Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zgrich F. Takens, Groningen
1580
Mario Milman
Extrapolation and Optimal Decompositions with Applications to Analysis
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Author Mario Milman Department of Mathematics Florida Atlantic University Boca Raton, FL 33431, USA E-mail: Milman @acc.fau.edu
Mathematics Subject Classification (1991 ): Primary: 46M35, 46E35, 42B20 Secondary: 35R15, 58D25
ISBN 3-540-58081-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58081-6 Springer-Verlag New York Berlin Heidelberg
CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready by author SPIN: 10130043 46/3140-543210 - Printed on acid-free paper
To Vanda
Preface In these notes we continue the development of a theory of extrapolation spaces initiated in [57]. One of our main concerns has been to connect the fundamental processes associated with the construction of interpolation and extrapolation spaces "optimal decompositions" with a number of problems in analysis. In particular we study extrapolation of inequalities related to Sobolev embedding theorems, higher order logarithmic Sobolev inequalities, a.e.convergence of Fourier series, bilinear extrapolation of estimates in different settings with applications to PDE's, apriori estimates for abstract parabolic equations, commutator inequalities with applications to compensated compactness, a functional calculus associated with positive operators on a Banach space, and the iteration method of Nash/Moser to solve nonlinear equations. Many of the results presented in these notes are new and appear here for the first time. We have also included a number of open problems throughout the text. While we hope that these features could make our work attractive to specialists in the field of function spaces it is also hoped that the central rSle that the applications play in our development could also make it of interest to classical analysts working in other areas. In order to facilitate the task of these prospective readers we have tried to provide sufficient background information with complete references and included a brief guide to the literature on interpolation theory. We have also tried to make the contents of different chapters as independent of each other as possible while at the same time avoiding too much repetition. Finally we have also included a subject index and notation index. It is a pleasure to record here my gratitude to a number of people and institutions who have helped me to complete these notes over
vnl
the years. In particular I am grateful to Bj6rn Jawerth with whom I have spend a lot of time over the years talking about extrapolation. My friends DMGSJ played also an important extrapolatory role, both professionally and otherwise. The first version of the book was written during a membership, partially supported by an NSF grant, at the Institute for Advanced Study. I am most grateful to the Institute and Professor L. Caffarelli, for their support and for providing such an stimulating environment for my work. The notes were further developed while I was visiting the University of Paris (Orsay), The Centre for Ricerca (Barcelona) and the University of Zurich. I am particularly grateful to Professors Herbert Amann (Zurich), Aline Bonami (Orsay), Joan C~rda (Barcelona) for their support and interest in my work.
Contents Introduction 1.1 A V e r y B r i e f G u i d e To T h e L i t e r a t u r e O n I n terpolation .......................
1 4
Background On Extrapolation Theory 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.2 More About the ~ and A methods . . . . . . . . . . . 2.3 Recovery of End Points . . . . . . . . . . . . . . . . . 2.4 The classical Setting of Extrapolation . . . . . . . . . 2.5 Weighted Norm Inequalities . . . . . . . . . . . . . . 2.6 More Computations of Extrapolation Spaces . . . . . 2.7 Notes and Comments . . . . . . . . . . . . . . . . . .
7 7 17 25 28 31 32 32
3
K/J Inequalities and Limiting Embedding Theorems 3.1 K / J Inequalities and Zafran Spaces . . . . . . . . . . 3.2 Applications: Sobolev Imbeddings . . . . . . . . . . .
35 36 39
4
Calculations with the A method and applications 43 4.1 Reiteration and the A method . . . . . . . . . . . . . 43 4.2 On the Integrability of Orientation Preserving Maps . 46 4.2.1 Background . . . . . . . . . . . . . . . . . . . 47 4.2.2 Identification of Sobolev Classes using A . . . 49 56 4.3 Some Extreme Sobolev Imbedding Theorems . . . . . B i l i n e a r E x t r a p o l a t i o n A n d A L i m i t i n g C a s e of a T h e orem by Cwikel 59 5.1 Bilinear Extrapolation . . . . . . . . . . . . . . . . . 60 5.2 Ideals of Operators . . . . . . . . . . . . . . . . . . . 67
x
CONTENTS
5.3 5.4
L i m i t i n g case of Cwikel's e s t i m a t e . . . . . . . . . . . Notes a n d F u r t h e r Results . . . . . . . . . . . . . . .
6 Extrapolation, Reiteration, and Applications 6.1 6.2
75
Reiteration ....................... E s t i m a t e s for t h e M a x i m a l O p e r a t o r o f P a r t i a l S u m s Of Fourier Series . . . . . . . . . . . . . . . . . . . . Extrapolation Methods ................. More On R e i t e r a t i o n . . . . . . . . . . . . . . . . . . Higher Order L o g a r i t h m i c Sobolev Inequalities . . . . Notes a n d F u r t h e r Results . . . . . . . . . . . . . . .
84 85 88 89 91
Estimates For Commutators I n R e a l Interpolation
95
6.3 6.4 6.5 6.6 7
69 71
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
75
Some O p e r a t o r s Associated to O p t i m a l D e c o m p o s i t i o n s 97 M e t h o d of P r o o f . . . . . . . . . . . . . . . . . . . . 102 C o m p u t a t i o n of f~ for e x t r a p o l a t i o n spaces . . . . . . 107 O t h e r O p e r a t o r s f~ . . . . . . . . . . . . . . . . . . . 111 Compensated Compactness .............. 115 R e l a t i o n s h i p to E x t r a p o l a t i o n . . . . . . . . . . . . . 118 A F u n c t i o n a l Calculus . . . . . . . . . . . . . . . . . 121 A C o m m e n t on Calderdn C o m m u t a t o r s . . . . . . . . 124 Notes a n d F u r t h e r Results . . . . . . . . . . . . . . . 125
8 Sobolev Imbedding
Theorems and Extrapolation Infinitely Many Operators 8.1 Averages of O p e r a t o r s . . . . . . . . . . . . . . . . .
of 127 127
9 S o m e R e m a r k s o n Extrapolation Spaces and Abstract Parabolic Equations 131 9.1
Maximal Regularity . . . . . . . . . . . . . . . . . . .
132
10 Optimal Decompositions, Scales, a n d N a s h - M o s e r I t eration 139 10.1 Moser's A p p r o a c h to Solving N o n L i n e a r E q u a t i o n s 140 10.2 Scales w i t h S m o o t h i n g a n d I n t e r p o l a t i o n . . . . . . . 142 10.3 A b s t r a c t Nash-Moser T h e o r e m . . . . . . . . . . . . . 145 Bibliography
149
CONTENTS
xi
Index
158
Symbols
161
Chapter 1 Introduction In the last thirty years or so interpolation theory has become an important chapter in the field of function spaces. The origins of the theory are the classical interpolation or convexity theorems of Riesz, Thorin, and Marcinkiewicz. These classical results were subsequently extended by many authors including Calderdn, Cotlar, Salem, Zygmund. The foundations of the general theory were laid down in the sixties by Aronszajn, Calderdn, Gagliardo, Krein, Lions, Peetre, among others. It has since been extended, perfected, and applied, by many mathematicians. We refer the reader to the section at the end of this chapter for a brief guide to the available literature. The theory has found many applications to classical analysis. In particular it has become an important tool in theories as diverse as partial differential equations, approximation theory, harmonic analysis, numerical analysis, operator theory, etc. In the last few years, mainly in collaboration with B. Jawerth, we have been developing a new theory of "extrapolation spaces" which somehow is the converse of interpolation theory. The classical framework of interpolation theory can be briefly described as follows. We are given a pair of compatible Banach spaces (X, Y) and we attempt to construct all the spaces with the interpolation property between them. The real and complex methods, for example, provide parameterized families of spaces (X, Y)O,q, [X, }I]0, with the interpolation property. In extrapolation we conversely ask: given a family of interpolation spaces can we reconstruct the originating pair? This
2
C H A P T E R 1. I N T R O D U C T I O N
question is, of course, directly related to best possible interpolation theorems. In practice, however, one is also very much interested in weaker formulations of this general question. Thus, given a family of estimates for an operator T acting on interpolation spaces, one wishes to "extrapolate" this information either as an "extrapolation theorem" (i.e. a continuity result for T, the model for which is provided by the classical extrapolation theorem of Yano) or an "extrapolation inequality" for T where the extrapolation estimate is usually based on the basic functionals of real interpolation (K, J, E functionals). In a sense one could also consider "extrapolation" as chapter of the theory of interpolation of infinitely many spaces, since one is trying to obtain information from an infinite family of spaces. The precise connection between these theories is an interesting open problem. The usual relationship between modern analysis and classical analysis is that the former provides a framework for consolidation, extension, and simplification of results of the latter. At times, however, it is the general framework that suggests the right questions, and the techniques of modern analysis sometimes also provide the answers. It is this symbiotic state of affairs that, from our perspective, makes the general field of interpolation theory interesting. In these notes we study the connection between optimal decompositions, extrapolation, and its applications to other areas of analysis. In particular we also explore the role that cancellation plays in interpolation/extrapolation theory. We have arranged the development of the theory visa vis concrete applications to classical analysis in order to emphasize our point of view. Thus the development of theory in this book is not "linear." A prospective reader interested mainly in the abstract theory could certainly skip the applications developed in each of the chapters. On the other hand we hope that the specific applications will be of interest to analysts working in different fields. As a consequence we have tried to make the reading of each of these sections as independent as possible from the others while at the same time trying to avoid too much repetition. Most of the results presented in these notes are new or have not appeared in book form before. It is hoped that they will serve as the basis of a larger, more detailed, and formal book. The author would therefore welcome suggestions, remarks and corrections.
3
The notes are organized as follows. In Chapter 2 we provide a detailed introduction to extrapolation theory (the reader is referred to [57] and [58] for further information and other specific applications). In this chapter we also provide detailed computations of extrapolation spaces for different scales. In Chapter 3 we discuss K/J inequalities in the context of limiting embedding theorems for interpolation scales with specific applications to Sobolev embeddings, in particular we relate these results to limiting inequalities by Kato and Ponce [63], and neale, Kato and Majda [5] concerning singular integrals. In Chapter 4 we study the A method of extrapolation, develop new tools to compute it, and apply these results to sharpen recent estimates by Mfiller [82] and Iwaniec and Sbordone [54] on the integrability of the Jacobian of orientation preserving maps. We also give an application to Sobolev imbedding theorems extending recent work by Fusco, P. L. Lions and Sbordone [42]. In Chapter 5 we study bilinear extrapolation and prove general bilinear extrapolation theorems of Yano type. As an application, involving the Schatten classes, we derive an end point inequality, due to Constantin [22], of a well known theorem of Cwikel [28] concerning the singular values of Schrodinger operators (which leads to Constantin's [22] limiting version of the collective Sobolev estimates of Lieb [70]). In Chapter 6 we give a number of new reiteration theorems for extrapolation spaces and apply our results to give a new approach to end point estimates for the maximal operator of Fourier partial sums due to SjSlin [95] and Soria [97]. In this chapter we also provide a new approach to the higher order logarithmic Sobolev estimates of Gross and Feissner (cf. [46], [41]). In Chapter 7 we study the role of cancellation in interpolation/extrapolation theory: we prove new commutator theorems and establish the relationship between commutator theorems and a functional calculus for positive operators on a Banach space. We believe that this connection could lead to applications in the theory of parabolic equations. In this chapter we also develop in detail an application to the theory of compensated compactness of Murat and Tartar, and, in particular, we indicate how commutator theorems can be used to obtain sharp integrabili~ theorems for Jacobians of orientation preserving maps. In the short Chapter 8 we consider families of operators an establish an abstract version of a Sobolev embedding theorem due to Varopoulos [102]. In
4
C H A P T E R 1. I N T R O D U C T I O N
Chapter 9 we indicate rather briefly the role of extrapolation spaces in the theory of abstract parabolic equations in Banach space and establish a number of new end point estimates. Finally, in Chapter 10, we discuss the iteration process of Nash/Moser in the setting of scales with smoothing and interpolation/extrapolation spaces, in particular we establish a precise relationship between the theory of scales of spaces with smoothing and interpolation scales as well as provide an interpretation of the paracommutators of H6rmander [50] in terms of optimal decompositions.
1.1
A V e r y B r i e f G u i d e To T h e Litera t u r e On I n t e r p o l a t i o n
This guide is intended to provide a rather incomplete and super brief guide to background literature on interpolation theory relevant to the developments in these notes. It is intended to help a newcomer to the field to begin to study the literature. We apologize in advance if your favorite book or paper is not quoted here. The classical reference books on interpolation theory include [13], [8], [101], and [67]. These books also develop many of the early applications of the theory to harmonic analysis, approximation theory, semigroups and partial differential equations. The book [65] also presents a detailed account of interpolation and its applications to operator ideals. More recent contributions are the research monograph [85] which develops a new approach to interpolation theory, the book [7] which presents a detailed study of rearrangement invariant spaces and the computation of K functionals, and the ground breaking treatise [12] presenting new important theoretical developments in the field. The monograph [57] presents the first detailed account of extrapolation theory and contains a bibliography of 60 items, as well as brief historical survey. Further results and applications of extrapolation are contained in [58] (which also contains an extensive bibliography) and the forthcoming paper [27] which deals with complex extrapolation. The theory of second order and commutator estimates in interpolation theory and their applications was developed in, among other
1.1. B R I E F GUIDE TO THE L I T E R A T U R E
5
papers, [89], [60], [25], [62], [61]. Finally we should mention that the recent book [26] that contains an extensive list of unsolved problems in the area.
Chapter 2 Background On Extrapolation Theory In this chapter we present in detail some of the basic results of the theory developed in [57], [58]. In order to make the presentation self contained we have included proofs to all the results presented. For the benefit of the reader we have also provided a rather brief introduction to interpolation theory.
2.1
Introduction.
We shall start by briefly recalling some basic definitions of interpolation theory. We say that a pair of Banach spaces A = (A0, A1) is compatible if there is a topological Hausdorff space ~ such that both A0 and A1 are subspaces of T/. If A = (A0, A1) is a pair of compatible spaces, then a space A is called an intermediate space between A0 and A1 if
A(it) = A o N A 1 C A C E ( A ) = A o + A, The spaces A and B are called interpolation spaces with respect to fi, a n d / ~ if for any bounded linear operator such that T : A --~ /} (which simply means T : Ai ~ Bi, i = 0, 1) we can conclude that T : A --* B. Moreover, if [[T[[A_.B _< max{I[THAo_.B~ ,[[TJ[AI_.B1} then we say that A and B are exact. An interpolation method is a functor defined on compatible pairs of Banach spaces such that
8
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
if fi, and /) are two pairs, then Jr(el) and jr(/)) are interpolation spaces with respect ft. a n d / ) , and ~'(T)=T for all T : r ~ /). A method j r which yields exact interpolation spaces is called exact; if moreover, for some fixed 0 C (0, 1) we have
for all T : fi. --*/), then we say that Jr is exact of exponent O. In extrapolation theory we start with families {A0}0ee of Banach spaces indexed by some fixed index set 19, (usually 19 = (0,1)), which are strongly compatible in the sense that there are two Banach spaces A and E such that A C Ao C I3, 0 q 19 (with continuous inclusions). Suppose now that {Ao}oee and {B0}0ee are families of strongly compatible spaces: Aa C A0 C 13~ and As C Bo C 13s for some spaces A~, ~ and Ab, Es, respectively. In this setting the i natural morphisms are the bounded, linear operators T : {A0}0ee --* {B0}0es; this means that there is an operator T : 13a --* 13s, whose restriction to Ao maps A0 into B0, with norm < 1 for each 0 E 19. We say that the spaces A and B are extrapolation spaces (with respect to the families {Ao}oee and {Bo}oze)ifA~ C A C E~, As C B C ~s, and T: {Ao}oeo ~ {B0}0eo ?- T : A ---* B (or, more precisely, that T has an extension which is defined on A and maps A boundedly into B). An extrapolation method g is a functor, defined on a collection dom(E) of families of strongly compatible spaces, such that C({Ao}oco) and s are extrapolation spaces if {Ao}oeo, and {Bo}oco E dora(E). In analogy with interpolation theory one can also define exact extrapolation spaces (rap. functors); in this case we require that IITIIA-.S < sup 0
IITllAo-.so.
The simplest extrapolation functors are the ~ and A methods. Let us now describe their natural domains. Suppose that {Ao}oee is a family of strongly compatible spaces such that the norms Mr.(O) of the inclusions Ao --* P~are uniformly bounded,
< 0r sup Mr.(O)= sup IIa[I------E~ o o IlallAo
(2.1)
2.1. INTRODUCTION.
9
Then we can form the sum S,o(Ao) of the family {Ao}oee Eo(Ao) is the Banach space of all x in ~ which have a representation x = S o ao (with absolute convergence in E), where ao c: Ao, 0 E 6), such that 9
II~ll~.(ao) = inf(~--~ II~011ao : 9 = ~0~0 } < ~ .
(2.2)
0
The hypothesis (2.1) implies that Ao (Ao) C Eo (Ao) C E . It can be shown t h a t if {Ao}oeo is an ordered scale (i.e. O = (0, 1), and Ao1 C A02whenever 01 > 0~), and g(0) a positive decreasing function defined on (0, 1), then to compute the spaces ~o(g(O)Ao) it is enough to consider the values of 0 in a fixed sequence {0v} C O. To illustrate this, we prove it in detail for the case where g(0) = 0 -~ for some fixed a > 0. Thus, we shall show that oo
(O-'~Ao) = ~ 2V'~A2-,,,~ 0
v~l
Indeed, let a E ~o(8-'~Ao), then there exists a representation of a = ~~176 1 a0~, with ao~ E Ao~, and such that OO
IlallEo(O-~Ao ) ~ ~ o; ~ Ilao~llAo~. v=i
For each tt = 2, 3,..., let E u = {t,: 0~ e [2-",2-~+1)}, and E = { u : 0~ ~ U,~__2E , }. Let
b2-, = I ~
t
vEE•
aO~ ,
Y]~veE a0~,
].t = 2 , ...
tt = 1
then, a = ~--~ b2-. tt=l
The norm of the first term is easily estimated using the triangle inequality and the ordering of the scale:
2 -~' IIb2_lllA~_l < ~_, O-~'llae,,lIA2_l < ~ O'~C'llae,,llAo,, 9 r,E E
vEE
10
C H A P T E R 2. B A C K G R O U N D ON E X T R A P O L A T I O N
Therefore,
]lb2-1l[~o(8-ctAo ) <_ cl[a][Eo(s-,,Ae) Similarly, Oo
OO
2""IIb~-~IIA~_~ _< ~ 2 ~ ~ g=2
Ila0~IIA~
v6E.
g:2
oo
2"" <_ :2"[[all~o(O-~Ao)
v=l
{ - l o g 2 S~,_
as desired. In particular, if {As} is an ordered scale, and if 0o 6 (0, 1), then, with norm equivalence, we have
(2.3)
z
06(0,So)
06(0,1)
The dual, A method, can be defined for strongly compatible families such that the norms Ma(O) of the inclusions A ~ Ao are uniformly bounded,
IlallA____~< s Ilatl~
c~,
(2.4)
= sup Ilal]A. < c~ }.
(2.5)
sup Ma(0) = sup Under this assumption we let A(Ae) = {a fi ~ A s : o
Ilalla(A.)
o
Observe that (2.4) implies that A C A(Ao) C ~ . It is not hard to see that ~ and A are extrapolation methods which, in a certain sense, are extremal. Moreover, they are dual extrapolation functors (cf. Chapter 4, (4.3)). Variants of the 2 method can be defined for compatible families of Banach spaces {A0}0eo such that the norms M~.(O) of the inclusions As --* ~ satisfy
( ~ o ( i z ( O ) ) v , ) l / v ' < o o , where 1 < p < o0, 1/p' + 1/p = 1. Then we define Ev(Ao ) by requiring
Ilal[~:,(Ao) -- inf
I]aoll~o
: a -- ~ ao, ao 6 As 0
In particular, E1 = E.
< oo
2.1. INTRODUCTION.
11
R e m a r k . In what follows we shall assume, unless explicitly specified, that we are dealing with strongly compatible families of spaces. Thus, a family of spaces shall mean a s t r o n g l y comp a t i b l e family of spaces, and in particular a B a n a c h p a i r or simply a p a i r .~ shall mean a compatible Banach pair. A function p on the half line is quasi-concave if p is increasing and ~ decreases. These functions play an important role in interpolation theory (cf. [8]). Let ~- be an interpolation functor, then its characteristic function is defined by the relationship (cf. [12]), 1
.T(C, C)= p(t-----~C,t > 0 .
(2.6)
If ~- is exact, then p is quasi-concave, and if .T is exact of exponent 0, then p(t) = Ct ~ Conversely, if p(t) is quasi-concave, then there are several exact interpolation methods with characteristic function p(t). Among these, fi-p,oo;g and -4p,1;J are extremal. Recall that /]p,oo;g is defined to be the collection of all a E E(/i) such that Ilall~,~;K = sup
,>o
K(t,a; ft) p(t)
< oo.
(2.7)
where the Peetre K functional is defined by
K(t,a;ft)=
inf {[[ao[lAo +tllalllA1 :a,e Ai, i=O, 1}
a=ao+al
The space Ap,1;g consists of all a C E(/]) such that
Hall.~p.1;, = inf f0 ~176J(t,u(t);fii)dt p(t) t < oo,
(2.8)
where the infimum is taken over all representations a = f o u(t)~ t (with convergence in E(.4), u ( t ) : (0, c ~ ) ~ A(fi.)strongly measurable), and where the J functional is defined by J ( t , a ; fi~) =
sup{][al[Ao,t[[all[A,}.
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
12
These functors are extremal in the sense that for any exact interpolation method .T with characteristic function p(t), then (cf. [85],
[12]).
Ap,1;j 1 ,~**(A)1_~Ap,oo;K.
(2.9)
It is instructive to outline a proof of (2.9) here. To prove the first embedding we fix u E A(A) and define T()~) = ~u for )~ C C. Then, IITIIc__,A, = Ilulla,, i = 0, 1, and, since .T" is exact,
I~111~IIT(~) = IITAIIs(~)
___max{llullA0 ,t Ilullal } IIAIl~(c,§
=
1
J(t,u; ft)-p-~ I)~1.
Therefore,
Ilull~r < p-~J(t,u;ft), If a C E(r has a representation a = inequality combined with (2.10) gives
Ilall~(~) ~<
Vt > O.
f~' u(t)~,
(2.10)
then the triangle
fo ~ J(t, u(t); ft) dt p(t) t
and taking the infimum over all representations a = f ~ u(t) ~, proves the first embedding. To prove the second, let us fix a E E(A) and t > 0, and define an operator T as a "linearized version" of K(t, a; ft) as follows: let T(a) = K(t, a; A) and extend T, using the HahnBanach theorem, to a continuous linear functional on E(A) with ITxl <_ K(t,x;fi~). Then, IITIIA,_.c -< t',i -- 0,1, and, using the fact that ~- is exact, we get that IITzllT(c,~c) < Ilxll~(a)- In particular, for x = a this implies
K(t,a; 7t)
p(t)
Ilall~r
Thus, taking the supremum over t > 0 proves the second embedding
of (2.9). We consider the classical set up of real interpolation as developed, for example, in [8]. Let fi, be a pair of Banach spaces. For 0 < 0 < 1
2.1. I N T R O D U C T I O N .
13
and 1 < q _< oo we let ]to,q;K be the space of all a r E(.~) for which
Ilallo,q; .=co,q
(t-~
q
< oo
where Co,q = ((1 - O)Oq)'h. The constant co,q has been chosen so t h a t the characteristic function of the (.,-)0,q;K m e t h o d is exactly t ~ Similarly, we let ]te,q;a be the space of all a ~ E(A) for which
IIall,o,q;, =
.
c0,qlnf
(t-~
q
< oo,
where the infimum is taken over all representations a = fOoo U ( t ) Tdt
(convergence in with (t)strongly measurable functions taking values in AoNA1) and where c;,q = ((1-O)Oq') -'/q', 1 / q + l / q ' = 1. Again the constant has been chosen so t h a t the characteristic function of the (., .)o,q;J m e t h o d is t ~ W i t h these normalizations we have the imbeddings (cf.[57]) 1 1 1 ao,1;J --+ Ao,q;J, AO,q;K ---+ ao,oo;K, AO,q;K --+ AO,r;K, q ~ r.
(2.11)
Given an interpolation functor Y we let be(A) ~ denote the closure of A(A) in .T(A). For a pair A, the aagliardo completion of Aj, j = 0, 1, which we denote by e~j, is the set of elements a C ~ ( A ) which are ~ ( / 1 ) limits of b o u n d e d sequences in Aj or, equivalently, for which
Ilall , =
sup t>o
K(t,a; i) t5
< oo.
We say t h a t a pair A is m u t u a l l y closed if Aj = f t j , j = 0, 1. For an element a C 2 ( A ) , the Gagliardo diagram P(a) of a is defined by P(a) = { ( X 0 , X l ) " 3aj C Aj s.t.
I[ajllA,
xj, j = o, 1, a = ao + al}
It is readily seen t h a t F(a) is a convex set of R ~. Its b o u n d a r y m a y contain a semi-infinite vertical segment a n d / o r a semi-infinite horizontal segment. The r e m a i n d e r of the graph will be the graph of a decreasing convex function xl = c2(x0). As it is well known (cf.
