Function Spaces, Interpolation Theory and Related Topics Function Spaces, Interpolation Theory and Related Topics Proceedings of the International Conference in honour of Jaak Peetre on his 65th birthday Lund, Sweden August 17Ϫ22, 2000 Editors Michael Cwikel Miroslav Englisˇ Alois Kufner Lars-Erik Persson Gunnar Sparr ≥ Walter de Gruyter · Berlin · New York 2002 Editors Michael Cwikel Miroslav Englisˇ Alois Kufner Department of Mathematics Mathematical Institute Mathematical Institute Technion Ϫ Israel Institute of Technology Czech Academy of Sciences Czech Academy of Sciences Haifa 32000 11567 Prague 1 11567 Prague 1 Israel Czech Republic Czech Republic Lars-Erik Persson Gunnar Sparr Department of Mathematics Department of Mathematics Lulea University of Technology Lund Institute of Technology/ Lund University 97187 Lulea 22100 Lund Sweden Sweden Mathematics Subject Classification 2000: 00B30; 46-06; 47-06; 46B70; 46EXX, 47BXX Keywords: Interpolation theory, function spaces, operator theory Țȍ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Ϫ Cataloging-in-Publication Data Function spaces, interpolation theory and related topics : pro- ceedings of the international conference in honour of Jaak Peetre on his 65th birthday : Lund, Sweden, August 17Ϫ22, 2000 / editors, Michael Cwikel … [et al.]. p. cm. ISBN 3110171171 (acid-free paper) 1. Functions spaces Ϫ Congresses. 2. Interpolation spaces Ϫ Congresses. I. Peetre, Jaak, 1935Ϫ II. Cwikel, M. (Michael), 1948Ϫ QA323 .F8652 2002 515Ј.73Ϫdc21 2002067478 Die Deutsche Bibliothek Ϫ Cataloging-in-Publication Data Function spaces, interpolation theory and related topics : pro- ceedings of the International Conference in Honour of Jaak Peetre on His 65th Birthday, Lund, Sweden, August, 17Ϫ22, 2000 / ed. Michael Cwikel … . Ϫ Berlin ; New York : de Gruyter, 2002 ISBN 3-11-017117-1 ” Copyright 2002 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permis- sion in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Preface The 65th birthday of Jaak Peetre is a natural time for a very special celebration. As July 29, 2000 drew closer, this was quite obvious to all of us who work in the theory of interpolation spaces, or apply this theory to other fields, and also to many working in other fields where Jaak has also left an indelible mark. So, indeed, on August 17–22, 2000, the Centre for Mathematical Sciences at Lund University held a conference in honour of Jaak’s birthday. This auspicious event was attended by 130 mathematicians from 19 countries and these Proceedings contain (mostly) contributions delivered at the conference. The conference was organized by Michael Cwikel, Björn Jawerth, Jacques-Louis Lions, Lars-Erik Persson, and Gunnar Sparr, with invaluable assistance from Genkai Zhang and also from Ann-Kristin Ottosson, Sven Spanne, Lars Vretare, and other members of the staff of the Centre for Mathematical Sciences at Lund University. To enable these Proceedings to better serve as a record of the conference, we have recalled and presented various details of the speakers and the program in some of the pages to follow. The reader wishing to further participate in the special spirit of the conference is invited to visit the website http://www.maths.lth.se/confer- ences/peetre65.html including the pictures on display there, and to remember or imagine the wonderful banquet in the Trollenäs castle, complete with medieval music and memorable speeches by Lars Gårding, Jaak himself, and others. Given the great depth and breadth of Jaak Peetre’s mathematical work, a detailed survey and discussion of it could fill an entire volume, and it would be larger than this one. We have confined ourselves to giving a brief summary of this work, including a list of Jaak’s publications. We are particularly grateful that Jaak himself agreed to augment our efforts here with an article giving some of his perspectives on the history of the theory to which he has contributed so substantially. It seems particularly fitting that the article appearing immediately after Jaak’s is a contribution from Jacques-Louis Lions, Jaak’s co-author for the creation of the “espaces de moyennes” in their illustrious seminal paper about the “real” interpolation method. As already mentioned above, Jacques-Louis Lions was also actively involved in the organization of the conference, even though he was not able to actually attend it. Very sadly, he passed away soon after we received this paper