Lecture 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].