14
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
[8]) Gagliardo diagrams are closely linked with the computation of K functionals. Indeed, let
D(a) = O(a) M {(Xo, Xl) E R2: xj > 0, j = 0, 1}, then, for each t > 0, K(t,a;it) = K(t) is the x0 intercept of the tangent to D(a) with slope - l / t , the corresponding xl intercept is K(t) Conversely, each point on the graph of qa intersects with the t tangent of slope - l i t for some values of t determined by the right and left derivatives of qa at x0, and thus Xo + txl K(t) We are now ready to discuss the strong form of the "Fundamental Lemma" or SFL (cf. [29], [12]). =
L e m m a 1 ( " T h e f u n d a m e n t a l l e m m a " ) . Suppose that 7t is a pair of mutually closed spaces. Then a E E(fi.) ~ if and only if there oo
is a representation a = fO u(s)d_~, with f0 min 1,
J(s,u(s);Ji)) ds < - T K ( t , a ; T i ) , t > O ,
(2.12)
S
for some universal constant 7. Proof. For the reader's convenience we indicate here the main steps of the proof given in [24] and refer to this paper for complete details. A different proof is given in [12] where references to other proofs can be found. For the classical form of the fundamental lemma the reader is referred to [8] . The approach of [24] sketched here provides the best known value of the constant G'. Namely, Va E ~(fi,)o, Vc > O, there exists a decomposition a = f o u(s)~-, such that f0 r162 min (1, ~} J(s,u(s); fl))dSs < 7 ( 1 + e)2K(t,a; 7t) with "y _< (3 + v/2). Let us also point out that the method of proof of [24] is a modification of the one originally given by Cwikel [29]. Referring to the notation introduced above let us define xoo = sup{x: (x,y) E D(a)}, x_~ = inf{x : (x,y) E D(a)}
2.1. INTRODUCTION.
15
yco = s u p { y : (x,y) e D ( a ) } , y-oo = i n f { y : (x,y) E D ( a ) } F r o m t h e g e o m e t r i c a l c o n s i d e r a t i o n s a b o v e it follows t h a t
xco= suptK(t,a;7t)= II liao
'
x_co
= l i m K(t,a;fi,) t-*O
y_co = l i m K(t,a;fii) K(t,a;fi) t-~co t ' yco = t-~colim t -Ilalla, W e c o n s t r u c t a s e q u e n c e of p o i n t s lying in D(a) following a n idea of G a g l i a r d o . F i x a n a r b i t r a r y p o i n t (x0, y0) o n D(a), a n d let r = 14-vf2 ( w h i c h j u s t h a p p e n s to be t h e o p t i m a l v a l u e in o u r c a l c u l a t i o n ) . For n > 0 c o n s t r u c t (x,,, yn) i n d u c t i v e l y so t h a t o n e of t h e following t w o a l t e r n a t i v e s holds: e i t h e r Xn = rXn-1, a n d Yn <_ r-lYn or xn >_ rXn-1 a n d Yn = r - l y n - 1 9 T h e process m u s t s t o p if for s o m e n we h a v e e i t h e r rXn-1 >_ xco or r-lyn_l <_ y_~. In this case we s t o p t h e c o n s t r u c t i o n of o u r s e q u e n c e at n - 1, a n d we let t h e c o u n t e r uco = n, o t h e r w i s e we c o n t i n u e t h e process i n d e f i n i t e l y a n d we set vco = oo. For n < 0 we p r o c e e d " b a c k w a r d s " a n d i n d u c t i v e l y c o n s t r u c t a s e q u e n c e (xn, yn) s u c h t h a t e i t h e r xn = r - l x n + l a n d r - l y n _< Yn+l or Xn ~ r-lXn+l a n d r-ayn = Yn+x. T h e process m u s t s t o p if for s o m e n < 0 e i t h e r r-lXn+l _~ X_co or ryn+a >_ yco h o l d s in w h i c h case we set v_co = n a n d we do n o t n e e d to define xn a n d Yn. O t h e r w i s e we set v_co = - c o . T o t h e s e q u e n c e {(xn, Y n ) } , _ = - l < n < , = + i we a s s o c i a t e a d e c o m p o s i t i o n of a as follows. G i v e n e > 0, for e a c h n, v_co < n < vco we c a n t h e n find, b y definition, a d e c o m p o s i t i o n !
a ~
a n 4- a n ,
such that
xn _
IlanllAo
(1 4- e)xn
and Yn _< Ila,' llA, ~ (1 4 - ~ ) y , . For v_co 4- 1 < n < vco we define un -- an - an-1 , we also let u ~ = a - a~oo-1, if vco < cx), a n d u~_~o+l = a~_oo+l, if v-co 4- 1 > - o o , a n d c o m p l e t e t h e d e f i n i t i o n by l e t t i n g Un = 0 in t h e r e m a i n i n g p o s i t i o n s , if any. B y c o n s i d e r i n g s e p a r a t e l y t h e s u m o v e r t h e n e g a t i v e integers, a n d t h e o n e over t h e p o s i t i v e ones it c a n be r e a d i l y seen t h a t ~_~co u,, = a, in ~ ( , 4 ) . W e also h a v e , by t h e t r i a n g l e i n e q u a l i t y ,
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
16
and the definitions, that II..lla0 ~ (1-4-r + r-1)Xn, and Ilu.llai < (l+r Using these estimates, and some careful analysis, the reader should be able to show that Vt > O,
min{llu.llao,tllu.llA,} ~ (1 -4- r
-4- 2vf2)K(t,a; 7i)
--OO
(otherwise we refer to [24] for the complete details). This is essentially a "discrete" version of what we wanted to prove. The "continuous" version we require is obtained as follows. Define for n E Z, S ~ = {u E Z : (1 +~)" <
Ilu"llA~
<
(1 + g)n+l},
Vn =
Ilu,,IIA, --
E
Uu,
uES.
and finally let
u(t)
oo
'~7~'~(log(1+
r
vnX((l+O.,(x+~).+i](t,.
--co
Then, it t
~-~
Vn ~ --00
it v ~
a.
--00
Moreover, ~,
/
(1+~) n + l
'}J(s' u(s); .4")~-=ds~ f
min{l, s
0
min{l, }}J(s,u(s);
7t)dss
-oo (1+~)"
t
< E min{1, (1 + e ) n+'} E
min{llu~llao,(1
+~)"+'llu~llA,}
uE Sn
--00
OO
_ E ( 1 -4-r -oo
E UE Sn
min{llu~llAo,tllu~llai}
OO
< (1 + e) ~ min{llu.llao,t[lu.llal} < (1 + e)2(3 + 2v/2)K(t,a;fi~) --OO
as desired. []
2.2. M O R E A B O U T THE ~. AND A METHODS.
2.2
More
About
the
E and
17
A methods.
It is easy to compute the ~ and A functors for certain scales of real interpolation spaces. For example, in the next result from [57] (a Fubini type theorem) the identification is easy because the ~ functor commu tes with (.,.) p,a;a functors, while the A functor commutes with the (., .)p,~;K functors. T h e o r e m 2 Let A be a pair of spaces and {pe}eeo a family of quasi-
concave functions. i) If the function p(t) = sup0 po(t) is finite at a point (and hence at all points), then
Y]o( Ape,1;J ) : Ap,1;J ii) If the function p*(t) = info po(t) is non-zero at a point, then Ao(A.o,o~;,~) = A r , o o ; K
Proof.
Ilall&,,;s
9
We prove only (i), the proof of (ii) is similar. _< Ilall~p0,~;~,0 c O, we clearly have
Since
On the other hand, combining
,,
-
p(t)
with the triangle inequality gives the converse inequality. [] In order to be able to compute these functors over other families of spaces we need to work a little bit harder. Given a function M(0), 0 E (0, 1), we associate/15/(8), the largest logarithmically convex minorant of M(O), and the concave function r(t) = inf0 M(O)t ~ with limt-.0 r(t) = 0, limt_.~ 5(t) = 0. Thus, r t has a representation (cf. [8])
r(t) =/o~176 min{1,rt-}d#(r) and d#(v)(= -vdr'(v)) is called the representing measure of r.
18
C H A P T E R 2. B A C K G R O U N D ON E X T R A P O L A T I O N
E x a m p l e 3 If M(O) - 1, then r(t) = min(1,t) and dp(r) = 51(r) (51 denotes the Dirac measure at r = 1). If M(O) .~ 0 -~' as 0 0 and ..~ (1 - 0) -a2 as 0 ~ 1 for some c~,, o~2 > O, then / log"it
r(t)
[
for larget
(2.13) tlog "2 ~ for small t
and consequently
dp(r) ,.~
{
(log" 1-1r) dT~ as r -+ oo (2.14) (log,~-I 1) dr as r -+ oo
I f one of the a~s is zero it should be replaced by a 5 measure. For example if (~1 = O, then we may replace the term corresponding to the behavior at infinity by 51(r).
L e t l _ < q ( O ) < _ o o , ~ = l - q , ! and let n(O)= f o ~ t - e r ( t ) ~ m(O) = mq(O) = ql/q(q')l/q'(1 - 0)0
t-Or(t)
Then, if we assume that M(O) is tempered, in the sense that M(20) M(O) for O close to 0, and M(1 - 2(1 - O ) ) ~ M(O) for O close to 1, we have the following (cf. [57]) P r o p o s i t i o n 4 With constants of equivalenoe independent of 0 M(e) ~ re(O) ~ n(e) Proof. Since i('I(8) = supt>ot-Or(t), we use (2.26) with r = cr = 1, applied to K ( t ) = r(t), and we obtain
~'/(O) _< (1 -O)/~ n(O) On the other hand since M(O) is tempered we have, for t > 0, r(t 2) _< cmin{1, t}r(t).
2.2. MORE A B O U T THE ~ AND A METHODS.
19
We deduce that (1 - 0)0 n(e) <_ c fo ~176 t -e min{ 1, t 12}r(t 89 d-[~
,.dr = ~M(O). < cM(O) fo oo t -~ min{1,t~}T Finally, the equivalence
_/~/(0) ~ m(O) is now trivial since
1 _<
q l l q q , l l q ' < 2. []
We are now ready to show a powerful extension of Theorem 2. T h e o r e m 5 Let 7i be a mutually closed pair of Banach spaces, and suppose that M(O) is a positive, tempered function, and let {~'o}o
be a family of interpolation methods such that the characteristic function of .~e is equal to t ~ Then, Vt > O,
llxllEo(,OM(O)ao,,;~) ~ IlxllEo(,OM(O).(~)) Thus, in particular ~(M(O)Tio,1;a) = ~-~JM(O)~'(A)) 0
Proof.
(2.15)
0
In view of (2.9) it is sufficient to prove that
IlxllEo(tOM(O).~o,,;.,) ~ c IlxllEo(,oM(O)~o,~o,K) The first step is to observe that if d~t is the measure representing r(t) = inf0 M(O)t e, and the pair ii is mutually closed, then S F L implies that
IlallEo(t~176
~ fo ~176
(2.16)
For our purposes here it is enough to show that the left hand side of (2.16) is dominated by the right hand side. For each fixed t, the o,1;j has characteristic function equal to po,t(s) = functor t~ (~)~ Thus, by Theorem 2 we have oo
Ilall~o(~OM(O)~o.A~
oo
0 S
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
20
< c
inf
< c
inf
/5 J(s,u(s);ft) /o
min{1,
d#(r) ds
J(s,u(s);fiI)min{1,
d#(r)
K
,a;/t)d#(r)
where the last inequality follows from the SFL. W i t h this estimate at hand we may now continue with
K ( - a;ft)d#(r) = t o /J*, /5 t-~ r
_<
t ~ fo ~ sup{ t -~ I(( t a; 2,) )d#(~) = ct ~ t>O
Ilall~0 ~ , , .
r
< ct~
'
'
Ibiho,~,.
where in the last step we have used Proposition 4, and the fact that M(O) <_M(O). Thus, we have obtained the following estimate
IlallE(,OM(O)~o,,;,) <- ct~
Ilall~o.~;K
Consequently, by the definition of ~ ,
IlallE(tOM(O3~o,,;~) ~ cllallE(tOM(O)~o,oo;~.) as we wished to show. [] R e m a r k . In particular, Theorem 5 applies to .To = [., .]0, .To = (., .)0,q;/(, 1 _< q < c~. R e m a r k . Let the form
p(t)
be a quasi-concave function, then inequalities of ,HaHA0,
K(t, Ta;fI) <_ IlallAoPtllallA ) or
K(t, Ta; ft) <_cp(~)J(s,a,A) are termed K / J inequalities. These inequalities are basic limiting estimates in the theory. W h e n combined with the representation theory of quasi-concave functions and the f u n d a m e n t a l
2.2. MORE ABOUT THE ~ AND A METHODS.
21
lemma they allow us to reverse the interpolation process. Let us briefly indicate how they occur naturally in the theory (cf. also Theorem 7 below). Let A a n d / ~ be Banach pairs, and let {po}oeo, {ao}oeo be two families of quasi-concave functions. Set, v ( s , t ) = inf0 p0(,) ~0(t)' then the following two conditions are equivalent for an operator T (cf. [57])
1 T : ~lpo,i;J ~ Bao,oo;K, 0 ~ 0
K(t, Ta;[~) < r(s,t)J(s,a;fi,), s,t > O. We shall now illustrate these results with some i m p o r t a n t examples. For a q R, let
p~(t) =
{(
1 + log t
0 < t _~ 1 l
(2.17)
Suppose t h a t fi. is a pair of mutually closed spaces. Then, using Theorem 2, we see that for a _ 0, we have, with equivalence of norms, •0<0<1 (0 -a Ao,1;j) = Apa,1;J. (2.18) On the other hand, by Theorem 5, it follows that for 1 < q < oc,
~-]0<0
~Xpa,1;j
(2.19)
Still, the actual explicit characterization of the spaces t h a t appear to the right in (2.18) and (2.19) necessitates some further work. In particular it is i m p o r t a n t to rewrite the norms of these spaces in terms of more readily computable functionals such as K functionals. Recall that a Banach p a i r / l is said to be o r d e r e d if A1 C A1. For an ordered pair ft., a >_ 0, and 0 < q _< oc, we let A(,);K be the space of all a C A0, such that
Ilall(.);
= f0'[(i + log 1 )"-'K(t,a; AI]-~dt < c~
(2.20)
if c~ > 0, and for a = 0 we let fi-(@K = AL In a similar vein, we define the corresponding J spaces for ordered pairs fi.. The space fi.(.);j, a > 0, is the space of all a C A0 for which
II ll( ),j = inf
[(1 + log
i-
<
(2.21)
22
CHAPTER 2. BACKGROUND ON E X T R A P O L A T I O N
where the infimum is taken over all representations a = f o 1 u(t)Tdt with u(t) : (0,1) --~ A1 strongly measurable. W h e n ~ = 0, we let .A(,);j = Ag. If, moreover, .4 is mutually closed, then the S:FL implies that A(~);K = A(~);J, and in this case we shall drop the subscripts K and J. Note that when fi, is ordered, we may replace the integrals f o in the definition of fi,p:,i;J by f : for any finite/3 > 0. In fact, when/3 _> l, we have equality of norms. Returning to (2.18) and (2.19), we see that
Y:o
(2.22)
with
Ilalla(~)
f0~(l + log T)
K( ,a;
,,,
c~>O (2.23)
K(1,a;A) ~
a = 0
* t ) - - po(t), t There is a dual version of this as well. If we let p~( then by a direct calculation p*~(t) ,~infe(1-O)-~t ~ and we have (cf.
[57]) m o < 0 < x ( ( l -- O ) ~
= Apa,oo;K
(2.24)
for a > O. In fact, if fi, is mutually closed, then we also have Ao
= Ap~,,oo;K
(2.25)
(for a much more general result concerning the A method see Theorem 21 in Chapter 4). In particular, using the fact that K ( t , f , LX,L ~) = t f**(t), we shall obtain precise characterizations of certain extrapolation spaces associated with L v spaces. Let a _> 0, and let 12 be a probability space. Then, (2.26) = L(LogL)"(12),
2.2. MORE ABOUT THE ~ AND A METHODS.
23
and,
Al
(2.27)
More generally, and for use later on, lets us show how one can extend these calculations to more general Lorentz spaces (cf. [72]). For a concave function ~o : (0, 1) --* R+, let A~,(0, 1) be the Lorentz space defined by the norm IlfllA~ =
11f*(s)dqo(s)
For a pair of Lorentz spaces A~I , A~2, the K functional is computed in [71]
K(t,f,A~o,, A~2) =
f*(s)dmin{~ol(s),@2(s)}
(2.28)
Using (2.28), we get,
llfll(^,,,^~)
~l(log t-) 1.~ - 1 f0 1 f*(s)d min(~l(s),t~2(s)}~
For example, if ~01(s) = s, and : 2 ( s ) = 1, s # 0, ~o2(0) = 0, then A~,, = L 1, and A~,~ = L ~, and (2.28) gives K(t,f, LI,L ~) = tf**(t). Let us show a computation of extrapolation spaces associated with Lorentz spaces which is going to be useful later. Theorem
6 (cf. [721)
~_,{O-1L(LogL)I+~ Proof.
} = LLogL(LogLogL)(T)
Let us first give a detailed proof of the known fact t h a t
L(LogL)I+~
= [LLogL(T),L(LogL)2(T)]o
(2.29)
with norm equivalence independent of 6. Note t h a t the spaces L(LogL)~(T),fl E [0,2], have absolutely continuous norms. Then, for 0 e [0,1], we have isometrically (cf. [141, page 125)
[LLogL(T),L(LogL)2(T)]o = [LLogL(T)]I-~
~ (2.30)
24
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
To compute the spaces appearing on the right hand side of (2.30) we use another technique of.Calderdn [14]. First observe that, since we are dealing with spaces on a finite measure space, only large values of the Young's functions involved are important. In our case we shall take x > e ~. Using Calderdn's notation we write L(LogL) ~ = A51(L1), where An(x ) = x(log x)O,~ r [0,2], for x _> e ~. Although an explicit formula of A~ 1 is not readily available, we have that ~O(x) = x(log x) -0 is equivalent to the inverse for large x. More precisely, an elementary computation shows that
X
-~ <_c2z(Az(x)) < x, fox- x > ee,/3 e [0,2].
(2.31)
Then, according to [14] page 166, we have, with norm equivalence independent of 0 E [0, 1],
Col(/l) = (All(L1)) 1-0 (A2I(L])) 0 --~- [ L L o g L ] 1-~
[L(LogL)2] ~
(2.32)
where
r
:
(A11)l-~176
(x)
(2.33)
Therefore, combining (2.33), and (2.31), we obtain, for large values of x, and 0 r [0, 1],
r
~ (x(log X)-I) 1-~ (X(1og X)-2) ~ ~ X (1og X
(2.34)
Consequently, combining (2.34) with (2.32), and then with (2.31) once again, we obtain (7.15). Thus, by (2.18)
~{O-1L(LogL)l+~
-~
{f/
= (LLogL, L(LogL)2)(,);K =
1
~0 K(t,f;LLogL, L(LogL)2) dt
- - "~ (:X:)} t
2.3. RECOVERY OF END POINTS
25
Since the spaces LLogL and L(LogL) 2 are Lorentz spaces the K functional can be computed using (2.28) and we get
g (t, f; LLogL, L( LogL) 2) ,.~ ~, folf*(s)dmin{ foS(1Tlog l)du, t foS [l +log l]2du} It follows that, for t < e -1,
K(t, f; LLogL, L(LogL) 2) "~ 1 --3t
fo
e -- "'"~'--
f*(s)(1 + log
1)ds + t i e-1
__ ,I,"~3t
[
if(s) 1 + log
+]2ds
Integrating with respect to -~ yields
1))ds, tlflI(LLogL,L(LogL)2)O); K ,'~ fOe-1 f*(s)(1 + log 1)(log(log 8 from which it follows that
IIfll (LLo.LX(Lo.L):)<,;~
IIf lILLo.L(Lo,(LogL) )
as desired. [3 We shall give a general version of Theorem 6 in Theorem 48 below.
2.3
R e c o v e r y of E n d P o i n t s
An important aspect of the theory developed in [57] are the so called K/J inequalities which not only can be used to identify extrapolation spaces but allow the recovery of the end points, that is they allow us to reverse the interpolation process. This part of the so called "recovery of end points" theory developed in [57] and [58]. We shall now introduce the idea of complete families of extrapolation functors. Let us say that a Banach pair ti, is regular if A(+t) is dense in Ao and An. We say that a family of interpolation methods {.T0}0eo is complete if whenever Jt and/~ are pairs of mutually dosed spaces and j is regular, it holds that. T : 9v0(+i) -+1 ~'0(/~),V0
e
O
': T : +i -+/~
26
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
It turns out that complete families can be characterized by their action on one dimensional spaces. This is the content of the next result. T h e o r e m 7 (cf. [57]) A family of interpolation methods {~'o}oeo
is complete if and only there exists an absolute constant such that Vt, s > O (2.35) inf po(t) < min{1, t e po(s) where po denotes the characteristic function of the functor .To.
;}
Proof. We shall only consider in detail here the sufficiency of (2.35) and refer to [57] for a proof of the necessity of this condition. What we use is the fact that if a E A(fi,), then
IlallAo /ll~m_~_~l~ ~
Ilall~,,,1;~ -~
Po \ll,,lla~ J Since po(t) is increasing and ~ sup 0
is decreasing we see that
p,(t) Ilallao (H~Hao) < J(t,a;fi) po \11~11.,,1)
Combining these estimates we get
Ilall~(,~(,)~, oo~) : sup po(t)Ilall~,, =.,< ~ sup po(t)Ilall~(~) '
0
'
'
B
<_suppo(t)[[a][.~.ol. J < J(t,a;A) 0
'
(2.36)
'
Suppose now that T : 9vpe(fi,) 1 ~-p0(/~) ' and the pair/3 is mutually closed, then the condition (2.35) implies that HTaHa<po(oB~o,~;K) = sup, supo K(s, Ta; [ ~ ) ~
1
K(s, Ta;[~) 1 8 9 = -~J(t, Ta;[~) sup min{1, 7}
>_
2.3. RECOVERY OF END POINTS
27
which together with
and (2.36) gives
J(t, Ta;B) < cJ(t,a;A). Thus, since A(fi.) is dense in Ao and A1, we obtain T : A ~ / 3 . D C o r o l l a r y 8 Any {~'0}o<0<1 family of interpolation methods, with
each .To exact of exponent O, is complete. Proof.
Compute inf
po(t)
-
inf
Cet o
t
- min{1, "--}, s,t > 0, .s
and apply the previous theorem to conclude. 12 An interesting application of Theorem 7, given in [58], gives a proof of the following version of a theorem of Stafney, which was originally proved for the complex method of interpolation. E x a m p l e 9 (Stafney, cf. [8], [58]) Suppose that 7t is a mutually closed regular pair and [ao,Ax]oo = Ao, for some Oo E (0, 1), then Ao
= A1. In fact this result holds for much more general functors. The argument of the proof of Theorem 7 shows that if {.~'0}0eo is a complete family of interpolation functors, with characteristic functions Po, then, for any Banach pair fi. which is mutually closed, we have, Ilallar ~ J(t,a; A) Va E A(A), Vt > 0. Dually, a similar argument yields that, for a complete family of interpolation functors {.T'0}oee, and every Banach pair A, which is mutually closed and regular, we have
IlallE(~0(07e(a)) ~
K(t,a; A)
Va E ~(.A),Vt > 0 (cf. the proof of Theorem 5).
(2.37)
28
CHAPTER
2.4
The classical Setting of Extrapolation
2. B A C K G R O U N D
ON EXTRAPOLATION
It is instructive to return to the classical setting of the extrapolation theorem of Yano, as developed in Zygmund's book [104]. Suppose that T is a bounded linear operator on Lv(0, 1) for p > 1 with I[TIILp__,Lp = C9((p- 1)-~), as p ~ 1, for some c~ > 0; then these estimates can be extrapolated to T : L ( L o g L ) '~ --~ L 1. There is also a dual statement for operators T acting on Lv(0, 1) for p close to or with IITHLp__,Lp = C0(p~), as p ~ cx~, for some a > 0; then T : L ~ ~ E x p L 1/'~.
To prove the first half of Yano's theorem, we apply the ~ functor to obtain T : ~--~((p- 1)-~L v) ~ L 1 p>l
and conclude by (2.26) that T : L ( L o g L ) ~ --* L 1
Similarly, the second half follows using the A functor and (2.27). The reader will observe that, even in this classical setting, we have obtained a much more general result through the use of the abstract methods. For example, the assumptions of Yano's theorem imply that for any/~ > 0, we have T : ( p - 1)-("+~)L p -~ ( p - 1)-~L p so that by extrapolation T : ~'-~(p - 1)-("+~)L p --+ ~ ( p p:>l
1)-~L p,
p>l
and consequently T : L ( L o g L ) '~+~ ~ L ( L o g L ) ~
This is part of a general theory of " d i v i s i o n of i n e q u a l i t i e s " developed in [57]. We also observe that the classical proof of Yano's theorem, as given for example in Zygmund's book [104], does not use the full
2.4. THE CLASSICAL SETTING OF E X T R A P O L A T I O N
29
force of the hypothesis on the operator T. We shall now show that, if we insist on using all the information provided by the assumptions, we can recover, via extrapolation, very sharp "rearrangement inequalities" for classical operators. Let A be a pair of mutually closed spaces, and let d# be the representing measure of r(t) = infe M(O)t ~ then as a consequence of SFL we have, as we have pointed before,
IlalIE.(~M(O&~),;~ ) ~
~0 ~176 K ( -rt, a; fl)d#(r).
For example, if M(O) ~ 0 -1, aS 0 --+ O, and M(O) ,.~ (1 - O)-1, aS 0 --+ 1, then
IlallEo(,eM(e)ae,~o),;~)~ i0ooK(s,a;Ti)min{1, t_}--ds 8
8
(2.38)
In the special case of rearrangement invariant spaces these methods lead to rearrangement inequalities for classical operators including the Hilbert transform or more generally Calderdn-Zygmund operators, etc. For example, as is well known, the Hilbert transform is an example of an operator acting on Lv(/iT') satisfying
p2
IIHfllp < -
c
p-1
Ilfllp,
1 < p < 0%
(2.39)
Since the pair (L 1, L ~176 is mutually closed, and the L p spaces can be obtained by real interpolation
L p = (il,L~176 We see that, with 0 = 1 - }, we have
C 0(1 - O )
Ilfll("l,~)o,~(o);K'0 ~ (o, 1)
By extrapolation, we get
IIHallEoOO(L,,Loo)o.~+);~ ) <_ IlalIEooOM(O)tL,,L+)o.~O);K )
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
30
Using (2.37) we see that the left hand side is equivalent to K(t, Ha; L 1, L~176while the right.hand side can be computed using (2.38). Thus, we have
K(t'Ha;LI'L~176 <- Cffo~176176176 Now, using K(t, Ha;L',L we finally obtain
t--} s
~176= t(Ha)*'(t),K(s,a; i l , i c~
t(Ha)**(t) <_c
--~
sa**(s),
s oo s a**(s) min{1, st }dss
=C(fota**(s)ds+t/~176
(2.40)
The inequality (2.40) is a well known rearrangement inequality of Calderdn-Stein-Weiss (cf. [57]). It is actually well known that it is possible to improve upon (2.40) using the fact that the Hilbert transform is of weak type (1,1) (cf. [57] for a much more general result). More generally, if M(O) ~ 0-4(1 - 0) -~, say, then using (2.14) we obtain
/o
7)
K(s,a;ft){(log +_t 4 - 1 ) + (log+ ~ 9-a}min{1 ' s
t_}ds
8
,3
(2.41) Thus, if T is an operator acting on LP spaces, that satisfies
pC~+fl
IlTfllp < c (p_ 1) 4 Ilfllp, 1 < p < oo
(2.42)
We obtain the rearrangement inequality.
tT(f)**(t) _
2.5. WEIGHTED NORM INEQUALITIES
2.5
31
Weighted Norm Inequalities
We develop further the idea that starting from a family of estimates, like (2.39), we can produce very precise estimates for a given operator, including weighted norm estimates. For example, Sawyer [91] has given necessary and sufficient conditions on weights, for weighted L v norm inequalities to hold for certain positive operators acting on decreasing functions. In particular, he has characterized the weights (w0, wl) such for a given p C (1, co), the operator
Sf(t)= fo ~ f(s) min{ 1_ t ,~1}ds satisfies
tlSfllLp(,oa) <_cllfllL.(~o) for all f decreasing. Thus, if T is an operator satisfying norm estimates of the type (2.39), we get
K(t'Tf;il'i~
K(s' f ; Ll' L~176min{ ~ ' l -< Cfo ~176 = cS( K(s' f;L1,L~176 S
Now, since K(s)/s is decreasing, we see that, if the weights (Wo, Wl) satisfy Sawyer's conditions, we get
fo~[K(t, Tf;L1, L~)]PWl(t)dt < c f~[K(s,f;LX,L ~ ds t J0 s )]Pw~ Sawyer's results also give sharp conditions for pairs of weights (Wo,wl) that imply weighted norm inequalities of (LP(w0), LP(Wl)) type, for operators defined like the right hand side of (2.41) on decreasing functions. Thus, the analysis presented here will also produce results in the situation described in (2.42). The same argument also applies to other weighted norm estimates for the operator S (e.g. (LP(w0), Lq(wl)) estimates, etc.). Finally we note that this argument is very general, and independent of the LP scale of spaces, and produces very sharp extrapolation theorems.
