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
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,.)+ 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
mand 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(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,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 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
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.
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 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_ 2
1.11.3. The Meaning o f the Theorems o f Riesz and Thorin for pi
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 ~
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 ~ 0, there 0
z = 2,
+ z,
1
-+
:= suph’(t,z;
exist zb
X)
, 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 CkO
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
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 , ~,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 ( (
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
.
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;), 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
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(* 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#
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 ( ~ 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: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
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 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
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 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 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)
,
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 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)
,
-2kI 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 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,
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 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( 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 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 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-%(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(X'>
.
The calculation of the right-hand side with the help of formula (3.7.41) gives
( D K A ( ; m ) ) ( 2z) I 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 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 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 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
. 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
(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 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 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 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 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
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
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 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 .
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 0,
0 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$(d)is a
Banach space. Besides, in view of the
previous inequality we have
-
Thus, the embedding
J6(2)
C(2)
is also established. Let us now verify the validity of the embedding (4.2.68)
A(d) +J 3 ( 2 ) .
Since A(6) # {0}, there exists a subset Ro and
If for
x := xno E @,n xE~ ( d we )put := p(Ro)-'X
.(W)
it follows that
x=J n Since, in addition,
udp.
. .x ,
c R, such that 0 < ~ ( R oll@l(Xl)5 2 11~11Ta(2).
Weput u ( t ) := -tf'(t). It follows from definition (4.2.70) that ~ ( tE) A(x'). Also, in view of (4.2.71). we have
1
(4.2.74)
dt
u(t) 7 = -
Rt
1 f'(t)dt
=z .
R+
Further, let
where the function and
'p
E C r ( l R + )has the support [2-',1],
is nonnegative
548
Selected questions in the theory of the r e d interpolation method
This equality together with (4.2.74) implies t h a t (4.2.75)
5
=
J
u(s)
ds s =
J mt
Rl
J
=
e(t>
dt
.
R 1
It follows from the assumption about the support of cp that 2t
I I ~ ( ~ L) (max I I ~ ~9)
J
IIu(S>llxldt= ( m m cp>Hi(IIullxl)(t).
t
Since the operator H1 is bounded in
Q1,
we obtain in view of (4.2.73) and
the choice of u
llcll*l(xd L 2(max 'P) IlHlll .
II"IIT&(R)
.
It remains to estimate t h e norm of ii in a o ( X 0 ) . Integrating by parts and taking into account the choice of u , we get
Hence it follows that m
m u Ilc(t)llxo I
ds
Id1 J Ilf(s)llxo y
= (max I d l ) ~ o ( l l f l l x o ) ( t )
t
a0 and (4.2.73),
Using the boundedness o f Ho in Il~llOo(X0)
I
we finally get
lcp'l . I l H O l l . I l f l l * o ( x o )
I2 max
19'1 I l H O l l
I
II"IlT~(d).
Together with equality (4.2.75), the obtained estimates of the norm of give
lI4lJ6(X) I 2(m=
IV'I
. llffoll + m a IcpI . llHIl>II+&(2) .
U
Red interpolation functors Thus, t h e embedding
549
T&(d) ~t J$(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 ~s, 7761, - .-
+
Here
P,
-+2 Po
Pl
(O
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-' 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 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@
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)*
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
_
"
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 we have
623
Inverse problems of red interpolation
+ t s ) 5 a, + nt
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 , )
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,, 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)
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 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,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
it. Choosing the number Nk
Nk-1
sufficiently large, we can make the
inequality
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