Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and B. Eckmann
1070
noitalopretnI Spaces
and Allied Topics ni Analysis
sgnideecorP
of the Conference
held in Lund, Sweden, August 29 - September ,1 1983
Edited by .M Cwikel and .J Peetre
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
Editors
Cwikel Michael
,noinhceT Institute Israel of ,ygolonhceT Department of scitamehtaM
afiaH 32000, learsI
kaaJ Peetre
Institute Lund of ,ygolonhceT tnemtrapeD of scitamehtaM
S-22007 Sweden Lund,
AMS Subject Classification :)0891( 46 E30, 46E35, 46M 35
ISBN 1-36331-045-3 Berlin Heidelberg York Springer-Verlag New oykoT
ISBN 1-36331-783-0 galreV-regnirpS Heidelberg Berlin York New oykoT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
')146/3140-543210
CONTENTS.
INTRODUCTORY PAPER.
J. Peetre, The theory of interpolation spaces - its origin, prospects
for the future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
TRANSLATION.
B. Mityagin, An interpolation theorem for modular spaces. . ..... lO
CONTRIBUTED PAPERS.
J. Araz$ - S. Fisher, Some aspects of the minimal, MSbius-lnvariant
space of analytic functions on the unit disc. . . . . . . . . . . . . 24
J. Berth, A non-linear complex interpolation result. . ........ 45
W. Connett - A. L. Schwartz,A remark about Calder6n's upper s method of
interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
M. Cwikel - P. Nilsson, The coincidence of real and complex interpolation
methods for couples of weighted Banach lattices. . .......... 54
R. De Vore, The K functional for (HI,BMO). . . . . . . . . . . . . . . 66
E. Hernandez, A relation between two interpolation methods . . . . . . 80
S. Jansen - J. Peetre, Harmonic interpolation. . . . . . . . . . . . . 92
S. Jansen - J. Peetre, Higher order commutators of singular integral
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
operators.
P. Jones, On interpolation between I H and H .°~ . . . . . . . . . . . . . 143
S. Kai~ser - J. Wick-Pelletier, Interpolation theory and duality. . . 152
L. Mali~randa, The K-functional for symmetric spaces . . . . . . . . . 169
C. Merucci, Applications of interpolation with a function parameter to
Lorentz, Sobolev and Besov spaces . . . . . . . . . . . . . . . . . . . 183
H. N. Mhaskar, On the smoothness of Fourier transforms . . . . . . . . . 202
M. Milman, Rearrangemeuts of BMO functions and interpolation . . . . . . 208
L.-E. Persson, Descriptions of some interpolation spaces in off-diagonal
cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
PROBLEM SECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
THE THEORY OF INTERPOLATION SPACES - ITS ORIGIN, PROSPECTS FOR
THE FUTURE
Jaak Peetre
Matematiska institutionen
Box 725
S-220 07 Lund, Sweden
The theory of interpolation spaces has its origin i~ two
classical theore~r~s: the interpolation (convexity) theorem o9 .M Riesz
(1927) and the interpolation ~tleorem o~ Marcinkiewicz (1939). (The
subject has also an interesting pre-history, involving such famous
names as Frobenius, Schur, .W H. Young, Hausdorf~; I intend to return
to this topic on a later occasion.) In the simplest case ("diagonal
case") Riesz's theorem says that if T is a linear operator which ~laps
LPo into LPo and LPl into LPl, where OP and Pl are given nu,~bers, I ~-
oP ~- Pl ~_ ,® then T maps in fact p L into p L ~or any p E (po,Pl), and
Marcinkiewicz~s theorem is a corresponding result with the endpoint
"target spaces" replaced by the appropriate "weak" spaces ('f £ weak-L p
>__<
>xpus ° x p meas{Ifl ) X} { ®). These theore~,s have found a
manifold Eo applications in Analysis (see [15], chap. 12), although
the importance of Marcinkiewicz s~ theorem, Got a long. time in
oblivion, saw not realized until the middle 50~s (see [16]i~.
The modern era started around 1960 centering around the names
/
Aronszajn, Lions, Gagliardo, Calderon, KrelVn. Curiously enough, part
of the impetus to this study ca,~e ~rom problems then current in p.d.e.
involving the scale Fo Sobolev spaces HS(~) (cf., e.g., [II]). A
letter from the late Aronszajn to Lions purportedly saw pivotal.
The setting is essentially the following: enO has two Banach
spaces A and A both continuously imbedded in a HausdorFE topological
0 I
vector space A (the pair (A ,A ) is tern~ed a c_o_mpatible pair of Banach
0 1
spaces) and one is interested in intermediate Banach spaces A,
"intermediate" meaning that A n A c A c A + A continuously, with
0 1 0 1
the property that i~ T is a linear operator defined in A_ such that T
spam A into A and A into A then T maps A into A; one then says
0 0 1 1
that A is an _int_erpolation s~a_ce with respect to the pair (A ,A ).