32
CHAPTER 2. BACKGROUND ON E X T R A P O L A T I O N
2.6
M o r e C o m p u t a t i o n s of E x t r a p o l a tion Spaces
We indicate here a few more computations of extrapolation spaces in the classical setting of L p spaces, using modern technology. In [43] we show that for an ordered pair ii. we have
1 K(t, f, A0, fi~(o);K) ~- t jfe_§
K(s,f, Ao, A1) ds s
(2.43)
It follows that
1 -~ds < oo} (Ao, A(o);K)(O);K = { f : J01 K ( s , f ) 1--~-
(2.44)
8
E x a m p l e 10 Consider the pair fi~ = (L 1, L ~176over a finite measure space. We have shown that fi-(0);K = LLogL. Iteration using (2.44)
leads to (L 1, LLogL)(o);g : { f / JO1 sf*'(s) ]~_ -ds s- ~ o o }
$
= LLog(LogL)
We can continue computing...
(L1,LLog(LogL))(O);K = LLog(Log(LogL))
2.7
N o t e s and C o m m e n t s
For the most part the results in this chapter come from [57]. The counterpart of Theorem 5 for the A method will be established in Theorem 21. The connection with Sawyer's weighted norm inequalities seems to be new and complements earlier work in [90] on the use of weighted norm inequalities in interpolation theory. In the classical context of LP spaces, different extrapolation methods have been used by Kerman [64] to obtain estimates like (2.40). For a detailed study of Lorentz-Zygmund spaces we refer to [6] and the references quoted therein.
2.7. N O T E S A N D C O M M E N T S
33
Other aspects of the theory of K / J inequalities are developed in Chapter 3 in the context of embedding theorems, and in Chapter 5 in the context of bilinear extrapolation. We have not said anything about non-linear extrapolation. It is easy to see that one can extrapolate operators that are K - linear or quasi-linear in suitable technical senses. We now indicate less standard conditions under which we can extrapolate non-linear operators. In the context of lattices, Cwikel and Nilsson [30] have obtained versions of SFL where the functions in the representation have disjoint supports. A typical application to extrapolation is the following (cf. [27]) T h e o r e m 11 Let 7t be an ordered pair of mutually closed Banach lattices on a measure space (f~,~, #). Let {.To}oce be a family of exact interpolation functors, with each .To of order O. Let ~ > O, then there exists an absolute constant c, such that if f E Eo O-~.To(ft), then, there exists a representation of f as a series, f = ~ = 1 f,~, where the functions fn have disjoint supports, and
OX~ Ilfll~o.r
-< c IIfllEeo-,,:~or
n=l
for some sequence of numbers {On} in (0,1). This result can be used to prove extrapolation theorems for nonlinear operators that are additive for disjointly supported combinations of functions. For example, we can readily prove the following C o r o l l a r y 12 Let A be an ordered mutually closed pair of lattices, X be a Banach space, and {.T0}oeo be a family of exact interpolation functors, with each .To exact of order O. Suppose that T is an (not necessarily linear) operator, T : .To(it) ---+X , V0 e O, and such that (i) For some a > O, Vr > O, VO E O,
supllsll~o(a)<,llT fllx <_ cO-~
(ii) /.f f
=
T f = EncY.1T A
~-']~ne~
fn, and the f ' s are disjointly supported, then
34
CHAPTER 2. BACKGROUND ON EXTRAPOLATION
Then~ T : ~ 0-~o(+~) -+ X 0
and there exists an absolute constant c > 0 such that sup
IITfllx ~ cr
IlJll~e(e-o.~e(~))<- "
E x a m p l e 13 The following operators satisfy the conditions of the previous Corollary. Superposition operators u ---+ f(x, u(x)), such that f ( x , O ) = O, integral operators of the form
K f ( t ) = f k(t,s,f(s))ds, with k(t,s,O)=O N O T A T I O N : We shall at times, when it is not important to be precise about constants, and the spaces are equivalent, drop the subscript K or J when dealing with real interpolation spaces. Whenever dealing with families of spaces we shall assume that they are strongly compatible unless otherwise specified.
Chapter 3 K/J Inequalities and Limiting Embedding Theorems In this chapter we study K/J inequalities in connection with limiting embedding theorems. We focus on the replacement of the classical "power type interpolation inequalities" by analogues using quasiconcave functions, and the relationship of these inequalities with extrapolation. In the classical theory of weak type estimates or more generally in classical real interpolation theory it is easy to characterize the continuity of an operator from spaces of the form fi~0,1;J to any Banach space Z. We exploit the fact that the extrapolation spaces, obtained by applying the ~ method to a family of real interpolation spaces, are of the form fi~p,1;J, in order to give a simple criteria for the continuity of mappings from extrapolation spaces to any Banach space. When we apply this criteria in the setting of embedding theorems we obtain estimates which are proving useful in the theory of partial differential equations. For example, we give a new approach to the results of Brezis-Gallouet [10], and Brezis-Wainger [11]. Also recent results concerning end point inequalities for singular integrals by Kato and Ponce [63], Beale, Kato, and Majda [5] can be treated in this fashion. More importantly, the methods developed here are very general and can be applied to other operators and other scales (e.g. Besov spaces associated to semigroups of operators, which we intend
CHAPTER 3. K/J INEQUALITIES
36
to discuss elsewhere.) Our presentation here has been influenced and complements results in the recent monograph by Taylor [99] where, among many other things, the uses of K/J inequalities and their applications to the theory of hyperbolic equations is presented.
3.1
K/J
Inequalities
and Zafran Spaces
Let )~ be a Banach pair, and let {Xo}o be a family of spaces. Let us consider the familiar "interpolation inequalities" of the form Ilxll0 _< c011xllo~-~
~
(3.1)
It is a well known fact (cf. [8]) that (3.1) is actually equivalent to the embeddings 20,,;J C X0 Even in this classical setting there are interesting variants of (3.1) containing logarithmic factors. As a motivation let us look at a scale .Ao,q;Kof spaces obtained by real interpolation. Then, we can prove the following
Ilxlla,,~K< ~011~lla~,,~(1+
log 1/p'
II~llal )
(3.2)
Indeed, for ~ E (0, 1), write
Using the fact that sup, g{8,:;+~) < Ilxllal to estimate the first $ term, and HSlder's inequality, and the definition of the ffiO,p;gnorm, to estimate the second term, we obtain Ilxll~o,1;K --- c0[llxlI,A 1-~ + Ilxll~.,.,r(log ~ ) ' / " ] , rll~ll.%,p;r l and choosing )~ = t II~IIAI j~-o, we obtain
ll~llao,1~K < cell~ll.~,.,K(1 + log'/~'( -
llxll, ))
ll~lla~.~,~
37
3.1. K / J I N E Q U A L I T I E S A N D Z A F R A N S P A C E S
E x a m p l e 14 L e t ~ be a probabilityspace, Ao = La(~), Aa = L ~ ( ~ ) , then L(p, 1)(f~) = (Ao, A1)~,i;K, and we have __
]Og.1/pt(
JlfllL(p,a) < c l l f l l L ~ ( l + - - o
IlfllL'
~
"llfllL~"
Let A be an ordered pair, and let Z be a compatible Banach space, we consider K / J inequalities of the form ,
Ilxllz <_ c x
,,AoJ~,~) ~r
in connection with extrapolation. Associated with this type of estimate is a construction due to Zafran (cf. [103], [57]) which we now recall. Let Ay be defined as the closure of A1 in A0 with the norm I1.11r given by n
Ilxllf = inf{y~' laklf(llxkllA,):x = ~ akxk, and k=l
k=l
sup l
IlxkllAo ~< X} (3.3)
1 it is shown in [57] that Then, if p(t) = i(~),
~,p,1;: = AI
(3.4)
The only serious issue in the proof is the fact that if x E A1, then in the computation of the norm Ilxll~p.,;~ we only need to consider decompositions with only a finite number of terms. More precisely, we have that if x E A1, then n
I[xll~pl.s ~ i n f { y ~ J ( 2 - k ' x k ; r k=a P(2-k) ' '
n
: x = E xk, xk e A , } k=l
We now record the following consequence of this result. T h e o r e m 15 Let ft be an ordered pair, and let f : [0, cx~) ~ [1, ~ ) be a quasi-concave function such that limt.-,~o f ( t ) = c~, limt--,oo $(t) = t O. Let p( t ) = I--(~,)' 1 and let X be a Banach space. Suppose that T is a bounded linear operator T : A1 ---* X , then T can be extended to a bounded operator T 9f~p,1;g ~ X if and only if there exists an absolute constant c > 0 such that Y x E A1 I[X[IA~ IITxllx _< cllxllAof(llxllAo,
(3.5)
C H A P T E R 3. K / J I N E Q U A L I T I E S
38
Proof. Given the equivalence (3.4) we have that T can be extended to/lp,1;j if and only if there exists a constant c > 0 such that Vx E A~ we have ]lTx]]x < cl]xll I. Now, suppose that (3.5) holds. Let n x = ~_, akxk, with xk e A1, sup IIxkllAo< 1 k=l
l
and
laklf(llxkllAo)
~ Ilxllf
k=l
Then, 12
IIT~llx ~ ~ laklllTxkllx~ ~ k=l
laklllT~kllx
k=l
Ilxkllal
< c ~ laklllxkllaof( -- k=~ Ilxkllao ) n
<_c ~ laklf(llxkllal) k=l
Ilxlll where the last inequality follows using the quasi-concavity of f. Conversely, suppose that T : fi,p,1;J ~ X, then, in particular, there exists a constant c > 0 such that
IITxllx _< r
Vx e A1
Now, the decomposition x = II~IIA~ II~llAo shows that
~,llxllA1, II~llJ --- 9 a o J ~ ) and therefore (3.5) holds, t:] Combining the previous theorem with the characterization of the spaces -Ap,1;J as extrapolation spaces gives (cf. Chapter 2, Theorem
2). T h e o r e m 16 Let (X,Y) be a mutually closed ordered pair, Z be a Banach space, and let {9v0}0~(0,1) be a family of interpolation functors such that each .To is exact of order O. Suppose that T is a bounded
3.2. APPLICATIONS: SOBOLEV IMBEDDINGS
39
operator T : Uo(X,Y) ~ Z, with IITIIF~(Xy)-~z ~ ~(o), then for f E X N Y, we have [[Tf[lz <_ c[[fllxr( ~ where T(S) = info{ s~
(3.6)
)
}.
By Theorem 2, we have, with p(t) = sup0e(o,x){t~
Proof.
-x },
IITfllz < IlfllE(~(0)ao,~;j) ~ I flla.1 j and i f f E
XNY,
then
Ilfll~.,1;j < inf{ g ( t ' f ; x ' Y ) - ,>o p(t)
Ilfllx } - oc ~
v~ II.tllrj
Thus, if we let 1
_ i~f{s0~o(O)}
then
= Ilf XTr IlfllY~ ' II-f~x j as desired. []
3.2
Applications:
Sobolev
Imbeddings
In this section we consider applications to Sobolev imbedding theorems. Let us record first the following general remark E x a m p l e 17 Let fi~ be an ordered mutually closed pair, and let X
be a Banach space. We consider the functions (cf. the discussion below Theorem 2) p~(t) ~ (1 + log e/t)-% a > 0, t E (0,1), i.e. 1 p~(t) ~ ~(,~, where f~(t) = log~(e 2 + t). Then, b~ Theorem 15, and the characterization of the extrapolation spaces ft(~), we obtain that for a linear operator T : A1 ---} X,
IIr~llx ~ ~11~11~o(1 + log II~llAo'llxllA~ ~ ,
'.. T : A(~) ---} X
C H A P T E R 3. K / J INEQUALITIES
40
In connection with Sobolev spaces let us show a result from [10] (cf. also [57]) E x a m p l e 18 There exists an absolute constant c > 0 such that V f 6 W~(R2), we have
IlfllL~ < c(1 + Proof.
log(llfllw~(;~)))1/2 IIflIw~(R~)
(3.7)
Observe (cf. [57]) that the embedding i : [WI(R2), W~(R2)]0 ~ L~176~)
has norm n/2
IITxlIL~o(.-) < clIzlILoo(R-)(1 + log( llXllw~(R") )) IIxlIL~(R-) -
Note that, by Sobolev's embedding theorem, (L~176 W~(/~)) is an ordered pair. It follows, from Example (17), that we also have T : (L~(R"), W;(Rn))(1) ---+ L~C(R~) In [99] it is shown that the result of Example 19 is a simple consequence of the following K / J inequality T h e o r e m 20 (cf. [99]) Let ~ > O, and suppose that s > n/2 + 6. Then, there exists an absolute constant c > 0 such that V~ > O,
IlfllL~(--) < c~6]lfllw~(- -) + c(log 1)llflbooo(.. )
(3.8)
3.2. APPLICATIONS: SOBOLEV IMBEDDINGS Proof.
41
The choice ~ -- IIIII,o (R,) ' transforms (3.8) into II/llw~(R-)
IlfllLcct.,,) < llfll.o (R,,)(x +
log(llfllw tR.)/llfll.o (R,,)))
(3.9)
The relationship to Sobolev's embedding theorems can be now be made explicit. Recall that we have (cf. [8])
W;(R '~) C B~,~,~/2(R") 0 n c~ 1 B+oo(R ) L~176'~) 0 > 0
(3.10)
(cf. [93], Corollary 31). Then,
[B~
s
n
)]e C
0 n [Booo~(R ) , B ~s - h i 2 ( R n) ] e
=
FlO(s-n/2)(Rn" ~ --~oo ,--,
and by (3.10) we get c(s)O -1
[B~
W;(R'~)]e C L~176
Thus, (3.9) follows from Example 17 and we are done. [] R e m a r k . Using the results of [93] we can also state vector valued analogues. The corresponding results for singular integral operators can be also obtained in this setting once appropriate assumptions are made on the Banach spaces.
Chapter 4 Calculations with the A m e t h o d and applications Although in previous sections we have focused mainly on computations with the ~ method it is easy to obtain, directly or by duality, suitable analogues for the A method. In this section we consider some calculations with the A method which are motivated by applications to the theory of orientation preserving maps, and Sobolev imbedding theorems. In fact, our elementary calculations can be used to sharpen recent results by Mfiller [82] and Iwaniec and Sbordone [54] on the integrability of the Jacobian of orientation preserving maps. We also consider a sharpening of recent results by Fusco, P. L. Lions and Sbordone on limiting cases of Sobolev embedding theorems.
4.1
Reiteration and the A method
We shall not attempt to present here the most general results but simply develop the necessary machinery to carry out the calculations in our examples. Thus, the methods are more important than the specific results obtained. The following easy result, which is the analog of (2.3) for the ~] method, will get us started. Let A be an ordered pair, let O = (0, 1), let M(0) be a positive continuous function such that M(0) -# 0, V0 E
CHAPTER 4. A METHOD AND APPLICATIONS
44
O, and for any Oo C O, let Oo = (0o, 1), then
Aoeo(M(tg)fltS,q(O);K ) = AOEOo(M(tg)flte,q(O;K)
(4.1)
In fact, we trivially have sup M(0)Ilall~o ~o~K -< sup M(/?)Ilall~e ~ o ~ '
0EO0
'
'
0CO
'
'
while on the other hand, sup M(O)llallo,q;K < sup M(O)llalle,q;~: + sup M(O)llallo,q;K. O
O0
O~
Now, since the pair is ordered and sup0~(0,e0)M(0) =
C(Oo) < ~ , we
have
sup M(e)llalle,q;~ <_ C(Oo)M(Oo)-~ sup M(e)llalla,q;~: eeos=(o,e0) Oo
and (4.1) follows. It is interesting to observe here that we could have derived this result by duality from (2.3). In fact, it is easy to see that for a family of Banach spaces {A0}0Eo, with a common dense subset, we have (cf. [571) Ao(A;) = (~--~(Ao))*. (4.2) o
In particular, we have sup(M(O)-l/i;,co;g) = (~-~'~M(O)7ta,1;g)*, O
(4.3)
0EO
where A* denotes the dual pair (Ao, A1). Thus, we could use (4.3) and the duality theory of the real method of interpolation to establish many results for the A method from the corresponding ones for the ~] method. One drawback of this method is that it may require some extra assumptions on the spaces. Let us now give a direct proof of the following analog of Theorem 5. T h e o r e m 21 Let A be a mutually closed pair, and let M(O) be a tempered positive function on 0 = (0, 1). Then for any family of interpolation functors {.To}oeo, with .To exact of order t?, we have: sup(M(O)-I.To(A)) = sup(M(O) -1/]o,~;K) O
O
4.1. REITERATION AND THE A METHOD
45
Proof. The assumption that the pair A is mutually closed is required since we shall use the SFL (i.e. the fundamental lemma in its strong form). To be precise we require the following consequence of SFL: fixS,1;K C fixe,1;J (4.4) with norm independent of O E 0. To prove (4.4) proceed as follows. Let fo ~ min{ 1' -st}j(s'u(s))dss
a = f~o u(s)~, with < c K(t,a).
Then, Ilallae.~ <(1-o)Ofo = (1 -
oo
~<
s-OJ(s,u(s)) ds 8 ~0~176
J(s'u(s))fo
~
t } t_ odt ds
min{l's
t s
9)0 fo ~ fo ~ J(s,
u(s))min{1,sst}dst_edtt
_< c(1 - 0)0 fo ~
K(t, a; ]t)t-ed---~
= cllallae.,; as required. Let us also recall from Chapter 2 the fact that if M(6) is tempered then, with r(t) = info M(O)t ~ we have
M(O) ~ [(1 - 0)0] fo~~
d-~.
(4.5)
With these auxiliary results at hand we may now complete the proof of the theorem. It is clearly sufficient to prove that
sup(M(O))-lllall&~;~a < c sup(M(O))-~llallao,~;~ #
O
Now,
sup(M(O))-lllallao.K < c sup(M(O))-l(1 O
O
-O)e
fo~176
-ads 8
CHAPTER 4. A METHOD AND APPLICATIONS
46
r.s)s < c sup[(1 - e)O](M(e))-a f0~176 ( -eds sup ( K ( t , .a)'~
-
o
_
(K(t,a))
< csup
r(t)
t>o
-
s ,>o _
r(t) ]
(by (4.5)) 1
= c sup K(t,a)[ ] t>o info{M(O)t e } = c s u p K(t,a)sup{(M(O)) -1 t -~ t>O
0
= csup (M(O))-I sup K ( t , a ) t -~ 0
t>O
= c sup(M(O))-l[la[[xo.oo;K 0
and the result follows. [] Let us remark that the same calculation proves that IlallA(t~176
~ sup[K(s'a;,>o r(~) fi')]
(4.6)
where r(t) = inf{M(O)t~ E x a m p l e 22 As we have pointed out in (2.27) it is easy to see that
Al
4.2
On the Integrability Preserving
of Orientation
Maps
Recently it has been discovered that the Jacobians of orientation preserving maps, and other related nonlinear quantities, enjoy better integrability properties than those known for the Jacobians of standard maps. The first results in this direction were obtained by Miiller [82], and have been extended in many different directions by a number of authors including Coifman, Lions, Meyer, and Semmes
4.2. O R I E N T A T I O N P R E S E R V I N G M A P S
47
[19], Iwaniec and Sbordone [54], Brezis, Fusco and Sbordone [9], Iwaniec and Lutoborski [53], Iwaniec and Greco [44], and many others. These developments have interesting applications to the study of the equations of non-linear elasticity, variational problems, compensated compactness, etc. We must refer the reader to these papers for a detailed and complete treatment. In Chapter 7 we shall give a more detailed account of applications of commutator methods to compensated compactness. In particular, there we prove some higher integrability theorems for Jacobians. In this section we show the relevance of the A method in this theory. 4.2.1
Background
Let f~ be a bounded open set in / ~ , f : ~ ~ R n be a smooth mapping, we say that f is orientation preserving if its Jacobian J f = det D f is nonnegative a.e.. A typical assumption on the smoothness of f is that f is in the Sobolev class W2(f~, R~). Observe that by Hadamard's inequality J f < IVfll ..... I v A I , and therefore by HSlder's inequality we have that f E W2(f~,/~) implies that J f E Ll(f~). It was recently discovered by Mfiller [82] that if f is an orientation preserving map then one has a better result T h e o r e m 23 Let f~ be a bounded open domain in I ~ , n > 2, and let f : f~ ~ R n be an orientation preserving map in the Sobolev class W2(f~, R~). Then ~/K C ft, compact we have that J f E L(LogL)(K). This result has now been extended and applied in many different directions by a number of authors. In [19] Coifman, Lions, Meyer and Semmes point out the r61e of the Hardy space H I ( R '~) proving, among other things, the following T h e o r e m 24 Let f : R ~ --, / ~ , f E W I ( R ~ , / ~ ) , then J f Hi (/iY').
E
The relationship with Mfiller's result is given by a theorem of Stein (cf. [98]) stating that if g >_ O, then g E H~oc(R~) .' ~. g E L(LogL)toc(R'~).
C H A P T E R 4. A M E T H O D A N D A P P L I C A T I O N S
48
R e m a r k . In Chapter 7, Theorem 78, we give a sharpening of Theorem 24, for smooth maps, using commutator techniques. In a different direction Iwaniec and Sbordone give in their paper [54] sufficient conditions to guarantee the local integrabillty of the Jacobian of an orientation preserving map T h e o r e m 25 Let B C 3B be concentric balls in R '~, and let f : 3B ~ R '~ be an orientation preserving map. Then, f e AsE[1,n)(n- 8)WI(3B, R n)
)" J f C LI(B).
Here we denote by A,e[1,~)(n - s ) W I ( 3 B , R ~) the space of maps such that sup ( n - s)][fl]w2(a/3,R,~) < oo
sEll,n)
It is shown in [54] that
W~,~(LogL)-I(3B, R n) C A,e[1,n)(n- s ) W I ( 3 B , R n)
(4.7)
W~n,zo(3B, t~n) C /\sc[1,n)(n- s)W: (3B,/I~ n)
(4.8)
where for a function space X, and a domain ft, we denote by W} (f~, R n) the class of maps f = (fl, ...f~) with components such that Vf~ C X(aB), i = 1, ...n. C o r o l l a r y 26 Let B C 3B be concentric balls in R ~, and let f : 3B ~ R ~ be an orientation preserving mapping, f E W~n(nogn)_,(3B, R '~) U W~..~(3B, Rn), then, J f c L I ( B ) . More recently, the author obtained the following (cf. [741, [75]) T h e o r e m 27 Let f~ be a bounded open domain of R '~, and let f be an orientation preserving map of class W~n(LogL)O(f~, R ~) then J f C L'~(LogL) TM ( K ) , V K C f~,V0 C R. R e m a r k . The case 0 = - 1 corresponds to the result of IwaniecSbordone, the case 0 E [-1,0] is due to Brezis, Fusco and Sbordone [9], while the case 0 = 1 is due to Greco and Iwaniec
[44]
4.2. ORIENTATION PRESERVING MAPS
49
Although we cannot go into the details here we would like to at least illustrate the import of these estimates in the study of the weak compactness of the Jacobian map. For example, as an application of his theory, Mfiller [82] proves that if {uj}jeN is a sequence of orientation preserving mappings, uj : f~ --+ / ~ , and uj ~ u (weakly) in W,i(f~,/iV'), then VK C Ft compact, we have
J(x, uj)----" J(x,u) weakly in LI(K)
(4.9)
To see the connection with the LLogL estimates we recall the classical criteria of de La Vallde Poussin stating that for a set K of finite measure, a sequence {fj}jeN is relatively weakly sequentially compact in L 1(K) if and only if there exists a positive function 7 defined on R+ with l!m~--.oo7(x)/x = oo such that sup 3
f..7([fj(x)l)dz
< or
(4.10)
Now, it was known (cf. [4]) that if u~ ~ u (weakly)in W,~(f~,/~) then J(x, uj) ~ * J(x, u) weak" in the sense of measures. Therefore, under the extra assumption that the maps are orientation preserving, we may apply Theorem 23 and de La Vallde Pousin's criteria to obtain the stronger conclusion (4.9). Theorem 27 combined with a general form of de La Vallde Pousin's criteria can be used to obtain similar convergence results for the Jacobians of orientation preserving maps in the W~n(LogL)O(fl , R n) classes (cf. [79]). 4.2.2
Identification
of Sobolev
Classes
using
A
In this section we give a characterization of the spaces that appear in Theorem 25. For the benefit of the reader we shall give a complete approach to the relevant part of the interpolation theory of Sobolev spaces. Working with the coordinate components we shall reduce ourselves to work with ordinary Sobolev spaces. Thus, we consider the Sobolev spaces W~(f~) = W ~ ( f t ) = { f :
O~ E LP(f~),
k}
50
CHAPTER 4. A METHOD AND APPLICATIONS
where f~ is a regular domain (which for our purposes we shall take to mean that all functions in W~(fl) admit extensions to functions in W~(R~), i.e. the existence of an extension operator), k > 0, 1 < p < co, and the derivatives are taken in the sense of distributions. The corresponding norms are given by
Ilfllw#-- ~
I~l
IID~fll~ 9
The following result is due to DeVore and Scherer [34], but we shall present here the proof given by C. P. Calderdn and the author in [151. T h e o r e m 28 If f~ is a regular domain, then
K(t,f, Wlk(f~),W~(f~)) ~ ~
g(t,D~f, Ll(f~),L~
I~l
~, ~
(D~f)'(s) ds
I~l
D~f(x) = ~
D(~+Of(y ) ( x -l!y ) t q- R~(x, y)
lal
l!
