Selected Papers of Antoni Zygmund Volume 3 Mathematics and Its Applications (East European Series) Managing Editor: M.HAZEWINKEL CenJre/or Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: A. BIALY NICKI-BIRULA. Institute 0/ Mathematics, Warsaw University, Poland H. KURKE. Humboldt University, Berlin, CD.R. J. KURZWEIL. Mathematics Institute, Academy 0/ Sciences, Prague, Czechoslovakia L. LEINDLER. Bolyai Institute, Szeged, Hungary L. LOVA sz. Bolyai Institute, Szeged, Hungary D. S. MITRINOVIC. University 0/ Belgrade, Yugoslavia S. ROLEWICZ. Polish Academy 0/ Sciences, Warsaw, Poland BL. H. SENDOV. Bulgarian Academy o/Sciences, Sofia, Bulgaria 1. T. TODOROV. Bulgarian Academy o/Sciences, Sofia, Bulgaria H. TRIEBEL. University 0/ lena, CD.R. Volume 41/3 Selected Papers of Antoni Zygmund Volume 3 edited by A. Hulanicki Unil'ersity of Wroclaw, Po/and and P. Wojtaszczyk and W. Zelazko Institute o/Mathematics, Po/ish Academy of Sciences, Warsaw, Po/and KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON Library of Congress Cataloging in Publication Data Zygmund, Antoni, 1900- [Selections, 19891 Selected papers of Antoni Zygmund I edited by A, Hulanicki and p, WOjtaszczyk and W, Zelazko. p, cm. -- (Mathematics and its applications. East European series) <U.S. set) 1. Harmonic analysis. 2. Zygmund, Antoni, 1900- I. Hulanicki, Andrzej, 1933- II. WOjtaszczyk, Przemys{aw, 1940- III. Zelazko, Wies{aw. IV. Title. V. Series; Mathematics and its applications (Kluwer Academic Publishers). East European series. OA403.Z9425 1989 515' .2433--dc20 89-20033 ISBN-I3: 978-94-010-6962-5 e-ISBN-13: 978-94-009-1045-4 DOl: 10.1007/978-94-009-1045-4 Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-ii'ee paper All Rights Reserved © 1989 by Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. SERIFS EDITOR'S PREFACE 'Et moi ..... si j'avait su comment en revenir, One service mathematics bas rendered the je n'y serais point aile: human race. It bas put common sense: back Jules Verne where it belongs, on the topmost shelf next The series is divergent; thCldorc we may be 10 tile dusty canister labelled 'discarded 1lOII- ICDIC'. able 10 do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'!tre of this series. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFO', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics." By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interreIations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. If anything. the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the non linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci- vi SERIES EDITOR'S PREFACE ate what I am hinting at: if electronics were linear we would have no fun with transistors and com puters; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre quently proved the shortest path between 'rea!' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading - fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five sub series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis cipline which are used in others. Thus the series still aims at books dealing with: - a central concept which plays an important role in several different mathematical and/or scientific specialization areas; - new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. These three volumes of selected papers by Antoni Zygmund -of which this is the third one -are the first of their kind in this series. More are scheduled to appear fairly soon. Often, collected or selected works are treated as separate projects on their own and are not placed in any series what ever (or are placed in a series of collected works); so possibly a few words of explanation, or at least motivation, are in their place. The interconnectedness of all things is something in which I (like Douglas Adams' creation Svlad Cjelli) firmly believe, especially in mathematics and science (though for vastly different rea sons). In fact this trust in the importance of interrelations is a main driving force behind the whole Mathematics and Its Applications series. Interconnectedness between things mathematical can be found in monographs, especially when written with that point in view, and in selected proceedings of conferences of an interspecialistic character, but it can also be found in the work of a