from him. There was also another tragic event which occurred just a short time before the conference. Yet another outstanding mathematician, Thomas Wolff, who was to have been one of the main speakers, perished in a road accident. On the day and at the time when Tom had been scheduled to speak, we honoured his memory by sharing recollections of him and recalling highlights of his work in a memorial session. The main speakers at the conference were Jonathan Arazy, Yuri Brudnyi, Mischa Cotlar, Ciprian Foias, Svante Janson, Nigel Kalton, Sergey Kislyakov, Peter Lindqvist, vi Preface Vladimir Maz’ya,Vladimir Ovchinnikov,Vladimir Peller, Richard Rochberg, Evgueni Semenov, Hans Triebel, Hans Wallin, and Nahum Zobin. In all, 73 oral presentations were delivered, a full list of which can be found at the end of this volume. We hope that this volume will give the reader valuable insights about new results and trends in some important fields of analysis which have all been strongly influenced by the work of Jaak Peetre. Finally we would like to express our gratitude to the Publishing House Walter de Gruyter, Berlin, for fruitful collaboration. Haifa, Prague, Luleå, Lund Michael Cwikel, Miroslav Engliš, Alois Kufner, April 2002 Lars-Erik Persson, and Gunnar Sparr Contents Preface .............................................................. v Hans Wallin’s opening address ........................................ ix M. Cwikel, L.-E. Persson, R. Rochberg and G. Sparr Jaak Peetre, the man and his work .......................................1 List of publications of Jaak Peetre .................................... 23 J. Peetre On the development of interpolation – instead of a history three letters.....39 J.-L. Lions Remarks on reproducing kernels of some function spaces ................49 A. B. Aleksandrov, S. Janson, V.V. Peller and R. Rochberg An interesting class of operators with unusual Schatten-von Neumann behavior .......................................61 J. Arazy and H. Upmeier Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains...............................................151 J. Brandman, J. Fowler, B. Lins, I. Spitkovsky and N. Zobin Convex hulls of Coxeter groups.......................................213 M. J. Carro and J. Martín On embedding properties of some extrapolation spaces .................241 M. Cotlar and C. Sadosky Revisiting almost orthogonality and eigenexpansions ...................249 D. Cruz-Uribe, SFO and M. Krbec Localization and extrapolation in Orlicz–Lorentz spaces ................273 M. Engliš Green functions for powers of the Laplace–Beltrami operator ............285 viii Contents T. Figiel and N. Kalton Symmetric linear functionals on function spaces .......................311 V. Gol’dshtein and M. Troyanov Axiomatic Sobolev spaces on metric spaces ............................333 S. Kaijser and P. Sunehag Interpolation of subspaces and the unit problem ........................345 D. Lukkassen and G.W. Milton On hierarchical structures and reiterated homogenization ................355 V.G. Maz’ya and I. E. Verbitsky Boundedness and compactness criteria for the one-dimensional Schrödinger operator ................................................369 C. Michels On Gaussian-summing identity maps between Lorentz sequence spaces . 383 E. Nakai On generalized fractional integrals on the weak Orlicz spaces, BMOφ, the Morrey spaces and the Campanato spaces...................389 L. Pick Optimal Sobolev embeddings – old and new ...........................403 S.Yu. Tikhonov Moduli of smoothness and the interrelation of some classes of functions . 413 H. Triebel Towards a Gausslet analysis: Gaussian representations of functions ......425 List of talks ....................................................... 451 List of participants ................................................. 455 Opening address by Hans Wallin, Umeå University Dear Jaak, Dear fellow mathematicians, We have come to this conference to celebrate Jaak as a mathematician and as our friend. Jaak is well-known and respected all over the mathematical world and it is a great pleasure for me to say a few words to him and about him at the opening of this conference. I would like to start by quoting the author G. B. Shaw: The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable. I think that a successful mathematician has to be a little unreasonable, and I think that Jaak has understood this. Jaak is a great mathematician