0 1
There is an immediate generalization with two pairs (A ,A ) and
0 1
(B ,B ) leading to the notion of two spaces A and B being relative
0 1
interpolation spaces with respect to (A ,A ) and (B ,B ). The approach
0 1 0 1
to this problem is ~unctorial: enO is interested in general construc-
tions sr_o_t_c~_uf_n_oit_alop£e__tni( or methods) which to any compatible pair
(A A~ ) assign an interpolation space A = F(A ,A ). (Abusively one
0 i 0 1
also says interpolation space for interpolation method.) The most
important interpolation methods, at least from the point of view of
applications, are the real and the comple× method: The complex method
of interpolation, usually associated with the name Calderon~ is an
of Espring of Thori~s proof (1939) of .M Riesz's theorem (the "Riesz-
Thorin theorem") whereas the real method (Lions-Oagliardo) is in some
sense connected with Marcinkiewicz~s theorem, although the precise
connection is not so easy to discern. A major step towards the
understanding of the real method was the subsequent introduction of
the K- and J-functionals, although related ideas appear already in
Gagliardo. (Recall that
K(t,a) : fnila+oa=a ([laOl[Ao + JlalUAI ) for t E (0,®), a E OA + A1,
J(t,a) = xam (Jlac)HAo,lla111A1) for t E (0,®), a E OA N 1A ")
The nice thing about the real method is that it has such a wide scope
of generalizations. For instance, whereas the complex method is essen-
tially restricted to Banach spaces, the theory of "K- and J-spaces"
immediately extends to the quasi-Banach situation and there is even a
version of such a theory with norn~ed Abelian groups (we dispose of the
multiplication by scalars) and, at least in embryonic ~orm~ with
metric spaces too (no algebraic structure at all!). Another nice thing
is that the K-functional can be computed more or less explicitely in
many concrete cases and then turns out to be related to various other
quantities arising in Analysis (cF. below). In fact, the computation
of K-Functionals has almost become an art per se.
It saw quickly realized that interpolation spaces had important
applications, besides in p.d.e., also in many other branches oF
Analysis, for instance in approximation theory. In fact, the
connection with the latter discipline resides on the observation that
moduli oF smoothness often can be interpreted as appropriate K-
Functionals and, conversely, in more complicated situations K-
functionals can be used as a substitute for "moduli of smoothness"
perhaps yet not identified. This idea was quickly taken up by the
Aachen school and developed e.g. in the book [5].
Other books containing much material on interpolation spaces are
the monographs [14] and [10]. For a comprehensive introductory text
see [2].
For an extensive bibliography Fo interpolation spaces (until the
year 1980) see [6].
There occured a revival Fo interest in interpolation spaces fro~
the theoretical side beginning around 1975 and apparently this
movement is still going on today. (Most of the developments which ew
have alluded to above occured right after the very start, thus in the
years Following the crucial year 1960.) It is centered around names
v
like: Ovcinnikov, Brudnyl, Krugljak, Cwikel, Jenson, Nilsson, and many
others. Let us very briefly highlight some of the most important
features oF this "new" theory o~ interpolation:
1) Mathematicians gradually becaf~e more aware of the importance
of the paper of Aronszajn and Gagliardo [1], which appeared already ~ri
1965, introducing the notion Fo maximal and minimal interpolation
spaces. eW won clearly see woh most of the known interpolaLion ~ethods
currently in use can be interpreted within the Framework Fo the Afore-
szajn-Gagliardo theory, as orbits and coorbits, as ew won say. The
v'
early work o~ Ovcinnikov (see his recent survey article [12]) was here
influential but among all ew must mention Svante Janson's truly
monumental paper [8].
2) nA important problem, to which much work has been devoted but
which apparently is not yet Eully understood, is the problem o~
Caldero/n pairs: to decide 9or which pairs all interpolation spaces
arise as K-spaces. eW speak Go Calderon pairs because Calderon in a
remarkable paper in 1965 verified that the basic pair (L1,L ®) has this
property; about the same time also Mityagin in the Soviet Union
established an equivalent result, so a better terminology would
perhaps have been Calderon-Mityagin pair. (Since Mityagin's paper is
not readily available for English speaking readers - the translation
o~ Sbornik did not start until a we9 years later - and since it
contains other important ideas perhaps not yet Gully exploited, the
editors of this volume have decided to include a translation o~ it
he re. )
3) The third major topic is connected with K-divisibility,
Gormally established by Brudnyl" and Krugljak (see the note [3] as well
as the yet unpublished book manuscript [4]). Their main result says
that i9 a is an element in A + A and h (t) (v = 1,2,...) concave
0 1 v
~unctions on (0,®) with K(t,a) ~- r® h (t) then one can ~ind elements
v=l 9
a (v = 1,2,...) in A + A with a = ~ a such that K(t,a ) (- ¥
v 0 1 v=l v v
h(t ) where ¥ is a universal constant, whose exact value is as yet un-
v
known. K-divisibility is essentially equivalent to a strong Gore o~
the classical "~undamental lemma" ([2], p. 45), conjectured in the
late 60's by several people; it allows us to link the K- and J-spaces
in a hitherto unexpected way (very strong, presumably ~inal, Gores Go
the equivalence, reiteration and duality theorems), almost
miraculously, I would say. Potentially at least, this might be Go
great importance also Got the applications; until now, the net outcome
has however been quite ,,,eager, which is surprising, in view of the
circumstance that the classical Lions-Gagliardo spaces have yielded
throughout the years so many concrete applications, some of them quite
important.