R~(~,y) = ( k - I~1)
fo tk-I~l-lD(~+Of(x+t(u-x))dt
I~+tl=k or
(x - y)t
h~(x,y) = (k-I~l) I~+tl=k
1
t! f (1-t)k-I~l-xD(~+~f(u+t(x-y))dt 0
4.2. ORIENTATION PRESERVING MAPS There exists an absolute constant C > O, such that
L e m m a 29
51"l-klR.(x,y)ldy <_C ~
sup5 - ~ 6>0
-yl<5
supS-~/ 8>0
51
./l~-yl<8
M(D("+Of)(x)
ic~+/l=k
5'~l-kli~(x,y)ldx
M(D("+Of)(y)
]~+tl=k
where M is the maximal operator of Hardy and Littlewood. Proof. We prove only the first estimate. By a translation and a change of scale, one can reduce to the case ~ = 1, x = O. Then we see that everything is a consequence of the following fact: if g is a positive measurable function and we set h(x) = f~ g(tx)dt. Then,
flx.<_lh(x)dx <__c Mg(O) To prove this we use Fubini's theorem and a change of variable u) to derive
(tx =
flx,<_lh(x)dx = fol flx,<_lg(tx)dx dt
--~9~01t -n ~xl(t g(•)dudt < c ig(O). []
P r o o f of T h e o r e m 28. Since ft is regular we may assume that f~ = R ~. The estimate
K(t,D~I, L1,L ~) <_cK(t,I,W~,W~) [~[_
is obvious. To prove the converse we may, by an approximation argument, restrict ourselves to smooth functions. Let f E Cok, and let t > 0 be given. For each a E Z~., with ]a] < k , let
E~, = {x : M(D~f)(x) > [M(D~f)]*(t)}
CHAPTER 4. A METHOD AND APPLICATIONS
52
and E = Ul~l_
Ilfoollw~ ~ c ~ [M(D~f)]*(t). I~l_
[M(h)]*(t) < ch**(t), and K(t,h, L1,L ~176 = th**(t), we get
tilfoolIws < c ~ K(t,D~f, L1,L ~) H
(4.11)
Define fl = f - f~o, then
Ilfillwt <- ~ IID"f XEIIL1+ ~ I~l___k
IIO"foo XEIIL1= 11 + 12.
I~l
To estimate each term in the sum 11, we observe that
IIO~f XEIIL' <--foC*(O"f)*(s)d~ <_c (D~/)*(s)ds (since (D"f)**(t) is decreasing) =cK(t,D"LLI,L ~) Having obtained the correct estimate for I1 we turn to I2. Again consider a typical term of the sum I2, then
IID~f~ X~IIL1___ct[M(D~f)]*(t) < c K(t,D~f, L1,L ~) and therefore, collecting these estimates, we have obtained
IlAllwt -< cK(t,D~f, LI,L~176
(4.12)
In view of (4.11) and (4.12), we see that we have constructed a decomposition f = f l + f ~ , with A E W ~ , foo E W~, such that
K(t,f; W~, Ws < Ilfxllwt + t llfoollw~
4.2.
ORIENTATION PRESERVING MAPS
53
< c K ( t , D " f , L 1, L~176 The theorem is proved modulo proving our assertion about the function f - . Recall that we are required to estimate the Lip norm of f in E c. We start with the higher order remainders. Let c~ E Z~., with ]a] = k - 1, and let x , , x 2 E E c, we estimate D ~ f ( x ~ ) - D " f ( x 2 ) . To do so let r = Ixl - x 2 I , and let Qi , be cubes centered at xi, i = O, 1, with sides parallel to the coordinate axes with length equal to ~ r. Then, by simple geometry, there exists a constant c,, independent of the x~s such that [Q1 rl Q2[ > c . I Q i l , i -- 0, 1. Let Hi = {y E Qi: I n " f ( x ~ ) - n " f ( y ) l
> A(t)N},
r
where A(t) = ~l~l
[H1 U H2] < ]Q1 A Q2[. Thus, it is possible to find points in the set Q1 f3 Q2 \ HI U H2. Let z be one such point, then we can estimate [ D " f ( x i ) - D ~ f ( z ) [ <_ r A(t)N, i = 0,1 and therefore we derive the desired estimate
ID'~f(xo) - D'~f(xl)] <_ Ix0- x~lA(t)N Vx0, xl E E. The lower order remainders can be now estimated using the formulae (cf. [981)
IR~(x,,x=)l <_ IR~(xl,z)l
+
v"
z_., I~+q___k-1
X2)[ '
Ix1 - zl' l!
and we obtain the required IR,(xx,x2)l < C l x o - xxlA(t) It remains, thus, to prove the assertion about the measure of the union of the sets Hi. It is at this point that the maximal operators of
CHAPTER 4. A METHOD AND APPLICATIONS
54
the previous lemma appear. In fact, to estimate the measure of Hi we use successively Chebyshev's inequality and the lemma to obtain IHil ~_ [ ) ~ ( t ) / ]
-1 /
JQi
]D~f(x') - n " f ( Y ) l d y r
~ M(D("+Of)(xi)]Q,[ I~+Zl=k ~_ C N-11Qi[ (since xi E E c)
Therefore, if we choose N so large that -~ < ~ , we accomplish our goal of showing that I//1 U//21 < [QI N Q2[. concluding the proof. D T h e o r e m 30 If ~ is a regular domain, 1 <_ Pl < P2 <_ oo, 1- = 1 p:
1 then p2 ~
K(t,f,W~l(a),W~,(a)) ~ E K(t,D~ 1
I.L
~ I~l_
{jo'~[(D"f)*(s)] p~ds} v,
Proof.
I~1
,~
(4.13)
It follows from Theorem 28 that (Wlk(a), W ~ ( a ) ) ~,v;K = W~(a)
(4.14 /
and therefore the result follows by Holmstedt's formula (cf. [8]). [:3 We consider now the identification of the class of Sobolev maps that is described in Theorem 25. T h e o r e m 31 u E A s E [ 1 , n ) ( n - - s)W~(3B, t ~ ) if an only if the distributional derivatives of its components, ~Oxi ~ are such that r~
sup p ( t ) [ ~ J 0
,~(o,1)
,,,
*
\ ox, ]
n
(s)ds+t{
~
\ o~, ]
(s)
ds}~] < o0
(4.15/ where p(t) - t-1(1 + log 1)-1
4.2. ORIENTATION PRESERVING MAPS Proof.
55
Observe that by (4.1)
~s~E,,n)(~- s)Wl(3B, R~) = ~t+,n)(~- ~)W)(3B, Rn). Now, by (4.14), reiteration, and Theorem 21 we see that A~e[~,n)(n -
s)wi(3B, R ~)
= ~0>~(~ - O)(WI(3B, R~), Wi(3B, R~))0,oo+r By (4.13) we have
K(t,f, W)(3B, R~), WI(3B, R~)) n
~,j Jo
[ \ Ox~ ]
Therefore the desired characterization follows from Theorem 2. [] Using Theorem 31 it is easy to give a proof of (4.7) and (4.8). Let us consider, for example, (4.7). Suppose then that u C W~,,oo (f~, Rn), that is
s
i,j
We must estimate each of the terms in (4.15):
p(t)t{~_
, \Ox~] (s) ds}-~ ~ct-'(1
1 l lo l ljn<
+log 7)- t
and
p(t),o ['~
t(O'+)*(s)ds < _ c t-'(i +
the result follows by Theorem 31.
1
1
7)-t
C
CHAPTER 4. A METHOD AND APPLICATIONS
56
4.3
Some
Extreme
Sobolev
Imbedding
Theorems This section was motivated by recent work by N. Fusco-P. L. LionsC. Sbordone [42] on extreme forms of Sobolev imbedding theorems. The main result in [42] is the following T h e o r e m 32 Let f~ be an open domain in t ~ with ]f~] < r E wl'l(f~) satisfy for some a > 0
=
0 ~
' /o
<~.
,
Let u
(4.16)
Then, if ~ - n-l+a' there exist constants cl(n, o'), c2(n, (7) such that
ClM I~1~
-
In this section we discuss the relationship of this result with extrapolation. First, the reader should have no problems in verifying that
Ilull~
< ~ r
u E Aa(1 - 0) "-~-=~ (W~(~),Wi(f~))a,oo; K
(4.17)
Let M r = M~(~) denote the space defined by the condition f E M.~ if and only if n
sup p,(t)[~_,[ t"-' f'(s)ds + t{ [1. t~(o,1)
i
(f*(s))"ds} 1-] < co (4.18)
J~ ~-~--
.10
where p.y(t) = t-l(1 + log ~) 1 -~ . Then, using Theorem 31 (or more precisely its proof) we find that (4.17) is equivalent to the condition Ou ox---7 E M.-~+~,i = 1..,n.
Let us now review the proof of Theorem 32 following [42]. We start with the known estimates for the potential operator
dy I f ( x ) = /~ I z -f(Yln_l Y
4.3.
SOBOLEV
IMBEDDING
57
THEOREMS
<(1--~
1-5
[[Ifl[Lq -- k~_--~_~]
]F~]"~-6 ]]fIILp
~
(4.19)
w h e r e 0 _ < ~ = !p _ ! q < !. n It is easy to see that (4.19) implies that I : (L 1, L '~)0,p(0);K -'4 (1 - O ) "-~ Z - c ~ ( L 1, L
)0,q(O);K
(4.20)
Thus, applying the A method, we arrive to n--1
I:
Ao(ii,in)o,v(o);
K --+
which gives :
n
I : L '~ ~
If we m u l t i p l y ( ! )
1
Ao(1 - O)-~-c,~(L ,i~176
(4.20)by ( 1 - 0 ) ~
e L'-I
and t h e n apply A we get:
I : Ao(1 - 0)~(L 1, i '~)O,v(0);K~ Ae(1 _ O ) "-~ - - ~ + ;"c , ~ ( L 1, L~ Therefore we get n
I : M~, ~
e L"-I+"
n
Using the fact that
[u(x)l _< I(IDul)(x) the discussion above gives the Sobolev imbedding theorem
W ~ (fi) c ~L~--=~ n
Note that, when a = 0, = w
(fi)
and we obtain Trudinger's Sobolev imbedding theorem.
Chapter 5 Bilinear E x t r a p o l a t i o n A n d A Limiting Case of a T h e o r e m by Cwikel In this chapter we discuss bilinear extrapolation. We provide a general approach to the extrapolation of bilinear operators including an extension of the classical extrapolation theorem of Yano. We also illustrate the use of bilinear extrapolation as a tool in analysis. In fact, in our main application in this chapter, we treat in detail certain bilinear operators, with values on Schatten ideals, studied by Cwikel [28]. We show that we can extrapolate from Cwikel's Theorem certain estimates derived by Constantin [22]. In turn the results of [22] have interesting applications: they provide an end point to the so called collective Sobolev imbedding theorems (cf. [70], [100]) and are useful in the theory of the Navier Stokes equations. In this chapter we also show that the Macaev ideals are extrapolation spaces associated to the Schatten Sp scale. In fact we show, following [58], that the Macaev ideals are the counterpart in the setting of operator ideals of the LLogL and e L spaces in the LP scale. Using this insight it is possible to formulate extensions of Cwikel's theorem in terms of the Macaev ideals and some variants of them.
CHAPTER 5. THEOREM B Y CWIKEL
60
5.1
Bilinear Extrapolation
We start our program developing in more detail some ideas already outlined in [58] for bilinear extrapolation in the setting of LP spaces. For conciseness sake we shall refer to [57], and [58] for proofs of the statements, whenever possible. T h e o r e m 33 (cf.[58]) Let A, B, C, be Banach pairs, and let T be a bilinear operator, defined on A(fi.) x A(/~) with values on a pair C. Let M(O) be a positive function of O, 0 C (0, 1), T(t)= infoM(O)t ~ Then, the following two conditions are equivalent for T :
0 T:Ao,1;J x Bo,1;J M(_+o)Co,oo;K, 0 < 0 < 1, ii) K(t,T(a,b);C) <- - cinf f0~176 f ~ J(s,u(s);A)J(r,v(r)'[3~r( t--]dsd~ ~ J \st/ s r ' where the infimum is taken over all possible decompositions a = b = f3 o v(r)
r
"
In what follows we need to impose more restrictions on the growth then, of the function m(o). Consider the function T(t) = inf0 t~ if r is twice differentiable, we can write r(t) = f0~176 min(1, t-r)( _r2T,, r()) drr. Now, suppose moreover that (-r2r"(r)) is itself a quasi-concave function, with lim-r2~'"(r) t---~0
= lim - r 2 r " ( r ) t---,r162
- 0.
t
Then, there exists a measure d# such that oo r dr r(t) = fO man(l, vt-)f0~ man(l, s)d#(s) r More generally, let us say that M(O) is presentable if the associated ~" admits a representation of the form: (5.1) for some positive measure d#.
5.1. BILINEAR E X T R A P O L A T I O N
61
Let us observe that (5.1) can be equivalently rewritten as = f0~176 f0~ min(1, ~)(2 + log ~ ) d # ( r ) ~
(5.2)
Under this assumption, and through the use of SFL we can prove that the conditions of Theorem 33 can be sharpened as follows T h e o r e m 34 (cf. [58]) Suppose that the assumptions of Theorem 33 hold and moreover, A, B ave mutually closed pairs, and M(O) is presentable. Then, all the conditions of Theorem 33 are equivalent to K(t,T(a,b);C) < Cfo r f o ~ 1 7 6 Vt > 0, a E A(.4.), b E A(/~), for some universal constant c > O. Proof. Let a = f ~ u ( s ) - ~ , b = f ~ v(u~ , I ~ , be representations provided by the SFL, then, by a familiar argument, we get -tdsdu s u K(t,T(a,b);C) < c 9 [ o ~ ~ J(s'u(s);fi')g(u'v(u);B)r(~s) Inserting the representation of r in this formula, and making a change of variables, shows that the right hand side is equal to
oooo
oooo
f f J(s,u(s);7t)J(u,v(u);[~) f f min(1, t ) min(1, ~)d~t(y)~~: d, dya~.~ds O0
O0
Now, interchanging the order of integration and using the special properties of the representations chosen for a, and b, we get that the last expression is equal to oooooo
(x)
0 0 0
0
f f f J(s,u(s);Ti) f J(u,v(u);B)min(1, t ) ~ m i n ( 1 ,
~)d#(y)3~ d~ds ~-
CHAPTER 5. THEOREM B Y CWIKEL
62
g(s'u(s);ft)min(l'sy
s
x b;[3)d#(y)dXx
x
=c
K(x,a; i)K(xy
dxx
As we wished to show. [] E x a m p l e 35 The previous discussion gives us a criteria for deciding which growth functions M(O) are presentable. For example, according to Example 3, if M(O) ~ O-~,a >_ 1, then log '~t t > 1 t t< 1
T ( t ) ,.~
so that we have -t2r"(t) ~ { al~ a-1
t - a(a -
1) l o g ~ - 2 t
0
t > 1
t
and we see that --t2r"(t) is quasi-concave. Therefore, M(O) is presentable in this case. E x a m p l e 36 A minor variation of the previous example is given by growth functions of the form i ( O ) ~ 0-"(1 - O ) - ~ , a , fl >_ 1. We claim that these growth functions are presentable. Let us consider the ease where a = fl in detail. Let Mk(O) = O - k ( a - 0 ) -k, k = 1, ..., and let rk(t) be their associated functions defined by rk(t) = info t~ Then,
rk(t) ~ min(1,t)(l + [logtD k.
(5.3)
In fact, in a less precise fashion, (5.3) has already been indicated in Example 3. For a proof, we check, by computation, that (5.3) actually holds for k = 1. Then, we see that the estimate is actually ,~ 1 k valid for all k, since Tk(t) ... (rx(tk)) . In the case k = 1, its readily seen that we actually have the representation r~(t) =
~oo
t min(1,s)dS. rain(l, s) s
63
5.1. BILINEAR E X T R A P O L A T I O N Consequently, we obtain the desired representation as follows
rl(t) =
min(1, ~) ~176
/o
7o
)d6x(r)dSs
with 61(r) the 6 measure at r = 1. Note that to(S) obtain, inductively, the representation
k > 1, we
k(t) fo
= min(1,s).
For
min(1, t)rk_l(s) ds. S
S
Therefore, by (5.3), Tk(t) ~
f0r162 rnin(1, !) rain(l, s)(1 q-II~
and we finally obtain the representation Tk(t)
~
fo
,
min(1, s)fo min(1, s)[(log+ s )k-2
,
o,k 2]_7
Similar, elementary, but more lengthy and tedious, calculations, which shall be left to the interested reader, allow us to show that the growth functions M(O) ~ 0-~'(1- O)-0, a,/3 >_ 1 are presentable. Moreover, the corresponding r~.O (t) = infe t~ - O ) -~, can be represented by O0
ra,o ( t )
[j~176 _ min(1, s)t f min(1, s)[(log+ r ),-2 + ( log+ -rl /~ - 2 ] d r_r d ss o
o
O, say, then oomin(1, ;t) min(1, s)(1 + log+ r) ~-2 drr dss
On the other hand, if/3 = r~(t) <_ c
and consequently( 4 (5.2)), r ~ ( t ) < - C f o ~ m i n ( l ' ~ )(2+
l~
) (1 +l~
drr
Lemma 37
(5.4)
Let A, B, be mutually closed pairs, and let T be a bilinear operator, defined on A(ft) x A([~) with values on a pair C'. Suppose further that T : ]to,1;a x/~o,1;J M__.~(~)O0,oo;K, 0 < 0 <
1,
CHAPTER 5. THEOREM B Y CWIKEL
64
with M(O) ,~ 0-" as 0 ~ O, (r >_ 1. Then, K(t,T(a,b),C) <_ c f ~ K(s,a; 7i)K(t,b , 9[ ~j ,
o~=1
cf~Of~OK(s,a;~)K(~,b;[~)(1 + l o g + r ) ~-2 ~,~r c~> 1 Proof. Let T,(t) = infe teO-", then by (5.4) and Theorem 34 we have, if a > 1, oo
oo
K(t,T(a,b),C) <_ c f f K(s'a;ft)K(t'b;[~)sr
(1 + log+ r) ~-2 drr dSs
0 0
On the other hand, when ~ = 1 we have
K(t,T(a,b),C) < c
K(s,a;2t)K(
,b;B) ds $
as desired. D We shall now state and prove a bilinear version of Yano's extrapolation theorem. Theorem
38 Suppose that T is a bilinear operator such that
T
: Ae,i;J
x B0,1;J M__~(~)Ce,oo;K, 0
< 0 <
1,
with M(O) ~ 0-" as O ~ O, a > 1. We also assume that the pairs A,B, and C, are ordered and mutually closed, and moreover that A, B, are regular. Then, T: A(~,) x B(~) --~ Co Proof. We consider first the case cr -- 1. We compute, using L e m m a 37,
IIT(a,b)llCo = K(1, T(a,b);C) < c]o~176
b;B )"ds T
5.1. BILINEAR EXTRAPOLATION
65
We split the last integral in two parts: from 0 to 1, and from 1 to o% and we estimate each of them. Then, we obtain,
/01K(s,a;ft)K( ! ,b;[~) ds- ~
/01K(s,a;ft) ds
IlbllB0
,S
8
Ilblls(,llall~(, and,
f~K(s,a;fl)K(~,b;B)dST -< Ila]lAo f~K(~,b;B )T ds Ilall~(,>llbll~(,) as required. Consider now the case a > 1. To proceed we need to integrate by parts. Observe that, for an ordered pair, the associated K functional of any element is constant for t _> 1, therefore the derivatives of all the K functionals involved will be zero for t > 1. Moreover, by the regularity of the pairs A, and/3, we ha~ee that the K functional, for any element in the corresponding spaces, satisfies lim~0 K(t) = 0. Also, to simplify the notation, we shall let k(s) = d(K(s)). By Lemma 37, we have
IIT(a,b)llco = K(1,T(a,b);C)
<
~K(,a;A)
(log + r ) ~ - 2 K ( , b ; [ 3 )
drds - I . r
8
Changing variables I =
/0 K l! ,a; ;/s
log
Integrating by parts, and using the fact that we have the estimate
b;B) duds
o,
k(u, b; [3) = O, u > 1,
) k(u,b;[~) dudss I
CHAPTER 5. THEOREM B Y CWIKEL
66
now, interchange the order of integration and repeat the same steps to deal with g ( 1 , a;.A), we get
I <_ c
/01
k(u,b;B)
i01
k(s,a;Tt)(log +
)ads.
The estimate (log + 5 ) 4 _< (1 + log })~(1 + log 1;)~, implies
s _
1
+log
1)~d s
which again integrating by parts can be seen to imply
I < cllall(~)llbll(~) as we wished to show. o As an example we write down a bilinear version of Yano's theorem, E x a m p l e 39 (cf. [58]) Suppose that T : LP(0,1) • LP(0,1) --+ LI(0,1), with norm Mp <_ c / ( p - 1 ) " , a > 1, as p ~ 1, then
T : LLog"L x LLog"L ---+L 1. As we have seen before a convenient way to summarize the information of the interpolation/extrapolation process is via K / J inequalities. For a bilinear operator T a K I d inequality, with function T, is an inequality of the form:
K(t,T(a, b); ~)
llall~ Ilbll~ t) <_ Ilall011bll0T(iTg~,, Ilblll'
Let us state, without proof, an equivalent condition to those in Theorem 33 in terms of K / J inequalities T h e o r e m 40 (cf.
[58]) Any of the conditions of Theorem 33 is equivalent to the K / J inequality K(t,T(a,b);C) < T ( t ) J ( s , a , Ti)J(s,b,B)
5.2. I D E A L S OF O P E R A T O R S
67
Using these methods one can , of course, deal with more general decay rates than powers. We shall not, however, pursue the matter any further here (cf. [58]). It is also interesting to observe that these methods allow us to prove a somewhat more general version of the usual bilinear interpolation theorem of Lions-Peetre-Zafran. E x a m p l e 41 (cf. [58]) Let A, B, C, be pairs of spaces with i mutually closed, and let p,a,7, be quasi-concave functions. Suppose that there exists a constant c > O, such that k / s , u > 0, we have p(s)7(u) < 1 1 _ ca(su).Supposefurtherthat ;1 = ~+~_-1, 1 <_p,q,r < cx). Then, if T is a bilinear operator such that T : A • B --* C, we have T : -4p,v;g • /~-y,q;J ~ C'~,~;K. The point is that the assumptions on T imply, as it is readily verified, the K / J estimate
J(us, T(a,b);C) < cmin{1,
5.2
Ideals
of
US
~J(u,a;A)J(s,b;[~).
Operators
Let H be a Hilbert space, let So~ be space of bounded operators from H to H. The Schatten ideals of operators Sp are defined as follows. A compact operator T E So~ is in the Schatten ideal Sp, if the sequence of eigenvalues {s,(T)}, arranged in decreasing order, of ( T ' T ) z/2, belong to lP. In this case we write
IITII
= II {s.(T)} I1,
Similarly, we say that T 6 Sp,oo if {s,(T)} 6 l(p, cx)). In the literature, a class of operator ideals, the so called Macaev ideals, has been singled out and studied, as suitable end point ideals for the scale of Schatten ideals Sp. The Macaev ideals are defined as follows S~
--
{T 6 S~o: ][T][s~
.:
sn(T)
s n=l
< oo}
n
and
SM = {T 6
Soo:
IITIIs = sup m >--l
E,~"=I s~(T) m 1 < Ert=l
-n
CHAPTER 5. THEOREM B Y CWIKEL
68
Now, it is well known that (cf. [651) [t]
K(t,T;S~,Sco) .w. ~_, s,(T)
(5.5)
where It] is the integer part of t. We want to work with the ordered pair (Sco, $1) and therefore we write
K(t,T; Sco, S1) = tK(1, T; S1,Sco) = t ~_, an(T) and therefore dt
1 [~]
folK(t'T;Sco'Sl)-t -= fo E s n ( r ) d t n=l
n=l
1 ~ s,~(T) m=l - ' ~ n=l co
co
1
= ~_, s.(T) ~_, m2 n--~l
)'tl,-~'n,
=_ ~ s.(T) n=l
n
In other words, SW :
(5.6)
(SCO, S1)O,1;K
Now, after the usual renormalizations, we have,
Thus, combining (5.6) with the theory of Chapter 2, we have (with p = 0-1)
(S~, S,)o,v(o):N F
#
= S~
5.3. L I M I T I N G CASE OF C W I K E L 'S E S T I M A T E
69
Moreover, if p(t) = t(1 + log 1), t < 1, then sup K(t, T; Soo, $1) _ sup t ~t§ ~n=l
,<_1
p(t)
~<_1
~n(T)
p(t)
-IITllsM
which combined with extrapolation theory gives A v ( p - 1)(Soo, S1)~,v:K = A0(1-O)(Soo, S1)o,v(O}:K = SM Thus, the S,~, and SM ideals play the role in the theory of ideals of the LLogL and e L classes. The following result is a consequence of the previous discussion.
Theorem 42 (cf. [58]) Let T : Sp ---+ Sp, then (i) if IITIIs,__+s,, < p a: p --+ oo, th~,~ T : S,. ~ So~ (iOif IITIIs,-s,
~< (p - 1)-1 as p ---+ 1,then T : Si ~ SM
Of course we can consider other extrapolation classes of ideals of operators to accommodate different rates of decay. This could be used, for example, to derive an extension of Cwikel's theorem in the next section. It is also interesting to remark that the theory of ideals presented here can be carried out in the setting of ideals of compact operators on arbitrary Banach spaces.
5.3
Limiting
case of Cwikel's estimate
We now turn to our main application. We shall extrapolate from a theorem due to Cwikel and derive from it the end point estimate obtained by Constantin [22]. Our approach suggests further extensions of Cwikel's theorem. We shall be dealing with L(p, q) spaces. In what follows we use the notation II.llp,q to denote the usual Lorentz norms, and II-II;,q to
C H A P T E R 5. T H E O R E M B Y C W I K E L
70
denote the usual quasi-norms defined using the distribution function or the rearrangement instead of the double star. We consider operators acting on L2(Td), defined by
Tf,g(h) = 9 ( ( - ~ r ) V ) ( f h ) This means that Tf,g is formally defined on (q0,r by the inner product of f ~ and (gr The main result of [28] states t h a t for 2 < p < o0, we have Theorem
43 The following estimates hold
IITs,.llsp, with % = ~ ~-~-~-1,
___ llfllpllgll;,oo,
,- + ~-2) p"
In order to formulate this result in term of interpolation spaces we need to be very specific about which norms we are actually considering. Using (5.5) and the reiteration formula of Holmstedt we see that the K functional for the pair ($2, Soo) can be c o m p u t e d by (cf. [651): 1
K ( N , T ; S 2 , S~) ~ Thus, we can write Sp,oo = ($2,
IIHIIsp,
(5.7)
that is
Soo)O,oo;K,
= sup{ N - ~ N
sk(T) 2
sk(T) 2
}, 1 - - = 0.
