Hardy Spaces on the Euclidean Space
"Still waters run deep." This proverb expresses exactly how a mathematician Akihito Uchiyama and his works were. He was not celebrated except in the field of harmonic analysis, and indeed he never wanted that. He suddenly passed away in summer of 1997 at
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Springer Japan K.K
AKIHITO UCHIYAMA
Hardy Spaces on the Euclidean Space
Springer
Akihito Uchiyama (1948-1997)
Library of Congress Cataloging-in-Publication Data applled for
ISSN 1439-7382 ISBN 978-4-431-67999-8 ISBN 978-4-431-67905-9 (eBook) DOI 10.1007/978-4-431-67905-9
Mathematics Subject Classification (2000): 46E30, 46E15, 46J15 Printed on acid-free paper @Springer Japan 2001 Originally published by Springer-Verlag Tokyo in 2001 Softcover reprint ofthe hardcover lst edition 2001 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
SPIN: 10843141
Foreword
This book is based on a type-written draft which Akihito Uchiyama {19481997} completed in September, 1990. The mathematics community was awaiting its appearance, but he passed away in the summer of 1997, before he could publish it. Because we believe that his book is still worthy, we have decided to proceed with its publication. The original manuscript was transformed into a TeX file with the help of Professors Shige-Toshi Kuroda and Akira Mizutani of Gakushuin University, along with their former students Ms. Makiko Kokubu and Ms. Motoko Oikawa. We thank them very much for their hard work. We also send our hearty thanks to Professor Peter Wilcox Jones, Uchiyama's Ph. D adviser, who kindly wrote a very impressive essay about Uchiyama's personality and mathematical works. Nobuhiko Fujii Akihiko Miyachi K6z6 Yabuta March 2001
Recollections of My Good Friend, Akihito Uchiyama Peter W. Jones
Akihito Uchiyama came to the University of Chicago in the early 1980's at a time of expansive progress in Fourier Analysis. The field had achieved explosive growth since 1970, when Charles Fefferman had proved HI, BMO duality, and the influential paper by Charles Fefferman and E.M. Stein had appeared in Acta Math. Chicago was one of the centers of Fourier analysis, and the home of Alberto Calderon and Antoni Zygmund. Calderon had recently proved a major result: the Cauchy integral is L2 bounded on Lipschitz curves with small constants. It was clear that this result had opened a new area of research in the theory of Hardy Spaces. Antoni Zygmund was old and infirm but made a point of coming to the department every day. He never missed a lecture in the Calderon-Zygmund Seminar. Robert Fefferman and William Beckner were the junior professors and there were always several Dickson Instructors in analysis. R. Narasimhan, the famous complex analyst, was also closely associated with the group. The atomosphere was lively, and the most talented young Ph. D.'s were always vying to get a job there. This was
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