single scientist, especially a great one, a point that needs no discussion in the present case; and, in fact, in such a case the refusal to restrict con siderations to the domain of one labelled, classified, and recognized subdiscipline may take on dramatic forms. Abel recommended to read (only) the works of the original masters. I have my own relatively modest collection of collected and selected works, and, with increasing awareness of just how valu able Abel's remark is, I find myself dipping into them more and more frequently. In addition, the great ones often have the invigorating habit of scattering their papers far and wide; add ups and downs in the popularity and status of journals, an increasing tendency to publish important material only in proceeding volumes, and finding all the relevant stuff of a particular author on a particular topic may well become something of a serious chore in itself. Moreover, if present trends continue, very few libraries will be able to afford (more or less) com plete collections of even only the more important journals, proceedings, and other edited volumes. All this makes judiciously chosen selected/collected works (and other ways of grouping interest ing material) particularly inviting and attractive. The shortest path between two truths in the Never lend books, for no one ever returns real domain passes through the complex them: the only books I have in my library domain. are books that other folk have lent me. J. Hadamard Anatole France La physique ne nous donne pas seulement The function of an expert is not to be more l'occasion de ri:soudre des problemes ... eIle right than other people. but to be wrong for nous fait prcssentir la solution. more sophisticated reasons. H. Poincare David Butler Bussum, September 1989 Michiel Hazewinkel TABLE OF CONTENTS BIBLIOGRAPHY OF ANTONI ZYGMUND ix PAPER [141] (with A. P. Calderon) 'A note on the interpolation of linear operations', SM 12 (1951), 194-204. [142] 'Polish mathematics between the two wars (1919-39)', in: Proceed ings of tke Second Canadian Mathematical Congress, Vancouver, 1949, pp. 3-9, Univ. of Toronto Press, 1951. 12 [143] (with A. P. Calderon) 'On the existence of certain singular in- tegrals', Acta Math. 88 (1952), 85-139. 19 [148] (with R. Salem) 'Some properties of trigonometric series whose terms have random signs', Acta Matk. 91 (1954), 245-301. 74 [150] (with A. P. Calderon) 'Singular integrals and periodic func tions', SM 14 (1954), 249-271. 131 [151] (with R. Salem) 'Sur un theoreme de Piatetc;ki-Shapiro', CRAS 240 (1955), 2040-2042. 154 [153] (with A. P. Calderon) 'On a problem of Mihlin', TAMS 78 (1955), 209-224. 157 [164] (with A. P. Calderon) 'Addenda to the paper "On a problem of Mihlin"', ibid., 84 (1957), 559-560. 173 [155] (with A. P. Calderon) 'A note on the interpolation of sublinear operations', Amer. J. Math. 78 (1956), 282-288. 175 [156] (with A. P. Calderon) 'On singular integrals', ibid., 289-309. 182 [157] (with A. P. Calderon) 'Algebras of certain singular operators', ibid., 310-320. 203 [158] 'On a theorem of Marcinkiewicz concerning interpolation of oper- ations', J. Matk. Pures Appl. (9) 35 (1956), 223-248. 214 [159] 'Hilbert transforms in En', in: Proceedings of tke International Congress of Mathematicians, 1954, Amsterdam, Vol. 3, pp. 140- 151, Erven P. Noordhoff N.V., Groningen, and North-Holland Publishing Co., Amsterdam, 1956. 240 [160] 'On the Littlewood-Paley function g*(O)', Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 208-212. 252 [162] (with A. P. Calderon) 'Singular integral operators and differen- tial equations', Amer. J. Matk. 79 (1957), 901-921. 257 The papers appearing in these volumes are numbered according to the comprehensive list of publications of Antoni Zygmtmd which are presented in chronological order beginning on p. ix. viii TABLE OF CONTENTS [166J (with M. Weiss) 'A note on smooth functions', Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 52-58. 278 [174J (with A. P. Calderon) 'Local properties of solutions of elliptic partial differential equations', SM 20 (1961), 171-225. 285 [179J (with E. M. Stein) 'On the differentiability offunctions', ibid., 23 (1963/1964), 247-283. 340 [180J (with A. P. Calderon) 'On the higher gradients of harmonic functions', ibid., 24 (1964), 211-226. 