(n=l
P
Let us also write Lp,Oo = / r~2~ , o o , roox ~ )O,oo;K
,
/9 = 1 -- -2
p
Then, since
[Ifll(L2,oo LOO)o.oo;K >
cllfll;,oo
where 9 = 1 - _2 p, we see that the estimate of T h e o r e m 43 implies
IITf,.ll(s2,soo)e,o~;.~ <_ callfllcZ~,Loo)o,oo;.r162
5.4. N O T E S A N D F U R T H E R R E S U L T S
71
where co = (1 - 0)-'(4(1 - 0)0-1)~ + (1 - 0 ) 0 - 1 ) (1-0)/2. Summing up, when 0 ~ 0, we have arrived to the estimate
IITs,~ll(s=,s~)o,=,,< <
O-89
(5.8)
Taking the infimum we may now obtain the K / J inequalities for the bilinear operator T.t,a 9 The corresponding function r is defined by
r(s,u,t)=
inf {0-1/2(t--) 0e(o,1) su
~ < 1+
el/:V/21og~_/:(t)
(5.9)
-
Therefore (5.8), (5.9), and Theorem 40, give
K (N, Ts,g , $2, Soo) < cllfll211gllL2. ~ { 1 + log~_/2(NIIfll~176176176
Ilfll~llgllL~,= ) }
-
and recalling the estimate for the K functional for the pair ($2, Soo) given in (5.7) and making the change of variable N --+ N 1/2 we arrive to
N
{ ~ , sk(Ts,g)2} 1/2 < cllfll211gll2,oo(1 + i=1
Ilflloollgll~ ~} log'+n(Nlnllfll~llgll~, ~,
(5.10) This is the statement of Constantin's theorem in [22] except for the fact that he states his theorem for vector valued functions. However, since Theorem 43 extends mutatis mutandis to the vector valued setting we obtain, likewise, the full strength of the result of [22]. The reader of these notes will have no problem in deriving the end point version of Cwikel's theorem that can be obtained from (5.10) (or equivalently (5.8)) and can be framed in terms of variants of the Macaev ideals discussed in the previous section.
5.4
Notes
and
Further
Results
The relationship between extrapolation and approximation theory is considered in [57], [18] and [58]. For a pair r we let
F(t,a;7t) = f oo if IlallA, > t IlallA0 if IlallA, < t t
C H A P T E R 5. T H E O R E M B Y C W I K E L
72 and define
Ao,+;F={Z:z--
u(s
, F(~,u(s);A)<~;llxllao,~,F < ~ }
with 1
These spaces coincide with the usual approximation/interpolation spaces for the usual range of the parameters. At the end points of the scales we have modify the definitions somewhat. Working with a "discrete" definition we let, for an ordered pair A, 1
Ilxll~o,~;~ = inf
2n(F(22",un)) q
: x = ~ u,.,, F(2
2n
,u,.,) < o,z}
and
Ao,q;F = {x: Ilxll~o,,;F < ~ } Similarly, for the E functional defined by E(t,x;7t)
= inf{llx - xlllao : IlxllIA1 ~ t}
we let fi, o,q;E = { x :
n - l ( E ( n , x ; ft)) q
< c~}
Then (cf. [18], [58]), ~lO,q; E ~ fftO,q; F
We cannot go into the details here and we refer to [58] and the references quoted therein. For example, in the setting of operator ideals these results imply a description of the elements of Macaev ideals in terms of the rate at which they can be approximated by operators of finite dimensional rank. For example, oo
oo
S~o - { T : T = ~ Tn, with rank(T,,) _< 22", y~ 2 n [[Tn[[s~ < oo}. n=l
n=l
5.4. N O T E S AND FURTHER RESULTS
73
However, the point we want to make is that the formal analysis works in general to describe extrapolation spaces in terms of degree of approximation by suitable subspaces. Thus, for example, using appropriate descriptions of Besov spaces as approximation spaces (cf. [8]) one can characterize extrapolation spaces for Besov scales in a similar fashion (for example operators of finite dimensional rank could be replaced here by entire functions of exponential type, etc). We think that these descriptions could be useful in the study of compression of wavelet decompositions and its applications to image compression. In this setting the subspaces from which we approximate are finite dimensional spaces generated by wavelets. For references to work in this area we refer to [38], [35] and [59]. There is forthcoming related work by Houdre (cf. [51]).
Chapter 6 Extrapolation, Reiteration, and Applications Although, as we have stressed in these notes, extrapolation spaces can be incorporated in an extended theory of interpolation spaces, they enjoy special properties which distinguish them from the classical spaces introduced by Lions and Peetre. A central issue in interpolation theory is the idea of reiteration. In this chapter we give several general reiteration theorems, for extrapolation spaces obtained by the ~ method, which are motivated by applications to classical analysis. In particular, we consider an application to the study of the of the maximal operator of partial sums of Fourier series and show that certain limiting inequalities can be obtained through a combination of reiteration and extrapolation techniques. We also consider applications to the theory of logarithmic Sobolev inequalities. We show how to derive the so called higher order logarithmic Sobolev inequalities from lower order estimates, and we indicate new estimates for scales of spaces closer to L 2 than the L2(LogL) '~ scale.
6.1
Reiteration
In this section we consider some simple, and useful, reiteration properties which are needed to treat our applications.
76
CHAPTER
6. R E I T E R A T I O N ,
AND APPLICATIONS
Let us recall a characteristic property of the ~ functor:
0
0<0o
where fi, is an ordered pair, Y:'0 exact functor, and - ~1 E L + . Moreover, if the pair +~ is mutually closed then we have,
E (tO(0)')E'0(A)= E ~:)(0)f~0,1;J = Ap,1;J, 0 0
te
with p(t) = sup --~. In particular, if ~(O) = 0 - " , with ~ > 0, we have p(t) = p , ( t ) (cf. (2.17)), and
E ( o-etA0,1;J) = E (0-c~fi0,1;J) ----~tpa,1;J -'~ Apa_1,1;K O 0
= (Ao, Aoo,1;J)pa,1;J Thus, combining the results above, we get Ap<2,1;J =
Ap~t_l,1;K
=
(Ao, ASo,1;J)pa,1;J
= (Ao, AOO,1;J)pa-I,1;K
More generally, for any 0 < 0o < 1, 0 < q < c~, = A o_1,1; = (Ao,
= (Ao, A0o,q) o,1;:
AOo,q)pa_l,1;K
This type of constancy was discovered in [43], in the context of the K method: Vq E (0, oo], V0 E (0, 1), (A0, A1)(1);K ~---(Ao, A0,q)(1);K
(6.1)
The novelty here is given by the fact that we are allowed to vary both parameters: (0, q) at the same time. Recall that in the classical
6.1. R E I T E R A T I O N
77
reiteration theorem (cf. [8]) we have that, if 0 < p,q, r <_ co, then V
e (0,1), ( Ao, A0,q) ,;p = ( Ao, ,3,0,~),;p. The corresponding limiting property for the J method has been known for a long time, namely Vq E (0, co], V0 E (0, 1), (A0, A,)0,1;j = ( Ao, fto,q)o,,;j = A~) Results of this type are very useful in the actual computation of extrapolation spaces as we shall show in this chapter. It will be convenient in what follows to use the following notation. Let A be an ordered pair, 0 < p _< co, then we let Ao,p;g = {x : x E Ao,
{/oI[K(t,x; fi)] p }i/p
thus Ao,p;g = -3-0);g, as defined in (2.20). E x a m p l e 44 Considerthe following proof (cf. [43]) of a well known interpolation theorem by Zygmund: If T is a linear operator defined on LI(0,1) which is of weak type (1,1) and of type (p,p) for some p > 1, then T maps LLogL into L 1. In fact, by hypothesis T : (L1,LP)o,t;K ---+(wLl,wLP)o,1;g Now, by the constancy property (L 1, LP)o,1;K = (L 1,/~)0,1;K
=
LLogL
while it is easy to see that
(wL 1, wLP)o,1;K C L 1 In a similar vein one can extend to general interpolation scales the following interpolation theorem by O'Neil [84]. E x a m p l e 45 I f T is a sublinear operator of weak types (1,p), (q,r), with 0 < p < r < co, 1 < q < co, then for 0 < 0 < 1, T : L(LogL) ~ ---+Lp,1/o.
78
CHAPTER 6. REITERATION, AND APPLICATIONS
Proof.
We apply the
(.,.)0,1;K
m e t h o d as in Example 44. Then,
since
(LP'~176176176
=
(LP'~176176176 C L p'I
we get
T : LLogL ~ L p'I Now, this estimate can be interpolated once again with T : L 1 ~ L p'~176 and we obtain
T . (L1,L(LogL))o,1 ---, (Lp,~176Lp,1)o,1
But, since (cf. Iv]) L1,L(Logn))o,1 = L(Logn) ~ and (Hhlder's inequality),
(L p'~, Lp'I)o,1
C L p'l/O
we are done. [] We recall other related end point reiteration theorems from [43]. Let A be an ordered pair of (quasi-) Banach spaces, then it is shown in [43] that
I K(u,f;7t) -~du K(t,f;Ao,(do, ml)o,1;K) ~ t jfe_~
(6.2)
and
I((t,f;(Ao, A1)o,1;K, A1) ~ / 1 K ( m i n { ~ - l ( t ) , u } , f ; f l ) d u dO
U
(6.3)
where c2(t) = t log(e/t). We now prove a more general version of (6.3) which we require for our application to logarithmic Sobolev inequalities. We refer the reader to [43] for a proof of (6.2). T h e o r e m 46 Let ft be an ordered Banach pair, 0 < p <_ oo,then 1
K(t,f;~io,p;K, A 1 ) ~ { f o l ( K ( m i n { ~ - l ( t ) , u } , f ; A ) ) P ? }
where
t[log
;
6.1. REITERATION
79
Proof. The proof is patterned after the corresponding one in [43]. Let f E Ao, t > 0, and let
L(f)(t)=
{/:
-}U
1
(K(min{~-x(t),u},f;Tt)) "du ~
Suppose that f = fo + fl, fi C Ai, i = O, 1, then
L(f)(t) <_ L(Io)(t) + L(I~)(t). Now, since the K functional is increasing we trivially have 1
L(f~
<- fo [K(s'f~
-
ds
= [[fH~-o.,;K
To estimate the second term we use the trivial estimate
K(s, fl; ft) <
sHfl[[A1, to obtain,
L(fl)(t)_ IlflllA1
(min{~o-l(t),u}) '
U
Splitting the last integral appropriately yields 1
= p-pqo- (t), +
(t)log(qo_,[r))~
Now, since ~--l(t)[log (~)]l/p
=
(qo--l(t))l--1/p[~o--l(t)log(e--'---~--]]l/P~_l (t),I./
= (~-'(t))'-'/vt lip <_t (where have used in the last step that ~ - ' ( t ) < t), we get
L(f~)(t) <_ctllk[lA, Consequently, we have obtained
L(f)(t) <_cK(t, f; fi*o,p;K, A1)
80
CHAPTER 6. REITERATION, AND A P P L I C A T I O N S
Conversely, let f = fo(t) + f~(t) be an optimal decomposition of f for the pair A, i.e. such that K ( t , f ; 7 t ) ~ Ilfo(t)llAo + tllfl(t)llA,. We claim that the decomposition f = fo(qp-l(t))+ fl(qa-l(t)) is optimal for the pair
(Ao,p;K,A1).
In fact,
fo(~O-' (t))[-~o,,;K = I + I I where
1
1 II =
1-1(0 K ( s , f o ( q ~
p
p
Writing fo(~-x(t)) = f - fl(~o-l(t)), and using the triangle inequality, we get I < L(f)(t)+
{So
1
K(S, fl(~o-l(t));ft) p
1
_~L,,),,~ + ii,,~ '(,,,hA1 {io~ '"' ~,~} ~ < L(f)(t) + p-llvK(qa-l(t),f;fi,).
Now, since K (s, f ) / s decreases, we have L
K(~--X(t),f~Ix)'~C(fO~O--I(t)(K(s,f~A))P~) and therefore I < cL(f)(t)
A similar argument also yields I I <_ cL(f)(t).
p < cL(f)(t)
6.1. R E I T E R A T I O N
81
Combining these two estimates we obtain, ))[I~o,~;K <_ c L(I)(t)
f~
On the other hand, since t E (0, 1),
tl[fx(cp-i(t))llA1 < t K(cp-l(t),f;7t) < K(qa-x(t),f;fi.) <_ cL(f)(t). We have thus obtained
f~
xo,p;K + tllfl@-~(t))llA1 < c L(f)(t)
which implies
K ( t , f , fi~o,v;K,A~) < cL(f)(t) and the result follows. [] The following application of the previous theorem will be useful later. T h e o r e m 47 Let A be an ordered pair, then
((Ao, A,)o,p;K, A1)o,v;K = { f :
( K ( s , f ; A ) ) p log
--<
oo}
In particular, ( ( Ao, A1)o,~.,~:,A1)o,,;K = ( Ao, A1)(2);K Proof. To simplify the notation we let p = 1. By (6.3) (the proof for p > 1 is the same, we just have to use Theorem 46 instead)
K(t,f;Ao, A1)o,1;K, A1) ~ K(~-~(t),f;7t)ln ~---=r~0 ~ +
~-l(t) f K(s,f;fi,)~o
Thus, integrating with respect to ~ on (0, 1) we find
IIflI((Ao,A1)o,1;K,a,)O.,;K ~
CHAPTER 6. REITERATION, AND APPLICATIONS
82
fo K(~-l(t)'f;A)lnqo-l(t------~ 1 dtt + fo 1 [ ~-~(t)K(s,f; ft. ds dt Jo t To evaluate the first integral on the right hand side we make the change of variable u = T-l(t), then we get
~1
l dt -K(~-l(t)'f;A)In~-l(t) t ~
~1 K(s,f;fi.) ( 1 + l o g 1) -d8s
The remaining integral is computed interchanging the order of integration, and we see that
fo~-l(Og(s,f;A)
-
g ( s , f; A ) l o g s + loglog
7
and the result follows. [] Another computation of extrapolation spaces, that can be obtained by these methods, is the following extension of Theorem 6. (The reader should compare the methods of proof of these two results). T h e o r e m 48
(cf. [72]) ((Ao, A1
)(1);K,(Ao, A1)(2);g )(1);K=
{f E (Ao, A1)(1);K: So'K(s,f;Jt)[log+(log sl--)]-~ Proof. The proof is by "ping-pong" iteration. In fact successively applying Theorem 47, (6.2), and (6.3) (applied to B0 = (Ao, A1)(1);K; B1 = (Bo, A~)o);K ) we get
K (t, f; (Ao, A1)(1);K, ( Ao, A1)(2);K) K (t, f; (Ao, A1 )0);K, ((Ao, A1 )(1);K, t
_~
A1)(1);K)
:; (Ao,
I io 1 K(nfin{~p-l(u),s}, f; 71)dss du u ,~ t ~e_§ Therefore,
IIfII((Ao,A~)o),K,(Ao,A, )(~),K)(O;K
83
6.1. REITERATION ..~
_§
K(min{qo-i(u),s},f;7t
8 U dt
which in turn is equivalent to the sum of two integrals = (I)+ (II), say. where,
(I)=
fO1K(s,f;A) ~ : 1 '~188d t -d'ad$ (s) Jo u s
( I I ) = foolf~-,/, 1 g(qo-l(ul'f;/t)l~
~-l(u) ldu
U
dt
iow~
II)< /ol
[
e
~-r(u)
](log 1) x du u
= fol [[K(qa-1 q~-I (u)'-f; (u) "A)] (l~ l/u) -ldu
On the other hand, ( i ) = fO g ( s , f ; / i ) I
<_
fl
1
du ds
(s) Io~ a u
K ( s, f; A) log + log
1
ds ..~
1
/ K ( s ' f ' A ) l ~ + (log + (log 1)) ds + f K(s'f'Ti)l~ + (log + 1) ds 0
0
and the desired result follows. [] Example 49 Considerthe pair ft = (La(T),L~(T)). Then, fi,(1);K = LLogL(T), A(2);K = L(LogL)2(T), and, by Theorem 48, (fi'0);K,-4(2);K)(~);K = LLogLLogLogL(T). Combining this calculation with i ( i o g i ( T ) ) 1+0 = [fi,(i);g, fi-(2);g]e (see (7.15)) and (2.19) we obtain Theorem 6.
CHAPTER 6. REITERATION, AND APPLICATIONS
84
6.2
E s t i m a t e s for t h e M a x i m a l O p e r a t o r O f P a r t i a l S u m s O f F o u r i e r Series
We give a summary of the relevant estimates for the maximal operator on partial sums of Fourier Series. The fundamental results in this area are due to Carleson [17] and Hunt [52]. T h e o r e m 50 Let S(f) = sup, ISn(f)l, where Sn(f) denotes the
n *h
partial sum of the Fourier series of f, 1 < p < cx~, then for every f of the form, f = gXF, with 2 -1 < g < 1, we have
sup t~(sf)'(t) <_ cp llfllp t>O where ~ = O ( ~ ) . Combining Theorem 50, the fact that Lv,~(T) C La(T), with norm O(v!1) as p ~ 1, and the extrapolation theorem of Yano, allows Hunt to conclude (cf. [521) T h e o r e m 51 S : L(Log)2(T) ~ LI(T). These results were improved by Carleson and SjSlin (cf. [94]), and many other authors. I refer to the papers [57], and [72], for a more detailed bibliography on this. In particular, relevant to the methods developed in these notes, chapter 5 of [57] deals with extrapolation of weak type estimates in this context. In [95] the following result is proved T h e o r e m 52 If f E L(LogL)I+~ 0<0<1.
then S f E L(LogL)~
A complement of this result was obtained in [97] T h e o r e m 53 If f E iLogL(iogLogL)(T) t h e n S f E L(LogL)-I(T). A perusal of the constants in the proof of Theorem 52 in [95] shows that
IT Sf(t)(1 +log+Sf(t)))O-ldt < CO-1 fT [f(t)l(1 +log+lf(t)l)i+~ (6.4) where C is a constant independent of 0.
6.3. EXTRAPOLATION METHODS
6.3
85
Extrapolation Methods
In this section, which is based on [72], we give a proof of Theorem 53 by extrapolation. In fact, we shall show that Theorem 53 can be extrapolated from (6.4). Let us first observe that
L(LogL)~
C L(LogL)-I(T)
In fact, we have [ If(t)l[1 + log +lf(t)ll-adt <- L [f(t)[[1 + log+lf(t)ll~ JT
(6.5)
We also need to quantify the relationship between different ways of measuring the size of a function in the
L(LogL)~(T)
spaces.
L e m m a 54 For 0 E (0, 89 let
Io(f) = llfIIL(Lo, L)I+O(T)= foI f'( t)[l+
logl]l+~
Then, we have f01 If(t)l[1 + log + If(t)l]l+~
<_cro(f)
where c is a constant independent of O, and lo(f) r0(f)
Proof. that
=
0
iflo(f)_ 1
(6.6)
We may suppose that lo = lo(f) < oe. It is readily seen
if(t) <_ lot -1 Therefore, we obtain f01 If(t)l[l+log + If(t)l]~
= [ .
J { ] (t)<_lot -1 }
f ' ( t ) [ l + l o g + f'(t)]l+~
_< fox f*(t)[1 + log + ~ll+~
(6.7)
86
CHAPTER 6. REITERATION, AND APPLICATIONS
We consider two cases. If lo _< 1, then we estimate the function [1 + log+t~ 1+~ separately in (0, 10) and ( le, 1), and we see that the right hand side of (6.7) is bounded by 4 lo. On the other hand, if lo > 1, then
tJ
~01f*(t)[1 + log + @]1+~ Now, since log (1 + log
< (1 + log lo)a+~
lo <_0-11 ~ and lo > 1, 0 9 (0, ~), we have
lo)a+~ <_ (1 + ~)a+~
<_ (2 lo)a+OO-aO-~ < c(lo)20-10 -~
The desired result follows since 0 9 (0, 1/2), and 0 -~ is bounded as 0--+ 0. D Using (6.4), (6.5) and Lemma 54, we have,
S: L(LogL)I+~
L(LogL)-I(T).
with
IT IS f(t)i[1 + l~
f(t)l]-ldt <- cO-1P~
(6.8)
as 0--+ 0. Note that the functional
f ---, l(f) = fT If(t)l[1 + l~
(6.9)
is subaditive on positive functions. Let us also note that S f >_O, and that, moreover, S is a subaditive operator, if a norm estimate were available instead of (6.4) we could extrapolate directly. However, in the situation at hand we need an extra argument to overcame the nonlinearity of the estimate. Let f E F.o~(o,a){O-aL(LogL)l+a(T)} = ~o~(o ~){ 0-1L(iogi)l+e(T ) }. Consider a nearly optimal decomposition f ' = ~0<~ fe such that
IIflIE.(O-'L(LogL)'+O(T)}} "~ Y~ O-a IlfallL(LoaL),+.(r) e<89
We further decompose f as follows. Let
f=fo+fa, with fo=
f0,fl= OEE
OEF
6.3. EXTRAPOLATION METHODS
87
where E {0 e (0,1): IIfoIIL i}. Applying the functional l defined in (6.9) to S f, and taking into =
account the sublinearity of the functionals, we get
l(sf) <_ l(s:o)+ ~(s:,) We now estimate each of these terms separately. By subaditivity, /(Sfo) < ~
l(Sfo)
OEE
<_ c ~_, O-1Fo(fo) (by (6.8)) OEE
< c ~ O-11o(fo) (by the definitions of E and
F)
OEE
< c ~_~ 0-~ IIfOIIL(Lo, L)O+,(T)~ IIflIE~(O-~L(LogL)'+O(T)}} 0e(o, 89 Similarly, since the functional f --+ [/(f)] 89is also subaditive on positive functions, we obtain
[l(Sf~)] 89< c ~ [l(Sfo)] 89 OEF
< c ~ o-~ [r(:0)] 89 (by (6.8)) OEF
c ~ o-~o-~ IlfolILr L)~+~
C Z 0-1 IIfOIIL(LoaL)~+O(T)"~ IIflIEo(O-'L(LogL)~+O(T)}} o~(o, 89 Consequently,
( )2 l(Sf) <_ c{IIflIEo{O_,L(LogL)a+O(T)} )-4- IIfIIEe(O-aL(LozL)a+O(T)} ) }
as we wished to show. []
CHAPTER 6. REITERATION, AND APPLICATIONS
88
6.4
More
On Reiteration
In this section we collect some auxiliary results needed for our applications to higher order logarithmic Sobolev inequalities below. We consider some calculations with the (., .)0,q;K methods. In fact, one could prove similar results for the more general (., .)p~,q;K methods, but we shall not pursue the most general results here. A typical calculation with the (., .)0,p;K functor is given by the following E x a m p l e 55 Let f~ be a probability space, then
(LP(~),L~ Proof.
= LVLogL(fl)
This follows readily from the well known fact (cf. [8]) that (K(t,f;LP, L~
p ..~ /otP[f'(s)]Pds
(a proof can be obtained by reiteration and (2.28)). Integratin 9 with respect to
-~ on (0, 1)
the result follows. D
Likewise, one can also prove that (LP(n), L~~
= (LP(Ft), L"(n))o,v;K, p < r.
(6.10)
More ,generally, we have L e m m a 56 Let ft be an ordered Banach pair, then
(Ao, Ao,q)o,p;K = (Ao, A1)o,p;K Proof. The proof follows mutatis mutandis the corresponding result for p = 1. In fact, it is enough to show that V0 E (0, 1) we have (Ao, .4e,oo;K)0,p;K C Ao,p;K- Suppose that f C (Ao,-40,oo;K)O,v;K, then by Holmstedt's formula (cf. [8]) we have
K(t, f; Ao, Ao,oo;K) ,~ t sup s-OK(s, f; A) >_ K(tl/o, f; A). J. te<s
6.5. LOGARITHMIC SOBOLEV INEQUALITIES
89
Therefore, 1
-
v
/0 (K(t'f;A~176176
d2t>-~ (K(tl/O'f;A))PT 1
dt
and the result follows. [] We shall not pursue the standard extension of the results presented for p = 1 to the case p > 1. However, for our application to Logarithmic Sobolev inequalities we do require the following E x a m p l e 57 (L2LogL, L~
= L2(LogL) 2. (All spaces in this example are based on a probability space ~).
In fact using successively Example 55, Theorem 47, and the estimate of the K functional in Example 55 we obtain,
Proof.
((L2LogL, L~176
= ((L2, L~176
1 t2 = {s:/o/0 = {f:
(s) 2
l__dtds log89 t t
=
< oo} < oo}
= { f : Llf*(s)2(logl)2ds < oo} = L2(LogL) 2 []
6.5
Higher Order Logarithmic Sobolev Inequalities
We consider some aspects of the logarithmic Sobolev inequalities of Gross [46] and the higher order estimates of Feisner [41]. Our contribution here is to show that extrapolation spaces can be used to provide a simple proof of the higher order estimates from the lower order ones. The twist is that this time the extrapolation spaces are obtained by interpolation. This gives a new approach to the estimates by Feisner [41], a related method based in the theory of
90
C H A P T E R 6. R E I T E R A T I O N , A N D A P P L I C A T I O N S
commutators has been given in [25]. Our method can be also applied to derive new results for logarithmic spaces that are closer to L 2 than L2(LogL) '~ type spaces. We discuss this briefly in the notes at the end of the chapter. A detailed study of the relationship between extrapolation and logarithmic Sobolev inequalities shall given elsewhere (cf. [78]). Let us introduce some notation and background. Our presentation here follows Bakry and Meyer [3]. Let /t be the standard gaussian measure on R, and let Pt be the semigroup of HermiteOrnstein-Uhlenbeck,
Pt = / R f(e-t/2x + (1
-
e-t)l/2y)/t(dy).
Then, Pt = fit+ e-~tdE~, with E~ discrete and concentrated on { ~ : n = 0, 1, ...}. The Riesz potentials associated with Pt are defined for Re(v) > O, by Rv = ] X-~dE~, J(0~oo) which correspond to the L 2 multipliers X-'X{a>0}(= $-VX{~>l/2}, in dEa), and are therefore bounded). It can be shown that, in fact
R~ f = F(v) -1
/2
t v-1 PrY dt.
