Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms
Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are di
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Managing Editor M.W. Wong (York University, Canada)
Editorial Board Luigi Rodino (Università di Torino, Italy) Bert-Wolfgang Schulze (Universität Potsdam, Germany) Johannes Sjöstrand (Université de Bourgogne, Dijon, France) Sundaram Thangavelu (Indian Institute of Science at Bangalore, India) Maciej Zworski (University of California at Berkeley, USA)
Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted.
André Unterberger
Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms
André Unterberger Mathématiques U.F.R. des Sciences Université de Reims Moulin de la Housse, B.P. 1039 51687 Reims Cedex 2 France [email protected], [email protected]
2010 Mathematics Subject Classification: 11F37, 11F72, 47G30 ISBN 978-3-0348-0165-2 e-ISBN 978-3-0348-0166-9 DOI 10.1007/978-3-0348-0166-9 Library of Congress Control Number: 2011935042 © Springer Basel AG 2011 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, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.
Cover design: SPi Publisher Services Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
To the memory of Paul Malliavin
Contents Introduction
1
1 The 1.1 1.2 1.3
Weyl calculus An introduction to the usual Weyl calculus . . . . . . . . . . . . . Two composition formulas . . . . . . . . . . . . . . . . . . . . . . . The totally radial Weyl calculus . . . . . . . . . . . . . . . . . . . .
13 13 26 38
2 The 2.1 2.2 2.3 2.4
Radon transformation and applications The Radon transformation . . . . . . . . . . . . . . . . . . Back to the totally radial Weyl calculus . . . . . . . . . . The dual Radon transform of bihomogeneous distributions The symmetries ν 7→ −ν and ρ 7→ 2 − ρ . . . . . . . . . .
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3 Automorphic functions and automorphic distributions 3.1 Automorphic background . . . . . . . . . . . . . . . . . 3.2 Automorphic distributions . . . . . . . . . . . . . . . . . 3.3 The Kloosterman-related series ζk (s, t) . . . . . . . . . . 3.4 About the sharp product of two Eisenstein distributions 3.5 The pointwise product of two Eisenstein series . . . . . 3.6 The continuation of ζk . . . . . . . . . . . . . . . . . . .
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