377 [182J (with E. M. Stein) 'On the fractional differentiability of func- tions', Proc. London Math. Soc. (3) 14A (1965), 249-264. 393 [187J (with E. M. Stein) 'Boundedness of translation invariant opera- tors on Holder spaces and LP-spaces', Ann. of Math. (2) 85 (1967), 337-349. 409 [192J (with M. Weiss) 'On multipliers preserving convergence of tri- gonometric series almost everywhere', SM 30 (1968), 111-120. 422 [196J 'On certain lemmas of Marcinkiewicz and Carleson', J. Approx. Theory 2 (1969), 249-257. 432 [202J 'A Cantor-Lebesgue theorem for double trigonometric series', SM 43 (1972),173-178. 441 [212J (with A. P. Calderon) 'A note on singular integrals', ibid., 65 (1979),77-87. 447 BIBLIOGRAPHY OF ANTONI ZYGMUND Abbreviations BIAP Bulletin International de l'Academie Polonaise des Sciences et des Let tres, Classe des Sciences Mathematiques et Naturelles, Serie A: Sciences M athematiques CRAS Comptes Rendus hebdomadaires des seances de l'Academie des Sciences (Paris) FM Fundamenta Mathematicae MR Mathematical Reviews SM Studia M athematica TAMS Transactions of the American Mathematical Society Zbl. Zentralblatt fUr Mathematik [M) J6zef Marcinkiewicz, Collected Papers, PWN-Polish Scientific Publishers, Warszawa, 1964. [S] Raphael Salem, (Euvres mathematiques, Hermann, Paris, 1967. Other abbreviations follow those of Mathematical Reviews. 1923 [1] 'Sur la theorie riemannienne des series trigonometriques', CRAS 177,521- 523; Errata, ibid., 804. [2] 'Sur les series trigonometriques', ibid., 576-579; Errata, ibid., 804. [3] (with W. Sierpinski) 'Sur une fonction qui est discontinue sur tout en semble de puissance du continu', FM 4, 316-318; reprinted in: Waclaw Sierpinski, (Euvres choisies, Tome II, pp. 497--499, PWN-Polish Scientific Publishers, Warszawa, 1975. 1924 [4] (with A. Rajchman) 'Sur les principes et les problemes de la theorie rie mannienne des series trigonometriques', Ann. Soc. Polon. Math. 3, 147-148. [5] 'Sur un theoreme de M. Marcel Riesz', ibid., 153. [6] 'Sur les series de Fourier restreintes', CRAS 178, 181-182. ix The pape. . reprinted in this three-volume work are marked with •. Those appearing in this particular volume are marked with ••. x ANTONI ZYGMUND [7] 'Sur une generalisation de la methode de Cesaro', ibid., 179, 870-872. [8] (with S. Saks) 'Sur les faisceaux des tangentes it une courbe', FM 6, 117- 121. [9] '0 module cia,glosci sumy szeregu sprzt;Zonego z szeregiem Fouriera' (On the modulus of continuity of the sum of a series conjugate to a Fourier series), Prace Mat.-Fiz. 33, 125-132 [in Polish, French summary]. 1925 [10] (with S. Saks) 'Un teorema suIle curve continue', Boll. Un. Mat. Ital. 4, 7-10. *[ 11] 'Sur la derivation des series de Fourier', BIAP, Annee 1924, 243-249. *[12] 'Sur la sommation des series trigonometriques conjuguees aux series de Fourier', ibid., 251-258. [13] 'Sur Ia sommation des series trigonometriques et celles de puissances par les moyennes typiques', CRAS 181, 1122-1123. 1926 [14] 'Sur Ia sommabilite des series de Fourier des fonctions verifiant Ia condition de Lipschitz', BIAP, Annee 1925, 1-9. *[15] (with A. Rajchman) 'Sur la relation du procede de sommation de Cesaro et celui de Riemann', ibid., 69-80. *[16] 'Sur un theoreme de M. Gronwall', ibid., 207-217. [17] 'Sur la sommation des series par Ie procede des moyennes typiques', ibid., 265-287. *[18] 'Remarque sur Ia sommabilite des series de fonctions orthogonales', ibid., Annee 1926, 185-191. *[19] 'Contribution it l'unicite du developpement trigonometrique', Math. Z. 24, 40-46. *[20] 'Sur la theorie riemannienne des series trigonometriques', ibid., 47-104. [21] 'Sur un theoreme de Ia theorie de la sommabilite', ibid., 25, 291-296. *[22] (with A. Rajchman) 'Sur la possibilite d'appliquer Ia methode de Riemann aux series trigonometriques sommables par Ie procede de Poisson', ibid., 261-273. *[23] 'Sur les series trigonometriques sommables par Ie procede de Poisson', ibid., 274-290. *[24] 'Une remarque sur un theoreme de M. Kaczmarz', ibid., 297-298. [25] '0 teorji srednich arytmetycznych" (On the theory of arithmetic means), Mathesis Polska 1,75-85 and 119-129 [in Polish]. 1927 *[26J 'Sur l'application de la premiere moyenne arithmetique dans la thoorie des series de fonctions orthogonales', FM 10,356-362.