In what follows all the spaces considered will based on d/t, unless otherwise specified. The following fundamental result is due to Nelson. T h e o r e m 58 Let 1 < p < q <_ 1 + et(p - 1), then Pt is a bounded contraction from L p to L q. A weak version (due to loss of information on the constants) of the logarithmic Sobolev inequalities of Gross can be now stated as follows T h e o r e m 59 R 1/2 is a bounded operator from L 2 to L2LogL. Our purpose in this section is to use Theorem 60 and reiteration theorems for extrapolation spaces in order to prove the so called Higher Order Logarithmic Sobolev Inequalities, due to Feissner (cf. [41]).
6.6. NOTES AND FURTHER RESULTS
91
T h e o r e m 60 For every n E N, R 1/2 is a bounded map,
R1/2 : L2(LogL) n ~ L2(LogL) ~+1 Proof. The case n = 0 is Theorem 60. To prove the case n = 1, we use the case n = 0, and the trivially verified (Minkowski's inequality) fact that R 1/2 : L ~ ---* L ~ Applying the (., .)0,2;g method we get
R1/2 : (L2,L~)o,2;K ~ (L2LogL, L~)o,2;K The desired result now follows from the identifications
(L2, L~176 and
= L2LogL (by (6.10))
(L2LogL,Lcc)0,2;K =
L2(LogL) 2, by Example 57
An induction argument concludes the proof for n > 2. [] R e m a r k . A simple modification of these ideas leads to a proof of Feissner's results for p > 2.
6.6
Notes
and Further Results
A systematic study of the reiteration properties of extrapolation spaces is given in [48]. In particular it is shown there how the Lorentz-Zygmund spaces can be obtained via reiteration theorems for extrapolation spaces. Recently, several authors (cf. [40]) have derived variants of the Sobolev imbedding theorem involving "double exponential integrability". We wish to indicate very briefly here how to approach such results via reiteration theorems for extrapolation spaces. Using the method outlined in the proof of Theorem 46 (cf. [43]) one can prove that, if A is an ordered pair, then
K(t,f,/ip,o~;K,A1)..~
sup
{ K ( s , f , Ao, A1)}
0<s<_el_§
p(s)
"
where p(t) = pl (t) = t(1 + log ~). (A similar result holds for p=).
92
CHAPTER 6. REITERATION, AND APPLICATIONS Now, recall that (L 1, L~176
= eL
Thus, we have:
f C (e L, LC~
< :"
sup fgf*(u)du } sup { 1 p(t) 0<,_
= sup
sE(O,1)
{1;
se(o,1)sup ~
sup
t> 1-----!---wa+log $
oo
l 1 /} t(1 + log {)
1
1
f*(u)du(1 + log ~) 1 + log(1 + log ~)
,
1
sup ~,f**(s) 1 + log(1 + log ~) 9e(o,1)
< oo
}
< oo
) < oo
It follows that (e L, L~176
= eeL"
Thus, interpolating in this fashion, known extreme Sobolev embedding theorems (cf. Chapter 4, Theorem 32, and the discussion following it), yields results of the type sought after in [40]. We hope to return to this point elsewhere. We indicate now an improvement to the logarithmic Sobolev estimates that follows through the use of the methods developed in this chapter. We try to get closer and closer to the space L 2, therefore we interpolate the estimates of Gross and Feissner using an extrapolation method, R1/2:
(L2, L2(LogL))o,2;K -~ (L~LogL, L2(LogL)2)o,2;K
The corresponding interpolation spaces can be identified through the use of techniques similar to those developed in the text. For example, the required extension of the reiteration formula (6.2) is given by
{
ds ] K(t,f, Ao,(Ao, Al)o,p:K) ,,m t f f 1/,p(K(s,f;A))PT~
l/P
(6.!1)
6.6. N O T E S A N D F U R T H E R R E S U L T S
93
An extension of Theorem 48, and Theorem 47, now yields, R1/2 : L2Log(LogL) ~ L2(LogL)(Log(LogL)). We can continue by iteration,
n , l~ : L ~ ,Log( Log(...Log L ) ~
L~(LogL)(,Log(Log(:..LogL))
n times
n
times
For this and other related results we refer to [78]. These results should be compared to those available by regular interpolation, namely R'I2 : L2(LogL) ~ -.-+L2(LogL) 1+~
Chapter 7 E s t i m a t e s For C o m m u t a t o r s In Real Interpolation In this chapter we study interpolation theorems, for certain possibly nonlinear operators, where cancellation plays an important role. The results are formulated in terms of commutators between a given bounded operator and certain operators ~ which are associated with interpolation scales in a natural way. The typical example we have in mind can be described as follows. Suppose that K is a CalderdnZygmund operator, and for b E B M O , let Mb be the operator defined by M b f = bf, then (cf. [?]) the operator [K, Mb] = K M b - M b K
satisfies It[K, Mb]fllLp < c IlfllLp , 1 < p <
~.
Note that each of the individual terms in the commutator is unbounded in Lv while the cancellation effect of the operation [., .] produces a bounded operator. This result is important in the theory of compensated of Murat and Tartar, as developed by Coifman, P.L. Lions, Meyer and Semmes in [19]. For example, it can be used to prove Theorem 24. Rochberg and Weiss [89], and Jawerth, Rochberg and Weiss [60] have shown that associated with the classical methods of interpola-
96
CHAPTER 7. ESTIMATES FOR COMMUTATORS
tion there are certain operators, O, generally unbounded and nonlinear, such that if T is a bounded operator in the scale, then the commutator [O, T], is also bounded in the scale. Again this boundedness is due to cancellation since each of the individual terms of the commutator is unbounded. For example, for (LP(wo),LP(wl)) a possible choice for O is O f = f l o g ( ~ , ), this fact combined with the relationship between the space B M ~ and the theory of A v weights gives the commutator theorem of Coifman-Rochberg-Weiss alluded above (cf. Example 69 below). The connection with compensated compactness goes further. Recently Greco and Iwaniec (cf. [44]) and the author (cf. [74], [75]) have established, through the use of commutator techniques, new sharp estimates for Jacobians. Iwaniec and Sbordone [55] have obtained new estimates for perturbations of Hodge type decompositions for vector fields, with interesting applications to the study of the regularity of solutions to variational problems, in particular the regularity of A - h a r m o n i c maps. Further results in this direction were recently given in [69]. For further development of the relationship between compensated compactness and interpolation theory we refer to [44], [74], [761, and the references therein. The theory has also been applied to the study of H p spaces in the work of Kalton (cf. [61], [62]); logarithmic Sobolev inequalities in [25]; ideals of operators in [61]. More recently, Rochberg [88], and the author (cf. [77]), have obtained higher order commutator estimates for the complex and real methods of interpolation. In the higher order theory the degree of unboundedness of the individual terms is greater as is the degree of the non-linearity. On the other hand, the higher amount of cancellation present allows one to obtain control on the norm of the higher order commutators. It is hoped that these new results will also find interesting applications. In this chapter we shall indicate some of the main lines of these developments. We also develop new connections of the abstract theory of commutators to extrapolation theory, and the functional calculus associated with a positive operator in a Banach space. We believe that our results in this direction could have applications in the theory of parabolic equations. We also formulate, through the use of extrapolation spaces, some new end point results.
7.1. SOME OPERATORS
7.1
97
Some Operators Associated to Optimal Decompositions
In this section we show how to construct operators in interpolation scales through the use of optimal decompositions. It will be convenient for our purposes here to introduce some special notation. For 0 9 R, 0 < q < or and for measurable functions on the half line, let 1
Thus, the interpolation spaces AO,q;K, 0 < 0 < 1, 0 < q < oo, are defined by
AO,q;K = {f 9 E ( A ) : IlfllAo,q;K = r
(K(., f, fi~)) < oo}
(7.1)
Let us say that a decomposition f = fo(t) + fl(t) is almost optimal, for the K method, if
IIfo(t)llAo + tllfi(t)llA, <~ cK(t,a;fi,)
(7.2)
where c is a constant fixed before hand, say c = 2. We then write D g ( t ) f = Dg(t; 7t)f = fo(t). For n E N, the operators fla,n;g = ft,~;g associated with this decomposition are defined by
(._1), ' (Io'( logt) "-: DK(t)f T f~~ DK(t))f~ t)
(7.3)
Similarly, one defines the corresponding operators f~ associated with the J and E methods. For a Banach pair fi,, let us say that u(t) is an almost optimal decomposition of f for the J method, if
In which case we write D j ( t ) f = u(t), and the corresponding f~cT,,;J = f~,~;j operators are defined by 1 r ~176
dt
Ft,~;sf = ~ Jo Os(t)f (logt)"--[-
(7.5)
98
C H A P T E R 7. E S T I M A T E S F O R C O M M U T A T O R S
For the E method we have a similar definition. Recall that
E(t,f;fi~) =
inf {]]f0]]A0 " f = f0 + fl} ]If1 IIA1 <~t
The corresponding interpolation spaces ftO,q;E, 0 < 0 < cx~, O< q < cx~, are defined using the quasi-norms
Ilflho,q;. = r
( E ( . , f , ft))
(7.6)
We say that fo(t) is an almost optimal decomposition, for the E method, and we write D z ( t ) f = DE(t; A ) I = f0(t), if
E ( t , f ; A ) ~ ]lDE(t)fllAo
(7.7)
Then, for n E N, we let ~A,n;Z = F~n;Z be defined by
f~n;Ef -- (n---i)' -1
(log t) n--1 D E ( t ) f ~
1
(7.8)
- f ( l o g t ) n - ' ( I - DE(t))f o Whenever no confusion arises we shall drop the subindex that indicates which method of interpolation we are using in the definition of the fl's. We give examples of computations of these operators in different settings. Example
61 We consider in detail the pair (L1,L~176 For the E
method (cf. [57]) we get an optimal decomposition writing
f = f o ( t ) + f i ( t ) , f o ( t ) = ( f - t)x{ifl>t},fl(t ) = tx{lfl>t } + fx{lfl
1 -
[l]l(logt)n_ldt,
1)! .,o
It is however more convenient to consider the almost optimal decomposition (cf. [60]) f = fx{I/l
7.1. SOME O P E R A T O R S
99
E x a m p l e 62 If we use the K method then the same considerations of [60] for n -: 1, lead to the general formula
fL~,Kf(x) = ~ . f ( x ) ( l n ( r l ( x ) ) ) " where the "rank function" ry(x ) is defined by
rf(x)
=
I{y: If(yl > If(x)l
or
If(yl
=
IS(~)l ,y -< x}l
E x a m p l e 63 The pair (L p~ L pl ). Let ft be a pair, Holmstedt's proof of the reiteration theorem (cf. [8]) implies that for [3 = ( ftOo,qo;K, ftO~,q~;K) we have
f~,~,B,K = 10o -- 011n ~~n,a,K
(7.9)
and a similar result holds for the E method (cf. [60], [25]). This remark shows, in particular, that we can compute f~K for (L p~ L pl) using Ezample 62. E x a m p l e 64 The pair(LP(wo),LP(wl)). Let X = X(O,c~) be a Banach lattice. A positive function w(t) defined on (0, cx~) shall be called a "weight". We can use weights to construct "weighted" versions of X . Namely, let X~o denote the space of measurable functions on ( 0 , ~ ) such that f w E X , with the norm given by IIfllx~ = I l f w l l x
Let wl be weights, i = 0, 1, then, it is well known, and easy to see, that K(t,f,X~oo,g~l) ..~ ]lf min{wo, twx}llx In fact, a nearly optimal decomposition of f is given by f = fx(~,ot~,~) Thus an f~K associated with the pairs (X,.,,o, X~,~ ) is given by f l f = f log Wo Wl
100
C H A P T E R 7. E S T I M A T E S FOR C O M M U T A T O R S
For more details see [60]. More generally for these pairs we have, ~)n;g f = c f
(log
(7.10)
Similar results hold for the complex method, in fact Rochberg [88] shows that, for the pair (LV(wo),LV(wl)), one can choose
f/,,vf=lf n!
log(
)
In the case of L p spaces we can also define the weighted spaces LP(wdx). Observe that since LV(w) = Lv(wa/Pdx), we have 1 ~)(LP(wod~),LP(wldx)): ~)(LP(wo),LP(wl)) as long as p < oo.
Example 65 Through the use of the Fourier transform we can reduce the computation of fl for the pair of Sobolev spaces (H 2, H -2) to the computation of fl for the pair (L2((1 + Ixl 2)), L2((1 + Ix12)-1)). Using the previous calculations and taking inverse Fourier transform we see that we can choose
f~f = clog(I + A)f where log(I + A) is defined using the functional calculus. For more details see [60], and for the definition of concave functions of a positive operator on a Banach space see (7.443 below.
Example 66 Through the use of the fundamental lemma, operators fl associated with the J method can be computed from the corresponding ones for the g method (cf. [24]) ~~n,K = -- ~'~n,J
(7.11)
The following result from [60] is the main abstract result concerning the boundedness of commutators on real interpolation scales (for the corresponding results for the complex method see [89] and [88]).
101
7.1. S O M E O P E R A T O R S
T h e o r e m 67 (@ [60.]) Let A, B, be Banach pairs, linear operator T : ft ~ [~ , and, moreover, if F the real methods of interpolation described above, TQF(A),I --QF(B),IT. Then, there exists a constant that
I1[T, fh,r]fllr(~)
T be a bounded denotes any of let /T, Q1,F] = c -- c(F), such
<- c Ilfllr(;).
(7.12)
Theorem 67 has been recently extended to higher order commutators by the author (cf. [77]). T h e o r e m 68 Let 7I and [7 be a Banach pairs, T 9 7I ---* [~ be a bounded operator. For n = 0, 1,2 .... define
C n f --
T
n=0
[T, Q1,K]f
n ----
[T, f~n,K]f + • I , K C n - I , K f
-[- ... -[- f l n - l , K C l , g f
1
n >_ 2
If F denotes either the E or J methods, we define
Cnf =
T
n=0
[T, fll,F]f
n = l . ~
[r, fln,r]f
- f ~ l , F C n - l , F f -- ... -- Q n - I , F C I , F f
n>2
Then, for 0 < 0 < 1, 1 < q < cx~, there exist absolute constants c = c(O, q, n, T) > O, such that, for all the methods of interpolation above, we have
lICnfll(.o,B,)o, q <_ cllfll(Ao,A~)o,q. Let us illustrate these results with applications to singular integrals. E x a m p l e 69 Let b C B M O , and assume that b has B M O norm sufficiently small. Then, e b and e -b are Ap weights and therefore if K is a Calderdn -Zygmund operator, we have, for 1 < p < (xD, K : (LP(ebdx),LP(e-bdx)) ~ (LP(ebdx),LP(e-bdx))
C H A P T E R 7. E S T I M A T E S F O R C O M M U T A T O R S
102
Since by Example 64 we can choose ~(LP(ebdx),LP(e-bdx));Kf = M b f = bf, we get [K, Mb] : i p = (LV(eb),LP(e-b)) 89 K ~ (LV(eb),iP(e-b))~,v; K = i p
II[K, Mb]fblLp <
cllfllLp
An homogeneity argument (i.e. replacing b by cb) shows that the assumption on the B M O norm of b is not a restriction on b. E x a m p l e 70 The higher order estimates that correspond to the previous example are given by
[...[[K, Mb],Mb] ...... ,Mb]: i p --* L v. (cf. [88]).
7.2
Method
of Proof
This section is devoted to the proof of Theorem 67. We shall also briefly indicate the strategy for the proof of Theorem 68, although we must refer the reader to [77] for the complete details. We introduce further useful notation in order to formulate a result from [80] for commutators of order 1. Let P and 15 be defined on measurable functions on (0, oo) by (7.13) and let
S ( f ) ( t ) = P (f) (t) + P ( f ) ( t ) In this context Hardy's inequality can be stated as
r
< CO,qr
0 < 0 < 1, 1 < q < oo
(7.14)
The following formula can be easily established by an elementary computation * 1
f~,~,Kf + ~! (logt)'~f
(7.15)
7.2. METHOD OF PROOF
103 oo
(n_l),(fot(logs)"-'DK(s)f~--~
(logs)"-l(I-DK(s))f~)
In particular, for n = 1, we obtain +
j (i-
-
Thus, for each t > O, (7.16) provides us with a decomposition of f~x,gf + (log t ) f . Let us now check that, in fact) this is a "good" decomposition, modu/o a bounded operator. In fact, we have
tDK(a)f
IIDK(s)fIIA~-7
ds
< fo' K(s, f; 7t) - 7
(by definition of
Dg(s)f)
Similarly, we see that
t
( I - DK(s))f
<_t A1
K(s,f;71ds 8
8
Consequently, we have proved that Vt > 0,
K(t,f~l,Kf +(logt)f;74)
(7.17)
where c is an absolute constant independent of t and f. Therefore, by (7.14), we get, for 0 < 0 < 1, 1 < q < r
r
+ (logt)f;A))
< c0,q Ilflla,,,~
(7.18)
Theorem 67, for the K method, can be now proved as follows. Let T : 71 ---}/~ be a given bounded operator, let f E 710,q;K, and let fl = ftl,g; we need to control
II[T, f~]fllBo,r = Ce,q(K( t, [T, f~]f ; [3)) To use (7.17) to the advantage we write
K(t,
[T, f~]f;/3) =
K(t,T(f~ f + f
< K(t,T(f~f + f
logt);/~) +
logt) -
(flT f + T flogt); B)
K(t, f h T f + Tflogt;[3)
CHAPTER 7. ESTIMATES FOR COMMUTATORS
104
~< IITII K(t, a l f + f logt; A ) + K(t, f t , T f + Tflogt; [3) Consequently,
r
K (t, [T, ftl,K]f;/3)) <_cCe,q(K(t,f~if + f logt; ft.)) + r
f + T flogt; B))
Now, we can apply (7.18) to estimate the first term,
+ f logt; fi~)) < CO,qIlflla,,~,K
r
Likewise, to estimate the second term we successively use (7.17), the boundedness of T, and Hardy's inequality (7.14) to obtain
r
f lt- Tflogt;B))~_< Cr (S (g(.,r f; B)) (t)) ~_
and we are done. In view of Example 66 the result just obtained for the K method implies the corresponding one for the J method. To complete the proof of Theorem 67 it thus remains to consider the E method. We record the following analog of (7. I5)
f ~ , E f - n~(logt)"f 1
= ( n - 1)!(ft
oo
DE(s)f(l~
(7.19)
fot(I--DE(s))f(l~
ds
)
Again we consider the case n = 1, and we observe that the previous formula gives us a decomposition of ~ l , E f - - log t f that allows us to compute the E functional of this element. In fact, note that
'!
ds Sal
<_
I 1 ( I - DE(s))flIA, S
<
t ds s--=t.
fo
8
Therefore,
E(t, f~l,Ef -- log t f; fi,) =
inf I1~111,1St
IIS~I,E:--
l o g t f - alllA0
7.2. METHOD OF PROOF
105
I
<_ ~I,Ef-- logtf- fot ( 1 - DE(s))f sds Ao = ft~176
<
ao
IIDE(~)fllAo d~ $
E E(s, f; 7t) ds
(7.20)
8
where the last inequality follows by the definition of DE(s)f. We use this estimate to take advantage of the cancellation as follows. We need to control r [T, 9tl,E];/3)). For this purpose let us write
E(t, [T,a~,Elf; B) = E(t, Tfll,Ef - logtT f + logtT f - fll,sT f; B) < E(t, Tf~l,Ef - l o g t T f ; B ) + E(t, l o g t T f - f~l,ETf;B) Then, by (7.20), and a variant of Hardy's inequality, we get
II[T,S~l,ElfllBo,q;E <- cllfllao,~,E concluding the proof of Theorem 67. The proof of the Theorem 68 follows a similar strategy, although the details are somewhat more complicated. Let us briefly indicate the main ingredients of the method. First we introduce iterates of the operators P,/5, S, as follows p(,~) (f) = p ( ..... p(p(f)))... ),/5(,0 (f) = ? ( ..... t5(/5(f)))... ) 9
Y n
9
n
times
times
Sn(f) = P(=)(f).4-/5(n)(f)
(7.21)
By computation, we recognize the familiar operators studied in Chapter 2, S,~(f) =
log(~)
min{1,
}I(~)T
106
CHAPTER 7. ESTIMATES FOR COMMUTATORS
By iteration of (7.14) we see that these operators are bounded in the r norms. Now, we select expressions that can be controlled by these operators. For example, in the case n = 2 we prove 1 2 K(t, (logt)12,,tff - ft2,Kf + ~(logt) f;fi,) < cS2(K(., f; 7t))(t)
In fact, multiply (7.16) by log t to obtain t
OO
logtfhf 4-(log t ) ' f = J Iog'Dg(s)f~ - f log t ( l - OK(s))fdSs 0
t
(7.22) Now, (7.15) for n = 2 gives t
fl2f + ~(logt)2f
oo
f log
-
0
-
s
t
(7.23) and adding (7.22) and (7.23) we see that, 1
l o g t ~ f + g(log t ) 2 f - ft2f Z
-
Thus, by the triangle inequality and the readily verified facts that
K(t, DN(s)f; ffi) <_K(s, f; ft); K(t, (I - DK(s))f; Ti) <_t K(s, f; fii) 8 (7.24) we obtain,
K(t, logt~f + l---(logt)2f - ~t~f; A) z
-~cL ~176
log! K(s,f;A.) dss
= cS2(K(., f; it))(t) as desired.
7.3. COMPUTATION OF fi
107
The general case follows thus a combinatorial pattern where the expressions to control are given by the computation of the powers of log~ = log t - log s. We obtain
K(t'a~f +P~(l~
=-1
(_1)~-1-~
kl
f + Y~ (l~
fln_l_kf;fit ) (7.25)
k=l
< cS,(K(.,f;Zt))(t) where p. =
n-1
n-l-k
1
+
1
Now, the proof Theorem 68 follows from (7.25) by a suitable modification of the arguments used to prove Theorem 67 (cf. [77] for the details).
7.3
C o m p u t a t i o n of f~ for e x t r a p o l a t i o n spaces
The formula (7.9) is not valid for extrapolation spaces. But using the proofs of the reiteration theorems given in Chapter 6 it is not difficult to formulate appropriate substitutes. In this section we indicate in detail how to do so, and compute the corresponding operators fl associated with certain extrapolation spaces. For a more detailed study of commutator theorems in extrapolation scales, and in particular in Orlicz spaces near L 1, and further applications we refer to [45]. Let A be an ordered pair, then the corresponding operator fl = fin for the pair (A0, A0,1;K) can be computed using a result of [43], which, for convenience, we state here as follows
DK(s, Ao, tto,1;K)f = DK(e-~,Ao, A1)f = DK(e-~)f
(7.26)
Thus, the corresponding operator associated with the pair (A0, fi,0j;K), which we shall denote fl(-1), can be computed, fl(-1)f=
1DK(e-~,Ao, A1)f
= ~-'DK(S)fsln!-$
More generally, the proof of (6.11) leads to the formula
DK(s, Ao, Tlo,p;K)f = DK(e-~,Ao, A1)f = DK(e-;#)f
CHAPTER 7. ESTIMATES FOR COMMUTATORS
108
and therefore the operator f~ associated with the pair (Ao, fi,o,p;K), p > O, is given by,
ny = fo D,~(,-~,Ao, A1)fdss _ p1 ~o~- ~D~(s)f.s Onds~)~/'" E x a m p l e 71 In particular, for the pair (L 1, L~176over a probabil-
ity measure space, the operator f~(-1) = f~, associated with the pair (L1,L(LogL)) = (La,(L1,L~ is given by f~(-1)f(x) = f(x)lnlnry(x)
where ry(x) is defined in Example 62 above. This process can be continued, thus we form A1 (A0, A1)0,1;K
A_~ =
n=0 n = 1
(Ao,(Ao, A1)o,1;g)o,1;g n = 2 .
.
.
.
(Ao, A-(n-1))o,1;g
,,
n
The spaces A_~ can be computed using the n th iterate of the (7.26) which leads to the following extension of the reiteration formula
(6.2): 1
K(t,f;Ao, A_~)~-t f e
1
f
_1. _-1_ t e un
1
.. f e
dul K(ul,f;Ao, A,)--duu..du~ Ul
__1. u2
(7.27) For example, if n = 2, then
K(t,f, Ao, A_2),~t
1
f
K(ul,f;Ao, A1)~d~
e_el/t 1
-e-X~ t f
e--eli t
(Ao, A-2)o,1;K = { f :
Ul
K(Ul,f;Ao, A1) d~' Ul
]o1 K ( s , f ; A o , A_2) ds s
< o0}
7.3. COMPUTATION OF f~
109
={f:]o e-~K ( s , f ; A o ,
1
A1)dln(ln ~) < co}
More generally, A-n = (Ao, A-(n-1))0,1;K = {f:
fo e" K ( u , f ,
Ao, A1)dln( ..... (ln(
~)))< co}
n times
where en = e x p ( - exp(exp(...exp(1)) .... ). E x a m p l e 72 In the case of (L 1, L ~176we obtain, by integration by parts:
(L',L~
= { f : fo e" f*(s) In( ..... (17(1)): ds
n times
that is (L',L~)(_n)
= L,Log( ..... LogL) Y
n times
The corresponding operators ~K = ~(-n) associated with the pairs (Ao, A-n) are given by: a(-'O f = f ( x ) !n(ln(ln(...(ln (rl(x)))..) n+l
times
The commutator theorem in this setting gives: if T is an operator bounded on the pair (A0, A_,,), then, we have, for n = 0, 1 ..... [T, fl{-n)] : ( Ao, A-n)O,q;K - , ( Ao, A-n)o,q;K. For other variants and applications of these theorems we refer to [45]. Let us now explicitly compute the interpolation spaces involved. Using [43], we have,
CHAPTER 7. ESTIMATES FOR COMMUTATORS
110
( i o , A_n)O,q;K ~
{f : {i[(logl)~
Ao, A_(n_l,)]qd-::}!q <
00}
Therefore, using (7.27) we are able to compute the corresponding spaces. The easier cases to deal with are when q 1, or 0 - q 1
(Ao, A-.)o,I;K = { f : f0
1(1)
log
0-1
K(t,f, Ao, A_(._x)ld---~ < oo}
(Ao, A-.)~,q;K = (Ao, A-(.-l))o,q;K E x a m p l e 73
Consider the pair (L 1, L~176then for 0 < 0 < 1,
(L1, L Log( .....~( LogL ))o:;K = L Log~ ( Log(.,]...( LogL ) ), n times
n times
In the same vein we can develop a formulae for the pair (A0,/~O,p;K)" We consider the higher order operators associated with extrapolation spaces. Again for simplicity assume that A is an ordered pair, then the operators fl,~ associated with this pair can be defined by
1Jol Dg(s, Ao, A1)f(logs)._ x dSs
f~"f-- (n - 1 ) !
