t o which the couple ( L , n L,,,L, L,,) (on R,)does not possess this property. On the other hand, it was established by Merucci [l] that the couple Lfi with N ; ( t ) := tPi(ln+t)ai, 1 5 p < 00, 0 < a; < 00 (z = 0 , l )
+
is a Calder6n couple. Only one “real” Calder6n couple is known t o exist among spaces of Naturally, the couple (Lipa,Lipp), 0
smooth and analytic function.
<
a, /? < 1, is K-monotone, but it (and similar couples) inherits this property from the functional parameters determining this couple (see corollary of Theorem
4.4.18by Cwikel). The description of this “real” C-couple is contained
in Theorem (Jones
[!?I).
The couple ( H , , H , ) o f Hardy spaces of analytic functions over the unit disk is K-monotone. 0
At the same time, it is known that (Cwikel [4]) the couple (Lp,Wpk)is not K-monotone for 1 < p
< 00.
When p := 1or
00,
the K-monotonicity
remains disputable.
The difficulties involved in this problem are clearly demonstrated by the following result obtained by Mityagin and Semenov
[l], whose analog is not
any longer valid for 1< p For 0
< 00. < 1 < k, one has C’ # Int,(C, Ck), while Wi # Int,(Ll, Wf).
4.7.3. Some Unsolved Problems
(a) Determine the exact constant in the interpolation inequality for the Cal-
der6n-Lozanovskii functor
(pc
(on the category of couples of Banach
lattices) (see in this connection Theorem 4.3.11).
Comments and supplements
685
(b) Determine the exact constant in the interpolation inequality for t h e couple (LipCr,Lipp), 0 < (Y < p < 1. (c) Describe all interpolation spaces for the couple (C, Lip 1). A “simpler”
problem: calculate t h e spaces o f the complex method Cd(C,Lip 1). (d) What necessary and sufficient conditions should be imposed on the couple
6 of the parameters of t h e K-method for which the general “com-
mutativity’’ formula obtained by Lions, viz.
i s valid? In this connection, see Corollary 4.3.6 and Remark 4.3.7. (e) Describe the set of minimal elements relative t o order relation
5
in
K the subcategory o f all relatively complete Banach couples. Consider the analogous problem for the set of maximal elements in the category
2.
(f) Determine the necessary and sufficient conditions on (pi, under which the couple (A,,,A,)
of the Lorentz spaces is K-monotone.
Consider the same problem for a couple o f Marcinkiewiu spaces.
(g) Prove or refute the following conjecture (Yu.A. BrudnyY).
If for the couple o f Orlicz spaces LR :=
( L N o ,L N , )
Boyd indices (see Theorem 4.4.36) holds
i
E (0, I}, then
CUN,
for the Shimogaki-
# p ~ at,
least for one
LR is not K-monotone.
(h) Prove or refute the following conjecture (Cwikel).
If the couple
x’
is such that the spaces
K-functors for all t9 E (0, l), then
(i) Let the symmetric space
(a,L,)
be K-monotone?
Cd(x’) are generated by the
2 is a Calder6n couple.
possess the Fatou property. Will the couple
686
Selected questions in the theory of the real interpolation method
(j) For what weights is the couple (@(w0),@(w1)) K-monotone? Here is an order continuous Banach lattice.
(k) Describe those couple K-monotone.
6,where
@; E Int(e,),
i = 0,1,
which are
687
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715
SUBJECT INDEX a-couple 311, 517 Abelian group, normed 310, 504 approximation family 314 - linear 567 approximation, best 567, 619 approximation property 193 associated lattice 422
convergence in measure 4 unconditional 565 weak unconditional 565 convexity inequality 10 coorbit 152 couple 9 1 B-isomorphism 96 B1-isomorphism 9 7 - Banach 1, 9 1 C-abundant 581 - Calder6n 578, 682 - complete interpolation 185 - dual 174 (F, 7)-adequate 579 - +-adequate 251, 578 finite-dimensional regular 211 - generalized - BY-isomorphism 107 - intersection 109 - sum 109 - generalized (Banach). 103 Holmstedt 606 - (K,y)-monotone 579 K-divisible 676 K-linearizable 464 - IC-monotone 579 - linear isometry 97 - linear isomorphic 96 - normed Abelian 3 1 1 normed 104 - completion 105 - ordered 9 1 quasi-Banach 312 - regular 104, 116 - relative complete 120 - universal 227 - weakly reflexive 635
-
-
-
Banach envelope 492 Banach family 110 - directed, limit 215 - intersection 110 - sum 110 Banach function lattice 246 Banach lattice - monotonically complete 642 - order continuous 642 Banach space - intersection 92 - quasi- 9 - sum 92 - uniform convexity 20 Bessel inequality, generalized 75
(C, y)-property 579 C-couple 578 C-property 579 Calder6n couple 578, 682 Calder6n-Zygmund operator 86 canonical representation 360 category 258 - small 260 Chebyshev's inequality 28 colimit 265 completion - abstract 104 - Cauchy 129 - relative 119. 120 condition (VP) 570 construction. Calder6n-Lozanovskii continuity, absolute 9 contraction 96
-
-
-
-
556
D-diagram 268 conjugate 276 - operator 278
-
716
Subject index