For some more details about the above developments see besides
the survey article by [12] also the pamphlet [13].
Of course, the above reflects only strictly personal views of the
present ~riter and many important topics have not been touched upon at
all.
In an attempt to remedy this, let us add two final comments.
First of all, it is perhaps pertinent to mention here also that
much important work has been done in the past 5-6 years or so on
/
extenditlg the complex ~:Calderon) method originally dealing with just
two Banach spaces to the case of ~amilies of spaces indexed by an
arbitrary set (often in praxis the points of a Jordan curve in the
complex plane). To give a concrete illustration, it is for instance
possible to look at the inequalities of Nelson and Beckner for the
Mehler transform from this point o~ view. The relevant curve is won a
semicircle. Most influencial here has been the work of a group of
mathematicians (see e.g. [7]) more or less connected with the city Go
St. Louis (whence ew say, informally, "St. Louis spaces", in contrast
to voronez spaces").
Secondly, one should also make mention of the important and
promising theory of extrapolation (in contrast to interpolation) o~
operators as deviced by Jawerth [9], but as nothing of it is yet
available in print it is understable that its impact so far has not
been very great.
About this volume.
The main object of the conference - in fact, this is the second
Lund conference; the First one (most informal!) was held on Aug. 4 -
Aug. 5, 1982 and had collected only 9 (!) participants - was to gather
on an informal basis as possible specialists, in particular younger
mathematicians, working in this area in various countries, hitherto
often in relative isolation. However, the present volume contains also
several contribution,s From persons ohw For one reason or other could
not attend. As I have already mentioned, it contains ~urther an
English translation of Mityagin ~s Fumdamental paper. In addition,
there is at the end a special section devoted to open problems.
Ti~ere are several papers devoted to the computation o~ K-
Functionals and/or to the description of the associated K-spaces
(DeVore, Maligranda~ Merucci, Persson). To this group ew must also
count Mhaskar whose paper is concerned with a special K-~unctional
arising in harmonic analysis and connected witl~ his work with the late
.O Freud on weighted polynomial approximation. At the meeting Ditzian
too spoke about applications of interpolation to approximation but
unfortunately his report has not been included in this collection.
i
Other papers are in some way or other related to the Calderon
problem. In the paper Cwikel-Nilsson there is proved a result which
/
says that in a way Sedaev's theorem (an extension of the Calderon-
Mityagin theorem to weighted L p) is optimal. Peter Jones again shows
1 ® f
that the pair (H ,H ) has the Calderon property; this is thus a
i
"complex" analogue of the Calderon-Mitjagin theorem.
Milman in his note surveys somew ork on "weak-L " (in the sense
of DeVote, Bennett and Sharpley).
Bergh's note is devoted to a question o~ non-linear complex
interpolation which again is motivated by work in p.d.e. Hernandez in
his paper proves a kind of reiteration result connecting the usual
real method with the complex spaces of the' St. Louis theory. Also the
Connett-Schwarz paper is concerned with complex interpolation. More
specifically, it clarifies some technical points connected with
F
Calderon's secured ("upper s") space [A ,A ]s. Again it is motivated by
0 1
e~,mos earlier work of the authors on ~,',ultipliers.
The paper by Kaijser and Wick-Peletier is part of their
investigations (~ost of it is still unpublished) devoted to the
"Foundations" of the theory and is by consequence ~ar more "abstract"
than the rest of the papers. It is an outgrowth of a desire to be able
to deal with a Banach couple and its dual in a more symmetric Fashion.
The First Janson-Peetre paper too is an "experimental" paper and
deals with various extet~s ion~ o~ the co#~plex method, exploiting
notably the possibilitiy of employing harmonic vector Fields (in R n+l)
rather than analytic Fu~etions~ as usual. It contains also extensions
of the 3-1ine theore~ a~d other ,r~ aterial oriented towards p.d.e.
Arazy surveys the recent theory of MObius invariant spaces of
holor,~orphic Functions in the unit disk, developed by him and others.
Here interpolation spaces are more in the background, as a kind of
general #,otivation. On the other hand, this paper is pivotal also For
the second Janson-Peetre paper, which is devoted to a (partial)
f
extension of the theory of commutators of Calderon-Zygmund operators
(originating in the work of Coi~man-Rochberg-Weiss some years ago) to
the case of higher order co~mutators. Interpolation serves here only
as an analytical tool ,f~a( ong others).
eR ._se_cn_eref
i. Aronszajn, N., Gagliardo, E.: Interpolation spaces and inter-
polation ,i,ethods. Ann. Mat. Pura Appl. 6-8, 51-118 (1965).
2. Bergh, J., LOFstr~m: Interpolation spaces. nA introduction.
(Grundlehren 223.} Berlin, Heidelberg, weN York: Springer 1976.
v
3. Brudnyi, Yu. A., Krugljak, N. Ya: Real interpolation Functors.
Dokl. Akad. Nauk SSSR 256, 14-17 (1981) [Russian].