The corresponding operators associated with a pair (Ao, A_m), are then given by 1
a(-")fE x a m p l e 74
1
(n -11!
~o1 DK(e_e-,... -~~ ,Ao, Al)f (logs) "-1 ds
S
For the pair (La, (L1, L~176
we have,
Ft(-m)f = f !og"(log(log...~(logr/(z) .... )
m+l
7.4.
fl
OTHER OPERATORS
7.4
Other
111
Operators
In the applications of the theory it is useful to consider other variants of the operators ~ discussed above. For example, in some important examples the "gain" that one obtains by the cancellation effect of [., .] is so subtle that it manifests itself only at the level of the constants appearing in the corresponding inequalities. This is precisely the case in the following result by Iwaniec-Sbordone (cf. [55]) T h e o r e m 75 (cf. [55]) Let 1< ri < cr
= 1,2, r E [ r l , r 2 ] and suppose that T : L ~ ( O , E ) ---+ L ~ ( ~ , E ) , where E is a Hilbert space. Then Vr such that ~2 ~---1 -< r -< ~~1- 1 , we have liT&
-
-%TII1-I-s: _
(7.28)
where S , f = ( f-J!-L]~: and c~ is independent of f . \I1111,7 J'
Letting ~ ---+ 0 in (7.28) we recover the Rochberg-Weiss commutator theorem ]]Tfi - ~Ti[ r < c~ [if lit with f i f = f log ]f]. Using our theory we can give a considerable extension of this result. In this section we develop this point in detail. The reader should also consult [55] to compare the methods of proof. Let a E ( - 1 , 1), a ~ O, and define f~E,.a = f~:a = a
DE(t)at"--[- --
I-- DE(t))at"
T h e o r e m 76 (cf. [76]). Let it and f7 be a B a n a c h pairs, T : 7t ---+ [7 be a bounded operator, then, there exists a constant c > 0 such that,
/ . f O + a > O,
II[n<,,
C
<- -6
(2c<~)~
i)x/q
CHAPTER 7. ESTIMATES FOR COMMUTATORS
112
It is easy to see that, according to our definitions, for any Proof. Banach pair/it, and for t > 0, we have ~ , / / a + aCe(t) =
= (~(ft~DE,fI(s)as'~ds -- fot(I-- DE,R(s))as " ds" s
where r Let s
= 1 - t". = a(fJ(I-
-7.)
DE(S))as"d--~
(7.29)
then
Ila~(t)llH, <-- I~1 (]o' I1(1 -- DE(s))a tlH1 8a 7)ds" <
I~1 to+,
(7.30)
- (c~+ 1)
Thus, letting c~ = [1~1 (,~ + 1)-11 and combining (7.29), (7.30), and (7.7), we get
E(c~t~+l,Ft~a + r
H) _<
I~l( f~176
Therefore, if T : J ~ / 3 , then we can estimate T~,aa; [3) as less or equal t h a n
E(cata+i,~,BTa -4-r
+
ds.
~7 )
(7.31)
E(2c~t ~+1, ~2~,BTa -
E(c~t~+l,T(f~,Aa + r
B)
Using the fact that T is bounded, and applying (7.31) to each of these terms we get
E(2c.t"+',~.,~Ta - T~.,aa;[~ ) <_clal( ft~E(s,a, Ji)s'~?), where c depends only on the norm of T on the initial pair. application of Hardy's inequality now yields
{Jo~176176 -<
-- Ta.,Aa;
cl~ {~~176176
[~)]qd-t}I/q
An
7.4. O T H E R O P E R A T O R S ~
113
and therefore we finally get
as we wished to show. [] We consider now in detail the special case o f / 2 spaces. The E functional for the pair (LI,L ~176is easy to compute (cf. [56] and Example 61) E(t,f,L',L
~176=
Al(s)ds
(7.32)
The interpolation spaces for the pair (L 1, Lr by the E method, can be determined using this formula. In fact, if we recall
Ilfll~ = {p f0 ~176 As(s)s p-1 d~ } '/P we see, using (7.32), and integration by parts, that IIflI(L1,Lcr
= [ ( P - 1)p]-I Ifftf~
A calculation using Example 61 gives
flof _- f ]fl '~ - f Let us set S o f = f lfl ~ , then we clearly have [T, ~o] = IT, So]. Now to apply Theorem 76 we let
0 a+l
-
-
r l+a -
1, then O + a -= r - 1 .
Then, the previous discussion gives r
IITSj- &TIII~ < c2~
1
(r-
1)Ilfllr
Raising both members of the previous inequality to the power 14~ T gives an estimate of Iwaniec- Sbordone type,
114
C H A P T E R 7. E S T I M A T E S F O R C O M M U T A T O R S a+l
,,:,,r
(7.33)
In order to obtain a version of Theorem 75 we argue t h a t
T kllfll~t'~] _ IT:rTI IIIITSlI:II,-/~ <- -
T ( ~~) ~
IT$1~T/ _ IT$1~T/ 111117 IITIII7 =I
+II,
- IT/I~T:II + IISII~ IIlfa 1-l-a
say.
Now, I is controlled by (7.33), while I I can be readily computed
WN, ) ~ -1
I I = llTfl]~]( ]]Tf[[~
Let x = IITfllr , y = Ilfllr, u = y / x , r = u ~+1 - u, a n d a s s u m e , as we may, that IITII~_.~ ___ 1, then u E [0, 1], and we have reduced everything to prove that there exists c> 0, such t h a t Vu E [0, 1]
_< cl~l
Ir
(7.34)
We study r using calculus, and we see t h a t (7.34) holds with c = (i-~)~-+f-. We conclude the analysis by observing t h a t the factor 1/(1 -I- or) is under control by r2/r. By collecting estimates we see that we have thus obtained an end point version of Theorem 75 by real methods. By reiteration we m a y obtain the full result. In a similar manner we may prove that, if a~;Ka = e~(
fo1 D K ( s l a s
-S
ff
( I -- D K ( s l l a s ~'ds ) 8
then, we have T h e o r e m 77 Let A , B , be Banach pairs, let T : 7t ~ [3, be a bounded operator, and let a E ( - 1 , 1 ) / { 0 } , 0 < /9 < 1, 0< q < c~, and suppose that 0 <0 - a < 1. Then, there exists a constant c = c(6, q) > 0, such that
7.5. C O M P E N S A T E D C O M P A C T N E S S
7.5
115
Compensated Compactness
In this section we discuss applications of the theory of commutator estimates to compensated compactness. Specifically, we derive estimates for Jacobians of smooth maps, and use commutator estimates to establish perturbed Hodge type decompositions for vector fields. We review briefly some basic results concerning the Hodge decomposition. We are trying to decompose a vector field F as F = Vu + H, where H is a divergence free vector field, that is divH = O. This is classically done as follows. Suppose that F E Lv(tP, tP), then select u to be such that Au = divF, i.e. by letting
Vu = KF where K is the matrix operator given by R1QR1
RI|
9. .
R I |
. o .
**o
P~ |
...
R,, |
and the R j , j = 1,...n, are the Riesz transforms. decomposition we seek is
R,,
Therefore, the
F=KF+(I-K)F Moreover, since the Riesz transforms are bounded operators on L v, 1 < p < cr we have the right control. For vector fields defined on a smooth domain ft, a similar result holds, and again Vu is given by a singular integral. We now consider operators of the form ~ o f = f r where the (generally non-linear) operator r is selected in such a way that for any bounded operator T on LP with p E (1, or the commutator IT, ~r is also a bounded operator. More precisely, this means that there exists a function 5, with lim~_.0 5(x) = 0, such that,
If[T, r
]If[], (llfll,)
CHAPTER 7. ESTIMATES FOR COMMUTATORS
116
We have considered a number of operators of this type in the previous sections. For example, in the Rochberg-Weiss theory we can take r = log If(x)l, and then 8(x) = cx. We can also deal with r
=
(log Ifl) ~ , 0 < ~ < 1
or more generally using the work of Kalton (cf. [61])
r
= (log Ifl)~ Ilog r i ( I f ( x ) l ) l ~ , 0
_< ~ , 3 ___~ + 3 ___ 1
where r I is the rank function defined in Example 62. T h e o r e m 78 (cf. [~4], [7~]) Let f :1~ ~ R", be a mapping of class
C~(1~,1~), and let f~r be a commutator on L", and let ~ be the function associated to f~r as above, then ~,, J ( x, f)r lD fl )dx <_ c6(ll lO fl llL. ) ll lD fl ll~: 1 Proof. We use the method of [44] and the notation of our previous discussion. Using a standard Hodge decomposition write
f~r
= Dg(x) + H(x)
(7.35)
e Ls(R~,GL(n)) a with g E Wl's(Rn,Rn), and H = ( I - K ) 1 2 r divergence free matrix-field, 1< s < c~. Note that since this decomposition is unique we have ( I - g ) ( D f ) = 0. The operator I - g is bounded on Lv for 1< p < c~, and therefore, by the theory of commutators, we get
ll(l- K)f~r
-
f~6((I- K)DI)IIL. <
c'~(IIIDIIIIL,,)
thus,
IIHII. ___c6(]llDflllL.)
(7.36)
Using the notation of differential forms, J(x, f)dx = dfl Adf2 A...dfn, and (7.35) takes the form
r
=dgk + hk, k = 1,..,n
(7.37)
7.5. COMPENSATED COMPACTNESS
117
where the hk are differential forms of degree one whose coefficients coincide with the entries of the k-th column of H. Computing, using (7.37) for k = 1, we get
fRos(x,f)r
= fR dg,^df^
. . . .
fn,h,Adf2^...^df,,
Given the assumption on the vector field f we see that, by Stokes' theorem,
fR~
Adf2A .... ^ d A = O
Moreover, the second integral can be estimated, by Hadamard's inequality and (7.36), as follows
/R,, hi Adf2A...Adfn <_/R,, IH(~)I ]Df(z)] ~-1 dx < IIIHIII~ IIIOflll~-'
_< c6(lllDflll.)IIIOflll~ -1 Combining these estimates the desired result follows. D The following application of Theorem 75 to Hodge decompositions is important in the study of the integrability properties of the Jacobian transformation, as well as in the study of regularity of variational problems (cf. [55]). T h e o r e m 79 Let B = B(a,R) be a ball in l ~ and let f E W : ( l ~ ) , r > 1. Then, for each r E (1 - r, 1) the vector field IVY]-* V f E L rr-w,( t ~ ) can be decomposed as
]Vf(m)I-'Vf(m) = Vg(m) + n ( x) , a.e. m E B where g E WX_~_;,(1~ ) and H E L ~ ( t ~ , t ~ ) is divergence free and such that IIIHlti~_~L <_ ct~lillvfilll~ -* Proof. The idea of the proof has already been shown in the course of the proof of the previous theorem. Thus, let f~,(f) = f If i-*, and using the operators defined in the introduction, we define now
H = ( I - K)f~,(Df) An application of Theorem 75 or Theorem 76 now gives
Illnlll-~, < cl~l IllVfllll, -* as we wished to show.
[]
CHAPTER 7. ESTIMATES FOR COMMUTATORS
118
7.6
Relationship
to Extrapolation
In this section we show some relationships between the theory of commutators and extrapolation. We derive a general functional calculus for interpolation scales which we then relate to the functional calculus of a positive operator on a Banach space. Since the methods are similar to those of the previous sections, we only indicate the main lines of the arguments here. In order to motivate our discussion we consider a K / J inequality associated with f~ = ~ ~ I , K . For simplicity, suppose that ft. is a m u t u a l l y closed ordered pair. Let x C A(fi.), then by Minkowski's inequality,
IlflXllAo = K(1,f~z;fi,) <
~o1
dr r~176 dr IIDg(r)xllAo-~-+J1 I[(I--DK(r))xlIAIT-
Therefore, using (7.24), and the fact that
K(r,x; A) _<min{llxllao,rllxllal}, we obtain
min{ 1, 1 } rain{ 1, "ll~lla, } d," IlazllAo < II~lla0fo~176 r II~llAo r which after a computation gives the
K / J inequality
IlflxllAo < Ilxllao(1 + log( IlxllA' ~ Ilxllao"
(7.38)
We shall now associate operators to quasi-concave functions as follows. Let ~o be a quasi-concave function defined on R+ such t h a t limt__.~ ~,(t) = 0, limt-.0 ~o(t) = 0, then, as we have seen before, ~o can t be represented by ~o(t) =
Z
min{1, t-}d~(r) r
Associated with ~ there is an operator 12~, defined by
a~(f) =
[1DK(r)f @ ( r ) - [~176 DK(r))f d.(r) dO
./1
(7.39)
7.6. RELATIONSHIP TO EXTRAPOLATION
119
: rjo, DK(r)f dis(r)_ rjt~o(i - Dte(r))f dis(r)- fq)(t) where q)(t) -= ft dis(r) We also have the corresponding Calderdn operators associated with ~, defined, for measurable functions, by
C~o(g)(t) = f0~ min{1, !}g(r)d#(r)
(7.40)
Observe that (7.39) implies that t
K(t,fl~f + c~(t)f) <_ f K(t, DK(r)f)d#(r)+ 0
oo
f K ( t , ( I - DK(r))f)d#(r) t
K(t,fl~of + O(t)f) <_ C~(K(.,f))(t). (7.41) Let ~b be a function space on /7, a "parameter" for the real method, and let si~;s,- denote the interpolation space defined by the condition K(t,x; ]1) C ~. Let us say that a concave function c2 is admissible for st~;s< if the operator C~ is bounded on $. Using the methods of the previous section we get, T h e o r e m 80 Suppose that ~o is admissible for/tr
, and let T be a bounded operator, T " ~t ---+ A, Then the commutator [T, fifo] =
Tfl~o - fl~T is also bounded in ftr Now we consider the operators ft~ in relationship with extrapolation. Let
with the natural quasi-norm. From the-above discussion we see that
K(t,a~,x;fI))-
I (t)l K(t, x;/1)] <_ cC~(K(.,f))(t)
Therefore, we readily get T h e o r e m 81 Let ~b be admissible for Ar
then
D ~ ( ~ ) ( / i ) - {x" K(t,x; fit) la)(t)[ C ~b}.
(7.42)
C H A P T E R 7. E S T I M A T E S F O R C O M M U T A T O R S
120
Let us now show a natural K / J estimate for ~v T h e o r e m 82 Let fit be a mutually closed, ordered pair, then for x C A(fit) we have
Ila~xllA0 _< ~II~IIA0~(~) Proof.
Let t = 1 in (7.42), then Ila~ll~o
_<
/7
< c fo~ min{1,
<_ cllxll~o fo~
min{1,-1 } K ( r , x ) d # ( r ) r
1} min{llxllAo,rll~llA~}d~(r) r
IIXlIA1 min{ 1, rllxllA~----~}d~(r) =
cllxllAo~(
IIXlIA0'IIxlIA~
[] This last estimate persists at the level of commutators T h e o r e m 83 Let T be a bounded operator T : fit -+ fit, where 7t is
an ordered, mutually closed pair. Then,
,IIxlIA1, ll[T,~dXltAo < clIxltA0~U~) Proof.
We readily get
K ( t , T F t ~ x - f ~ T x ; fit)) _< C r Now, letting t = 1, we arrive to
I[[T, adxllAo <_ cC~(K(.,x; ]t)))(1) and the computation on Theorem 82 finishes the proof. [:1 We should interpret Theorem 83 as an extrapolation theorem for the nonlinear operator IT, ~ ] which is in general not bounded on A0. Because the c o m m u t a t o r is not in general a linear operator we cannot extrapolate directly.
7.7. A FUNCTIONAL CALCULUS
121
We consider in detail the results for ~. We proceed directly, as in the proof of Theorem 83 and arrive to
II[T'~]flIA~ <-- cf0 ~176 min{l' 1}K(r'f;~t)drr
--< c[][fl](D + IIfHA0] Thus, we see that and we have proved the following [T, fl]: fi-(1) ~ Ao The general case, for a general ~a, can be now obtained in a familiar fashion and we shall skip the details.
7.7
A Functional Calculus
One could view the theory of Fly operators as a possible way of constructing a general functional calculus on real interpolation scales. In specific applications we could use this theory to see, for example, which functions operate on certain function spaces. We shall presently show that this general functional calculus is in fact closely related to the functional calculus associated with positive operators on a Banach space. The commutator theorems that result are then expressed, and classified, in terms of the growth of the concave functions of the operator. One can expect such results to have applications in the theory of abstract parabolic equations. In our theory the representation that we use for a quasi-concave function ~ plays an important auxiliary role. For other applications, however, we need sometimes to select different representations, which of course should not affect the final estimates. An example of this situation occurs in the functional calculus of positive operator in a Banach space, which we now consider in detail. Let T be an operator acting on a (complex) Banach space X;. T is positive if Vt > 0, the resolvent R(t) = (T + tI) -1 exists, and moreover, there exists a constant c > 0 such that
IIR(t)ll < c(1 + t) -1
122
CHAPTER 7. ESTIMATES FOR COMMUTATORS We consider quasi-concave functions represented by:
~p(t) =
t(ts + 1)-Ida(s)
(7.43)
where da(s) is a measure such that fo~(1 + s)-lda(s) < cx). Then, one defines the bounded operator (cf. [87])
o(T -1) =
R( )eo(8)
Associated with a quasi-concave function we have two other quasiconcave functions: ~o*(t) = ~(0' t and ~(t) = v-(-~)" 1 The identity cp(t)cp*(t) = t, and the positivity of T, allows us to form the operator ~(T-1) -1, and we are thus lead to define following [87], T(T) = ~5(T-1) -1, and finally ~o(T) = T93*(T -1)
(7.44)
The concave function ~5*(t) has the representation
~*(t) =
/j t(s + t)-lda(s) ,,~ /0 min{1, -t}do-(s) 8
and computing ~*(t -1) we arrive to the formula
~(T)x =
sT(T8 21-I)-lx -do(8) , x 9 D(T) 8
The correlation with optimal decompositions is given by the fact that for the ordered pair (X, D(T)) the optimal K decompositions are given by
nK(t)x = tT(tT + I)-lx ( I - DK(t))x = (tT + I)-lx. In this way, we see that ~o(T)x = fo~DK(t)x da(s)s
7.7. A FUNCTIONAL CALCULUS
123
84 Let ~ be the quasi-concave function represented by (7.43). Let d#(s) = d~(~) and define
Theorem
$
t
F(t) = f0~176 min{1, ~}d#(s)
Then, ~(T)x - f~rx = c x where fir denotes the operator associated with the pair (X, D(T)). Since f ~ min{1, ~}&r(s) ~ f ~ ( 1 + 8)-1d(7(s) Proof. see that f ~ d~(s) = c < oc. Now, s flrx = folDK(t)x &r(t) t
f~(I-
<
o o , we
DK(t))xda(t)t
So that formally, we have,
~(T)x - flvx = x foo &r(t) t
--
CX
as we wished to show. [] The point is that, under the notation of the previous theorem, we have, for an operator H acting on the pair (X,D(T)),
[H, iz(T)]x = [H, c 2 ( T ) - c I ] x = [H, flr]x Thus, our commutator theorems imply that we have control of the commutators formed between H and concave functions of a positive operator. There is, moreover, an explicit relationship between the growth of the concave function of the operator T, and the type of interpolation scale we use to control the commutators. The K / J inequality of Theorem 82 in this setting takes the form of a "moment inequality"
II (r)xll < cllxll ( ) -
II I1
These inequalities are well known in the case ~(s) = s ~ 0 C (0, 1) (cf. Krasnoselski and Sobolevskii [66], for logarithms see Sobolevskii
124
C H A P T E R 7. E S T I M A T E S F O R C O M M U T A T O R S
[97], and for the general case of concave functions we refer to Pustilnik (cf. [87], corollary 6)). The extrapolation theorem giving the commutation relationship for ~v seems to be new even in this classical setting. Conversely, once this connection is understood one can see that many aspects of the theory developed in [87] (and the references given there) can be naturally extended to the general setting of the functional calculus associated with optimal decompositions in abstract interpolation theory. This includes the observation that the comparison theory for concave functions acting on a positive operator can be extended to the general setting of optimal decompositions and operators f~,l, f~o2. In conclusion we should also point out that the relationship between the functional calculus for positive concave functions of a positive operator and the operators f~ had been conjectured earlier in [6O]
7.8
A Comment tators
on Calder6n
Commu-
There are other types of operators associated with optimal decompositions for which we can obtain commutator theorems. The motivation for what follows is the formula for the Calderdn commutator (cf. [211) T = c
OtM, Pt T +
PtM, Ot T
where a E L ~176M~ is multiplication by a, and Qt and Pt are multipliers defined by Pt = I ( I + t202) -1 and Qt = t D ( I + t202) -I where D is on the Fourier side multiplication by ~. Observe that if f E L 2, we have f = c fo ~~Q 2t f -~ dt
(7.45)
It is not difficult to see that (7.45) is a nearly optimal decomposition of f with respect to the pair of Sobolev spaces (W~ 1, W~); i.e.
125
7.9. N O T E S A N D F U R T H E R R E S U L T S
for this pair we can choose D a ( t ) f = QZt. One also notices that P~ is related to Q~ through the usual relation between the DK(t) and D j ( t ) nearly optimal decompositions: d
=
(with c = - 2 , in this case). This leads to the question of how much of the analysis of [21] can be pushed to this abstract setting (and conversely!). For example, in the former direction, R. Rochberg a n d t h e author have shown, and this is not hard, that if A is a Banach pair, and T : 3, ~ /l, is a bounded operator then the operator
is bounded on the real interpolation spaces A0,q.
7.9
Notes
and Further
Results
In [69], Li, McIntosh and Zhang provide another approach to estimates for Jacobians through the use of weighted norm inequalities. A general overview of recent results in nonlinear partial differential equations, using the methods of real harmonic analysis, can be found in Mfiller's survey [83]. Although commutators of a bounded operator on an interpolation scale with the Ft~ operators are not necessarily bounded, there is some cancellation which allows for the following type of results considered in [77]. Let 0 < 0 < 1, 1 _< q _< oe, a C R, and define 1
Define interpolation spaces Ao,logO,q;K, 0 < 0 < 1,0 < q <_ ec, by
126
CHAPTER 7. ESTIMATES FOR COMMUTATORS
It is not hard to see that, if T : A --+ /) is a bounded operator, then Ilog sl~-I min{1, ! } K ( s , f ; i l ) ds K(t,[T, an]f;[~) <_c fo ~176 S
Therefore, we obtain ]]IT, Qn]f]lBe,q;~: -< c nfn~e,~ogn_l q;n Observe that the case n = 1 corresponds to Theorem 67. See [77] for the details. The results of Section 7.5, were motivated, in part, by a joint project with Professor L. Caffarelli on the real Monge Ampere equation and Brennier rearrangements. It was Professor Caffarelli who suggested to me the study of the theory of compensated compactness, and Hodge decompositions, with the hope that we could adapt parts of it to our problem. The later part of that project I still have to accomplish. The relationship between optimal decompositions and a functional calculus for a positive operator can be extended in several directions. For example it would be of interest to relate our work here with the recent results of Dote [39], and Cowling, Doust, Mcln-
tosh, and Yagi [23].
Chapter 8 Sobolev Imbedding Theorems and Extrapolation of Infinitely Many Operators The Sobolev embedding theorems play an important role in these notes as a source of examples and applications. In this brief chapter we take a different tack we model from a form of the Sobolev embedding theorem an abstract interpolation theorem that suggests the possibility of a more general theory of interpolation. The relationship between these results and the theory associated with the operators fl~ of Chapter 7 ought to be investigated. In forthcoming work [78] we show that this set up can be used to study logarithmic Sobolev inequalities.
8.1
Averages of Operators
From the abstract point of view the general question is as follows. Suppose that we have a family of operators T = {T(t)}t~(0,~), each acting on a Banach pair ft,; then under what conditions can we assert that an appropriate average of this family is bounded on intermediate spaces? There are many possible variations on this general question.
128
CHAPTER 8. SOBOLEV IMBEDDING THEOREMS
Let us start by saying what types of "averages" we have in mind:
T = average(T(t))=
T(t)d#(t) ~0~176
in a suitable sense. For example if d#(t) is a probability measure, and T(t) = T, VT, then the answer is trivial: T is bounded on the interpolation spaces for the pair A. Another trivial result is: if f~' mo,q(t)d#(t) < oo, where rno,q(t) = IIT(t)llao.q__.~O,q then average(T ) is bounded on A0,q. The next result is stated in such a way as to make it easy to compare directly with an abstract version of the Sobolev embedding theorem. This remark should explain our somewhat awkward choice of parameters. T h e o r e m 85 Let A, B, be Banach pairs, and let {T(t)}te(o,~) be a continuous family of contractions, T ( t ) : fi, --~/~, such that for some
# > 1, we have
IlT(t)fllB, < ct-"/211fllAo Then, if we let formally, T-1/2 =
T(t)t-1/2dt
~0~176
we have, VO such that (1 - 0)/t > 1,
T-l/2 : fftO,q ~ B,7,oo VO such that ( 1 - 0 ) # > 1, 3' = [ 1 - ( 1 - 0 ) - 1 / z - 1 ] 0 + ( 1 - 0 ) - 1 # -1Proof. Let f C fi~0,q,we estimate the K functional, K(t, T-1/2f, B1). For M > O, to be specified later we split
T-1/2f = ]oMt-'/2TIt)f at + E
BO,q,
t-'/2T(tlf at = go +
and estimate each of these terms.