decomp,,ability 597, 598 decreasing rearrangement 48 E-functional 289, 49 &-method 289, 522, 668 - parameter 522 &-method on A , parameter &-method on B , parameter epimorphism 261
523 524
factor-diagram 275 Fatou property 9, 378 &method 249 Fourier operator 75 function - concave 290 - convex 289 - distribution 48 - equimeasurability 49 - Lipschitz 301 - locally p-integrable 6 - measurable, space 3 - metric 4 - order 5 - ,u-measurable 3 - pquasi-concave 440 - proper 290 - quasi-concave 291 - quasi-power 387 - cone 613 - linear continuous 102 functor 261 - admissible family 143 - intersection 144 - sum 144 - &invariant 498 - bounded interpolation 141 - complete interpolation 287 - computable 216 - computable on a couple 226 - continuous embedding 141 - dual 197 - equality 142 - equivalence 142, 264 - exact interpolation 140 - fundamental function 145, 146, 167 - modified 438 - generating 155, 164 - maximal 155 - minimal 155
- interpolation 140 - K-monotone 353
-
K-monotone i n the side sense minimal 212 minimal on a couple 226 natural transformation 264 nondegenerate 510 of power type 8 171 - of type 4 171 - real interpolation 505 - reflector 107, 263 - reflexive 287 - regular 143, 510 - relative complete 143 - strict interpolation 633 - superposition 145 - weak compactness 636 functor morphism 264 functorial operation 143
353
Gagliardo's diagram 72, 85 Grothendieck constant 37, 85 Grothendieck's inequality 37, 84 Hankel operator 489 Hardy space 90, 473, 684 Hilbert transform 80, 86 Holder's inequality 8 Holmstedt couple 606 homomorphism - bounded 311 intermediate space 113 dual 174 regular 116 - relative complete 120 interpolation constant 113 interpolation of operators, theory of 1 interpolation property of &type 137 interpolation property of power @-type 137 interpolation space 1, 113 - computable 287 - exact 114 maximal 115 minimal 115 - exact, K-monotone 503 - K-monotone 503 - theory of 1 interpolation type 4 115
-
-
Subject index interpolation, nonlinear isomorphism 261
717 493
J-functional 248, 289 3 - m e t h o d 247, 360 - discrete 3 8 1 - parameter 364 Jd-method 381 Jensen’s equality 12 I<-divisibility 74, 325, 550 I<-divisibility constant 325 - absolute 335 I<-functional 72, 289 K-method 246, 338 - parameter 3 4 1 I<-monotonicity 251 I<-orbit 353 I<-subcouple 465 Kolmogorov’s inequality 86 Kp-functional 305 /-couple 312 L-method 505, 668 limit 264 linear space, quasi-normed 312 Lipschitz function 3 0 1 Lipschitz mapping 232 - strong 233 logarithmical convexity 13 Lorentz space 471 lower bound, exact 5 L,-functional 305 majorant, least concave 291 mapping - Lipschitz 232 - monotonic 49 - re t ra c t i v e 159 - strong Lipschitz 233 Marcinkiewicz space 472 Marcinkiewicz’s theorem 66, 69, 71, 85 measure complete 3 - harmonic 42 measure space 3 method 245 - complex interpolation 249 - interpolation 245 - lower complex 249
-
-
nonuniform interpolation 245 real interpolation 9. 675 telescopic 549 - uniform interpolation 245 - upper complex 249 method of averages 541 - parameter 541 method o f constants 511 - parameter 511 method o f means 543 method o f traces 545 - parameter 545 Minkowski’s inequality 8 minorant, greatest convex 293 modulus o f continuity 3 0 1 modulus of convexity 20 monomorphism 261 morphism 258 - conjugate 276 - inverse 260 - universal 262 p-almost everywhere 3 p-measurable set 3 Nevanlinna class norm, monotone
40, 85 523
object 258 isomorphic 261 operator ideal 498 operator - admissible family 42 - bounded 493 - Calder6n-Zygmund 86 - conjugation 77, 86 - dilation 67 - dual 188 - embedding 100 - Fourier 75 - Hankel 489 - identification 103 - image 99 - injective 100 inverse 96 - kernel 98 - linear continuous 95 - maximal 27, 79, 86 - multilinear 50 - nonlinear 493 - nuclear 488
-
-
718 operator (Conl’d) quasiadditive 26, 496 quasilinear 26, 28 - subadditive 26 - sublinear 26, 29 - surjective 100 orbit 146
-
pestimate lower 606 upper 606 parameter - nondegenerate 388 - power 387 - quasi-power 387 Poisson kernel 4 1 projection 100 pullback 266, 274 pushout 267. 274
-
q-couple 312 q-decomposability 605 quasinorm 9 - monotone 522 regularization 116, 275 reiteration theorem 347 retract 159 Riesz constant 87 Riesz’s theorem 13, 16, 70, 84, 88, 89 Riesz-Thorin theorem 34, 40, 44, 71, 89 g-finite measure 3 simple function 9
Subject index singular number 487 Smirnov class 41, 85 Stoltz domain 474 subcategory 260 - complete 260 - F-stable 252 - factorizing 212 - interpolation-stable 252 - reflective 263 - small 168 subcouple 9 1 - complemented 101 subdiagram 274 support 6 tensor product 277 B-valued 277 0 power type 116 three circles theorem 3 1 three lines theorem 32 transposition 92 triple 136 - &invariant 500 - exact &invariant 500 - exact interpolation 136 - exact Zi-monotonic 503 - interpolation 136 - I<-monotonic 503 - optimal interpolation 138 - exact 5
-
weak approximation condition weight 12 weight space 12
594