Let us observe first that from T(t) : A1 --:, B1, T(t) : 7to ---* B1, the estimates we have available and interpolation we have IIT(t)l[a,,q-,B, _< co,qt-~'('-~
8.1. A V E R A G E S OF O P E R A T O R S
129
Consequently,
119,11.1 -< ellf IAoq
t-ll2t-'(l-~
<_cllfll~o, M~-"(1; ~ To estimate obtain
go ~o~ we use the fact that
IIT(t)llao,~-~Bo,~ <- e, and
Ilgoll~o,q -< cMlllfllao,~ 1
Therefore, if we choose M = t.(T~-o, we obtain K(t,T-1/2f,[~o,q, B1) <_ go Aoq + tl]glJlBx 1
c t~
Ilfll~o,q
In other words, we have shown that r -1/2" Ao,q -+ (Bo,q,B1)~,oo
= B..l,oo
with 7 defined as in the statement of the theorem. [] C o r o l l a r y 86 Suppose that # > 2, then under the assumptions of Theorem 85 we have T-l~2 : A1/2,2 ~/~1/tz+1/2,2 Proof. We use the reiteration theorem. Let 0o, 01, be such that 0o > 1/2, 01 < 1/2, and 1/(1 - 0 i ) < #, fl E (0,1) such that (1 -/3)00 +/301 = 1/2, then the 7o, 3'1 corresponding to the 0i satisfy (1 - fl)70 +/~'Y1 = 1 / # nt- 1/2. [] It is not difficult to write down a rather more general version of this result for T -~ but we shall not pursue this here. We do turn however to the Sobolev e m b e d d i n g theorem of Varopoulos [102] that was our original motivation. E x a m p l e 87 (cf[102]) Let e -Ht be a symmetric Markov semigroup on L2(f~), and suppose that for # > 2 we tgave
Ile-m/llL~ _< c t-,~llfllL: Then, the fractional power, H -1/2 = f ~ t-1/2e-ntdt, is a bounded operator H-l~ 2 : L 2 ~ L.L~
130 Proof.
CHAPTER
8. S O B O L E V
The assumption
on
IMBEDDING
THEOREMS
{ e - H t ) implies that
_/a
lle-m flIL~ <_ c t ~ ltftlLX In fact, by duality e - H t : L 1 ---+ L 2, with norm smaller t h a n or equal than ct - e4, then, since e - H t = e.-Ht/2e-Ht/2, we get
Therefore, if we apply Theorem 86 we obtain: H-1/2 : L 2 = (L 1 , L oo) 89 ~ (L1,L~176
Now, (LI,L ~ ! !
)~+2 ,2 c ( L 1 , L ~ 1 7 6
and the desired result follows. []
= L~
1 r
1 2
1 /~'
Chapter 9 S o m e R e m a r k s on E x t r a p o l a t i o n Spaces and A b s t r a c t Parabolic Equations Interpolation spaces play an important role in the theory of abstract differential equations in Banach spaces, and there is a substantial literature devoted to this field (cf. [311, [2], and the references therein). Here we shall only briefly begin to explore the role of extrapolation spaces in the theory of parabolic equations in Banach spaces. We shall consider the limiting cases of the theory of maximal regularity for these equations on L v spaces (cf. [32], [36]). In order to apply our theory and be able to extrapolate from known estimates in the literature we need to be very careful about the norms that we consider. One could also obtain weaker results through the use of reiteration theorems, as in Chapter 6. R e m a r k . In a number of papers Amann, and DaPrato and Grisvard (cf. [2], and [33]) have studied a notion of "extrapolation spaces" of an entirely different nature than those considered here.
132
9.1
CHAPTER 9. A B S T R A C T PARABOLIC EQUATIONS
Maximal
Regularity
Consider the formula for the mild solution of the parabolic initial value problem u ' = A(t)u + f(t) (9.1)
u(0) = uo
where A is the generator of an analytic semigroup S(t), with dense domain DA, on a Banach space E, and Uo belongs to an appropriate subspace of E. The solution is formally given by,
u(t) = S(t)uo +
Z S(t - s)f(s)ds
(9.2)
Thus, a central problem in the theory is to give a meaning to each of the terms in (9.2). This problem can be often formulated in term of the continuity of the underlying operators
Suo(t) = s(t)uo,
rf(t)
=
s(t-
)f(s)es
in suitable function spaces. The delicate part of the problem is to deal with the operator 1-', which should be considered as an abstract singular integral operator. Interpolation spaces play a key role here since they are the natural ambient for the study of the operators at hand. We need to explain carefully how these spaces are constructed in our setting in order to be able to use estimates available in the literature. Recall that if A is the generator of an analytic semigroup then, there exist 0 E (~, ~r), c > 0, such that, if arg(z) < 0, then Vx E E,
z ( z I - A ) - ' x E < cllxllE
(9.3)
It is also well known that there exists a constant c > 0, such that the following estimates hold Vt > 0, Vx E E, IIS(t)xllE _< cllxll~
(9.4)
tllAS(t)x]lE < c]lxllE
(9.5)
Consider DA with the norm given by
Then, it is well known and readily seen (cf. [81, [87]) that
9.1. MAXIMAL REGULARITY L e m m a 88
Let R(t) = (A + tI) -1, then K(t,x;E, DA) ~
Proof. then
133
HAR(1)XHE
If x = f0 + fa is a decomposition with
(9.6)
fo E E, fl E DA,
IIAR(1)X,,E <_ ,,AR(1)fo"E + "AR(1)fl"E Now, write AR(~)fo = f o - ' and (9.3), we get
7R(;)fo, then by the triangle inequality, ' 1
IIAR(~)folIE < ciifo[l~ and
IIAR(~)fllIE ~ ctllAf, llE <_ct IIflIDA Therefore,
IIAR(~)xlIE <~c GIf011E4 tllf, llDA) and taking infimum over all decompositions, we get
IIAR(~)xlIE <_r
DA)
In the opposite direction, write
z = AR( )x + -~R(-[)x Then,
K(t,x;E, DA) <_ AR(1)x E + t < 2 AR(1)x E []
The spaces (E, characterized by
oA,0q
DA)o,q;K , 0 < 0 < 1,1 < q < or can be thus
----
:
{/o llAR, x
l'q <
OO}
134
CHAPTER 9. A B S T R A C T PARABOLIC EQUATIONS
The corresponding norms are given by
where co,q = ((1 -O)Oq) 1/q . For 0 = 0, we have
(E, DA)o,q;A. = {x.
(llAR(t)xlIE)q T
< oo}
Observe that compared with the spaces DA(O, q) as defined in [36], we have (E, Z)A)o,~;,,- = D ~ ( 1 - o,q) and
Ilxll(~,~A)~,q~ = c0,q IIxll~A(l_0,q) Let us also remark that the pair (E, DA) is mutually closed. For example, (cf. [13]) we have
l i m K ( t , x ; E, DA) ,~ lim t [[AR(t)x[[ E = I[X[IDA t-~O
t
t ~oo
Let us now return to the study of the operators S, and F. As we indicated above the operator S is not difficult to control, for example from (9.4) and (9.5) we see that
t llAS(t)xll E < cK(t,x; E, DA)
(9.7)
Therefore, integrating (9.7) with respect to t, we see that
IISxlILle0,r,DA) _< clIxlI(~,DA)(1)
(9.8)
Let us now illustrate the role of extrapolation spaces showing an end point result for F T h e o r e m 89 Let T > O, then if f E LI(O,T;(E, DA)o)), we have that A P f C LI(0, T; E)
9.1. MAXIMAL REGULARITY Proof.
135
Let us define the operator
FAr(t) = A(rf(t)) It is shown in [32] and [36] that FA: LX(O,T;(E, DA)o,1;K) ~ L'(O,T;(E, DA)o,1;K) with
HFAHL,(O,T;(E,DA)o,1;K)..~LI(O,T;(E,D,~~)o,ac0;K)-1, as 0 ~
0
(cf. [36], page 72). By extrapolation,
FA: ~ (~LI(O,T;(E,DA)o,1;K))-+ L'(O,T;E) It remains to identify the corresponding extrapolation space. Now, since for any p a i r / t we have (cf. [8])
K(t,f;L'(O,T;Ho),LI(O,T;H1)) ~ ~oWK(t,f(x);[t)dx
(9.9)
We see that
LI(O,T;(E, DA)o,1;K)--: (LI(O,T;E),LI(O,T;DA))O,1;K Therefore, ~ ( 1 L 1 ( 0 , T; (E, DA)o,1;K)) 0
0
= ~(~(LI(O,T;E),LI(O,T;DA))O,I;K) = (LI(O,T;E),LI(O,T;DA))o,r,K (by (2.22)) = (LI(O,T;(E, DA)o,I.,K) (by (9.9)) and the result follows, n As a consequence we have the following regularity result (cf. [36], Theorem 21)
136
CHAPTER 9. A B S T R A C T P A R A B O L I C EQUATIONS
C o r o l l a r y 90 Let f E (LI(O,T;(E, DA)o,1;K) then and moreover d r f ( t ) = A r f(t) + f
~Ff(t)
~ LI(DA),
One could prove a whole family of end point results for F in a similar fashion. For a more complete exploration of applications we refer to work currently in progress (cf. [2]) We conclude this chapter with an example illustrating how extrapolation spaces in this setting can be computed in a concrete situation. Let L be a second order elliptic operator on R ~, given by n
Lu =
~
Dj(aij(x)Diu) + ~ bi(x)D~u(x) + c(x)u
l <_i,j<_n
i=1
where Di - Oxi a and the coefficients aij, bi and c are uniformly continuous and bounded. Let
A: DA C L~(R n)
~
LI(Rn)
be the operator defined by Au = Lu, and where DA is defined by
DA = {u E C~](Rn) NLI(Rn) : Lu E L I ( R n ) } where U stands for uniformly continuous and bounded. Moreover, let A1 be the closure of B in L 1. Then Cannarsa and Vespri [16] have shown that A1 generates an analytic semigroup. In [37] it is shown that
(LI(Rn),DA1)O,1;K = ( L l ( / ~ n ) , W2(Rn))o,1;K , 0 • 0 < 1, 0 < q < cx).
(9.10) From this result it is easy to derive a characterization of the corresponding extrapolation spaces. T h e o r e m 91
(L1,DA1)O,1;K = (L 1, Wt2)o,1;K = {f'][f[[L1 +
fo w,(t, f ) Tdt < oo}
where co,(t, f) denotes the L ~ modulus of continuity of f.
9.1. MAXIMAL REGULARITY
137
Proof. Let 9 E (0, 89 by (9.10), and the reiteration property (6.1), we have
(L 1, DA1)0,1;g =
(L 1, (L 1, DAI )8,1;K)O,1;K
= (L 1, (L 1, W21)~,I;K)O,1;K and by reiteration (cf. [101]) - (L1, (L1, W: )0,,1;K)0,1;K __--(L 1, W1)0,1;K and the result follows. [] In conclusion we remark that another potentially interesting aspect of the use of extrapolation methods in this area are the constancy properties described in Chapter 6. The standard assumptions in the theory require either that the spaces DA(t) are constant in which case automatically all the interpolation spaces for the pair (E, DA(O) are constant in t, or directly assuming that suitable interpolation spaces for this pair are constant. As we pointed out in Chapter 6, for extrapolation methods we need less stringent conditions to guarantee the constancy of the spaces.
Chapter 10 Optimal Decompositions, Scales, and Nash-Moser Iteration In this chapter we consider the relationship between interpolation and the implicit function theorems usually associated with the names of Kolmogorov-Nash-Arnold-Moser. These results play an important role in many problems of non-linear partial differential equations. We show a connection between optimal decompositions and iterative methods to solve non-linear equations in Banach spaces. Implicit function theorems have been studied in the very general framework of scales of spaces (cf. [47] and the references quoted therein). There have also been a few contributions using interpolation theory (cf. [68]), but as far as we know there have been no attempts to relate the methods of the theory of scales with real interpolation theory through the use of optimal decompositions. Our program in this chapter is as follows. First we review Moser's [81] original approach to solve non-linear equations and point out the close relationship between Moser's methods and the real method of interpolation. In the following section we relate the concept of a scale with "smoothing" with optimal K and J decompositions. It turns out that smoothing allows one to produce, by rescaling, optimal decompositions for all interpolation spaces between any two spaces in the scale. We use this idea to identify the ad-hoc spaces that crucially intervene in the Nash-Moser theorem as formulated by
140
CHAPTER 10. OPTIMAL DECOMPOSITIONS
Hormander [49] as those obtained by the (., .)8,00 method. Finally in the last section we review very briefly Hormander's approach to the Nash-Moser theorem using paracommutators. These operators are directly expressed in terms of optimal decompositions.
10.1
Moser's Approach to Solving NonLinear Equations
Although the main import of the iterative methods is to solve nonlinear equations we start by recasting the simplest possible iteration method of Moser for solving linear equations. A perusal of [81] shows that although this paper was developed independently from the theory of interpolation spaces, which was in a developing stage at the time, there is a close contact between these theories. Indeed, since then other authors have developed this remark in different directions (cf. [68]). We shall now rephrase the setting of Moser's theory [81] using interpolation theory. Let {Xe}ee[0,1] be an ordered, decreasing, scale of spaces. We assume, moreover, that the scale behaves almost as an interpolation scale in the sense that the spaces in it satisfy
(Xo, Xl)o,1;j C X8 C_ (Xo, Xl)8,oo;K
(10.1)
Note that (10.1) will be satisfied if the spaces Xo, 0 E (0, 1) are constructed by letting
7 (Xo, Xl) with {9r0)0r a family of exact interpolation functors with each 9% exact of order 0. Let L : X1 ~ X0 be an operator, and define the KL functional by: KL(t,g, Xo, Xa) = inf { l l L w - gllxo + tllw][x~}
w~X1
The spaces ~(L) ~,q;K are defined by imitating the classical definition, thus, for example, g E (Xo , X1 ~(L) /O,oo;Kmeans that 3C > 0 such that Vt e (0, 1),
If~ (t, g, )Co, Xl) _< Ct ~
10.1. MOSER'S APPROACH
141
This can be reformulated as saying that g 6 (X0, ~Yx r l O ~(L) ,oo,K (0, 1), 3Dl
-'9 ,'- Vt 6
gL(t,g, Xo, Xr) ~ IILDgL(t)g--g[Ixo + t[IDgL(t)g[[xa <_ c tOIIgll~L2;g This forces, in particular, the convergence of LDg(t)g to g, as t ---* 0, at the rate t ~ while at the same time we have control on the growth of IID~:L(t)g]]x,. Thus, {Du,~(t)g} forms a family of "approximate solutions" to the equation
Lf = g The issue then becomes the convergence of DKL (t)g, this necessitates a rescaling of the parameters, i.e. the use of other spaces in the scale and what, after discretization, essentially amounts to the J method: that is the study of the convergence of the sequence through the study of the speed of decay of successive differences. Let us plot through the details in the classical case, i.e. when L = Identity, to see that in this case the study of the convergence of the family of approximate solutions really amounts to (a weak version of) the fundamental lemma of interpolation theory. Indeed, the argument that Moser gives actually proves that (X0, Xa)O,o~;K C (Xo, X1),,I;j C Xs, if s < 0. In fact this is best understood through discretization. So let g 6 (Xo, X~)O,~o;K and let t ~ 2 -", g, = DK(2-")g, and study the convergence of g = ~,~--1 gn+l g, 9By the triangle inequality -
-
I I g - + l - g.llxo -< ~2 -"~ while, IIg~
- ~llx,
-< c2 -"(~
which implies Ib.+a - g.llx. _ c(Ib.+a
- g . l l x o ) 1-" ( l b . + ,
- g.llx,)"
Hgn+I --g.I[X, <_ C2-nO(1-")-n(O-1)8 = C2"(~-~ This geometric decay shows that E,~ 1 g,+l - g, converges in X,. In fact the estimates above show that J(2-~,gn+l - g n ; X o , X1) <_ 2 -n~
142
C H A P T E R 10. OPTIMAL DECOMPOSITIONS
and therefore g E (Xo, X1)s,1;J. With suitable assumptions the analysis also works for operators L acting on scales. In particular, let us consider the embedding
( Xo, X1 "~(L) ~(L) then That is, we try to prove that if g is in the space :Xo [ , X 1)O,oo;K the equation L f = g, has a solution f E (Xo, X1)8,1.,j. The only change needed is an appropriate condition in order to be able to estimate Jig,+1-g,,]]Xo by I[Lg,+l- Lgnl[Xo, (which of course was not needed in the case L = identity). Thus if we assume, for example, that L is an operator such that L : X1 --~ X0, and such that q c > 0, such that
Ilzllxo _< CIIL llxo, W e x , the argument above works. Thus, we should see these results as extensions of the fundamental lemma. This description suggests a number of projects (eg. extend the notion of approximate solution, extend the reiteration argument, extrapolate the end point results, compute the K functionals for specific operators, etc)...
10.2
Scales with S m o o t h i n g and Interpolation
We study scales of spaces with an additional structure provided by "smoothing" and we relate them to optimal decompositions for interpolation spaces. Let us recall the concept of quasilinearizable Banach pair (cf. [86], [81) D e f i n i t i o n 92 Let 7t be a Banach pair. We say that fi~ is quasilinearizable if there exist families of linear operators Vi(t) : E(7t) A(A), i = 0, 1,t E R+, and moreovevVt > O, we have,
Vo(t)f + V~(t)f = f
IIVo(t)fIIAo<_ cIlfllAo , llVo(t)fIIAo<_ ctllflIA~
10.2.
SCALES
WITH
143
SMOOTHING
t ] l V l ( t ) f l l A 1 ~_ cllfllAo,tHg~(t)fllA1
<_ ct]lfllA1
where c is an absolute c o n s t a n t
The motivation for this definition is given by the readily verified fact that if A is quasilinearizable then K(t,f;fit)
~ IIVo(t)fllAo + tHV~(t)fllA 1
In the case of an ordered pair .~ we only need families of operators {V~(t)}tei, i = 0, 1, satisfying all the conditions above, with I = (0,1]. We now turn to the setting of the analysis in [50]. Let {Ea}a>0 be a decreasing family of Banach spaces with injections E b C E ~ of norm one, when b _> a. Assume that a family of continuous linear operators {S(0)}e>l, is given such that s ( 0 ) : E ~ --+ E
= N E~ 0 _> 1 a
IIS(0)/llb _< ctl/ll~-, b _
(10.2)
a
(10.3)
IlS'(O)fllb <_ c ob-a-lllflla , u e E~
S(O)u --+ v in E ~ as 0 --+ oe
:. v = u
(10.4)
As a consequence of (10.2), (10.3) and (10.4), we get (cf. [49]) ][S(O)f[[b <_ c O b - a l l f l [ ~ ( b - a) -1, a < b
flu
- S(O))fllb <_ c
Ob-~llfll~(a
-
b) -1, a > b
(10.5) (10.6)
A consequence of (10.5) and (10.6) is the logarithmic convexity of the norms of the spaces in the scale: (cf. [49])
II/llc _< callfll2llfH~ -~
(10.7)
where c = Aa + (1 - A)b, cA < [ ( b - a)A(1 -/~)]-1. A scale {E"},>0 associated with a family of operators {S(0)}o>, with properties (10.2), (10.3), (10.4), (10.5), (10.6), (10.7), is usually called a B a n a c h scale w i t h s m o o t h i n g . Associated with {E~}a>0 Hormander defines the spaces {E.~}~>_o as follows.
CHAPTER 10. OPTIMAL DECOMPOSITIONS
144
D e f i n i t i o n 93 For a > 0, let E~. = {u E E ~ : 3 M > 0 such that I[ull0 < M, IlS'(O)u]10 < M 0 -~-1, Ils'(O)ul[~+l g M, O > 1} and [lull~ is the infimum of all the admissible M's. Our first observation is that smoothing provides optimal decompositions for all interpolation scales between two fixed spaces in the scale. L e m m a 94 Let {E~}~>_0 be a Banach scale with smoothing. Then, (i) For 0 < a < t5 the pair (E '~, E~), is quasilinearizable with Vo(t) = I - S(t-'Y), Vl(t) = S(t-~), 7 = 1 / ( f l - a ) , t E (0, 1]. 5i) ForO <_ a < /~, 0 < 0 < 1, 0 < q <_ 0% f E ( E ",E~)o,~;: can be represented by f = S(1)f + ( - 7 ) folS'(t-~)ft-'r dtt
and letting D j ( t ) f = (-7)(t-'YS'(t-'r)f)
Ilfllo,~;~ Proof.
~
{ fol[t-~ J(t, D j ( t ) f ; E", E~)]qd--~}1/q
(i) We verify all the conditions of Definition 92:
I1(I - S(t-~))fll.
___211fll. (by the triangle inequality and (10.2))
I1(;- s(t-~))fll~ <
(/~- c0-1t-lllfl}a (by (10.6))),
tllS(t-~)flla < ctt-lllfll. (by (10.5)) tllS(t-~)fll ~ < ctllfll a (by (10.2)) (ii) Recall that by the fundamental lemma t~t(DK(t))= Dj(t), and use (i). We now identify the Moser-Hormander spaces E.~ as interpolation spaces. P r o p o s i t i o n 95 E.~ = (E ~ E~+l)z~r,oo;J.
10.3. A B S T R A C T N A S H - M O S E R T H E O R E M
145
In view of Lemma 94 (ii), if f E E,~ then, with D s ( t ) f t - T S ' ( t - ~ ) f , 7 = 1/(a + 1), Proof.
=
IIDj(t)fllo <_ cM t - ' t "d~+~), IlDj(t)fll~+l < cM t -~ Therefore,
t-~/(~+l)J(t, D j ( t ) f ; E ~
TM) < cMt-~/(=+')max{t~/(~+l),tt-'/(~+i)} < cM
and taking the supremum over all t < 1, we obtain
Ilfll(EoE~+l)~,oo; ~ < cM Conversely, the argument also shows that Ilfll(EOEa+l)_~r,oo;j <_ M implies IlfllE.~ < cM. We say t h a t a Banach scale {Ea}a_>0is compact if for every pair of indices a < fi the injection E z C E ~ is compact. It is readily seen using L e m m a 94 and the known theorems on interpolation of compactness t h a t (cf. [49]): If there exists a pair of indices a l < fil such that the embedding EZl C E ~1 is compact then the same is true for all the embeddings E ~ C E ~ , c~ < ~. E x a m p l e 96 (cf. [49]) Let K C t ~ be a compact set, let X E C~~ t ~ ) be such that x = 1 in a neighborhood of g , and r E C~~ t ~ ) which is 1 in a neighborhood of the origin, and let ~o = r ~oo(x) = O'~qo(Ox), ~ > 1. Set Sou = X(~o * u), then the scale of Holder spaces {H=}~>o is a Banach scale with smoothing given by the {So}.
10.3
Abstract Nash-Moser
Theorem
Let { E6t }0>0, {Fb}b>0 be Banach scales with their respective smoothing operators SE, SF. The following results are established in [49]. Let q~ : E ~176 N V ~ F ~ , where V is a neighborhood of 0 in E.". Assume that:
CHAPTER 10. OPTIMAL DECOMPOSITIONS
146
i) 9 has a differential r for u E E ~ V, and r has a right inverse r such that for certain a l , a2, with 0 < 41 < ~ < 42 we have:
IIr
--- ~ y~(1 + II~'llA,(~))llglls,(~)
where 41 < a < a2, u E E ~176 Cl V, g E F ~176 the sum is finite, and Aj(a), Bj(a) are increasing linear functions. ii) The map (u,g) ~ r is continuous from E ~~ N V --* E ~ . iii) max{crl, #} < a < 42
iv) Bj(~) < ~ - . + ~, for 4~ < . < . : v)Aj(a) + B~(a) < 4 + ~, for 41 _< ~ _< -~ vi) The scale {E ~} is compact. Then r E E.~, if g E F.~. Moreover, suppose that r > 0 is small enough so that Aj(a) + Bj(a) + r < a + ~, then there exists a neighborhood W~ of 0 in E.~-~, such that if u E IV., g E F.~, then the operator
Tr
=
/1
r
at
is well defined, and the equation
- r
= (Tr
- Tr
has a solution u E W~ for all g E F.~ with sufficiently small norm. The crucial issues here are: i) to use the assumptions in order to establish the mapping properties of Tr and ii) to use the compactness together with the Leray-Schauder fixed point theorem to actually solve the equation. Suppose further that
vii) r = o viii) II(r162
< c E(1 +llullm, + I1"I1,.~) II~'-v II.,; Ilwllr-~",
for u, v E E ~ Cl V, w E E ~176 where m~, m~, ,,tjm E [al,
4 2]
ix) 4 > ~ax{m~ + m~'}/2,4 > max{r~; + . ~ ] + m7}/3 Then for every f E F.a with sufficiently small norm one can + find u E E." with small norm such that u (t) = r ft r ds ~ to u (in E.~) and r (t)) ---} f (in F.~) as t ~ cr where the convergence is strong in E ~ (resp F b) when a < 4 (resp b < ~). At this stage it is routine to generalize the results of [49] for more general interpolation scales of functors with characteristic functions
10.3. A B S T R A C T NASH-MOSER THEOREM
147
given by quasi concave functions more general than powers. Since we do not have any application for such results at this time we shall leave these developments for another occasion.
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Index C Calderon- Zygmund Operator, 40 conventions notation, 34
functor interpolation, 7 fundamental lemma, 14
G
decomposition almost optimal, 98
Gagliardo completion, 13 diagram, 13
E
H
D
estimates weighted norm, 31 exact exponent, 11 extrapolation exact, 8
F F- method interpolation, 71 family strongly compatible, 8 function characteristic, 11 presentable, 60 quasi-concave, 11 tempered, 18 functional J, 11 K, 11
Hilbert Transform, 29
I ideals operator, 67 inequalities division of, 28
K/J, 25 rearrangement, 29 integrability double exponential, 91 interpolation functor characteristic function, 11 complete, 25 exact, 11 of exponent, 11 exact, 8 space, 7
J Jacobian, 47
159
INDEX
K
S
K/J inequality, 20
8cale
M Macaev ideals ideals of operators, 67 maps orientation preserving, 46 Sobolev, 47 measure representing, 17 method extrapolation, 8 minorant largest logarithmically convex, 17
N notation conventions, 34
O Operator Calderon-Zygmund, 29, 40 orientation preserving map, 47
P pair compatible, 7 mutually closed, 13 quasilinearizable, 142 regular, 25
R reiteration constancy property, 76
ordered, 9 with smoothing, 143 Schatten classes ideal of operator, 67 semigroup Hermite-Ornstein-Uhlenbeck, 90 SFL, 14 Sobolev Logarithmic Inequalities, 89 Spaces, 40 space exact, 7 extrapolation, 8 Hormander, 144 intermediate, 7 Lorentz, 23 rearrangement invariant, 29 Sobolev, 47 Zafran, 37
W weight, 99
Symbols Ae,q;J
AO,q;K Ae,q Ap,1;a
it p,oo;K Ar A_-(~);j A(~);K A_(s) AS
-~O,q;K DA Dj(t) DK(t) ~, fl~
2.1 2.1 2.1 2.7 2.1 2.1 7.6 2.2 2.2 2.2 3.1 6.1 9.1 7.1 7.1 7.1
A(.)
E(.)
Ep(.) ExpL ~ m(7,) ~ J(t,a; ft) K(t,a;A) Kn(t,a; fi) A~ L(Logn) ~ LP(LogL) ~
r(a)
w:(a,R~) S~,S~,SM
2.1 2.1 2.1 10.2 2.2 2.1 2.1 2.1 10.1 2.2 2.2 6.4 2.1 4.2.2 4.2.1 5.2
Printing: Weihert-Druck GmbH, Darmstadt Binding: Theo Gansert Buchbinderei GmbH, Weinheim
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