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A Koksma–Hlawka Inequality for Simplices

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A Koksma–Hlawka Inequality for Simplices Springer INdAM Series Volum e 3 Editor-in-Chief V. Ancona Series Editors P. Cannarsa C. Canuto G. Coletti P. Marcellini G. Patrizio T. Ruggeri E. Strickland A. Verra For further volumes: www.springer.com/series/10283 Massimo A. Picardello Editor Trends in Harmonic Analysis Editor Massimo A. Picardello Dipartimento di Matematica Università di Roma “Tor Vergata” Rome, Italy ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-88-470-2852-4 ISBN 978-88-470-2853-1 (eBook) DOI 10.1007/978-88-470-2853-1 Springer Milan Heidelberg New York Dordrecht London Library of Congress Control Number: 2012948292 © Springer-Verlag Italia 2013 This work is subject to copyright. All rights are reserved by the Publisher, 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 any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to Alessandro Figà-Talamanca on the occasion of his retirement Preface This book collects some of the scientific contributions of the participants in the Conference in Harmonic Analysis held at the Istituto Nazionale di Alta Matematica from May 30th to June 4th, 2011. It illustrates the wide range of research subjects developed by the Italian research group in harmonic analysis, originally started by Alessandro Figà-Talamanca, to whom the Conference was dedicated on the occasion of his retirement. In 1978, the mathematicians in this newly formed research group started a cycle of conferences to present and share their research progress. These conferences were held in different places almost every year until this Conference, the thirty-first of the series, and the first whose Proceedings are published. This book outlines some of the impressive ramifications of the mathematical de- velopments that began when Figà-Talamanca brought the study of harmonic anal- ysis to Italy; the research group that he nurtured has now expanded to cover many areas, and therefore this book is addressed not only to experts in harmonic analysis, summability of Fourier series and singular integrals, but also to experts in potential theory, symmetric spaces, analysis and partial differential equations on Riemannian manifolds, analysis on graphs, trees, buildings and discrete groups, Lie groups and Lie algebras, and even far-reaching applications such as cellular automata and signal processing and connections with mathematical logic. In the last decades, Alessandro Figà-Talamanca has worked on harmonic analysis on trees, and his influence on ongoing research is also underlined by the fact that several contributions to the present volume, even those dealing with completely different subjects, are related to trees or similar discrete structures. Rome, Italy Massimo Picardello vii Acknowledgements We acknowledge with pleasure the hospitality of the Istituto Nazionale di Alta Matematica and the financial support of the Gruppo Nazionale per l’Analisi Matem- atica, la Probabilità e le Applicazioni. We are very grateful to the referees for their accurate reading of each contributed paper and for their valuable suggestions. The Editor is very grateful to all the authors for the patience that they have shown in kindly and promptly replying to his countless requests for revisions. ix Contents The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees ....................... 1 Jean-Philippe Anker, Pierre Martinot, Emmanuel Pedon, and Alberto G. Setti Invariance of Capacity Under Quasisymmetric Maps of the Circle: An Easy Proof ............................... 27 Nicola Arcozzi and Richard Rochberg A Koksma–Hlawka Inequality for Simplices ................ 33 Luca Brandolini, Leonardo Colzani, Giacomo Gigante, and Giancarlo Travaglini A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices .................................. 47 Michelle Bucher, Marc Burger, and Alessandra Iozzi The Algebras Generated by the Laplace Operators in a Semi- homogeneous Tree ............................ 77 Enrico Casadio Tarabusi and Massimo A. Picardello Surjunctivity and Reversibility of Cellular Automata over Concrete Categories ................................. 91 Tullio Ceccherini-Silberstein and Michel Coornaert Pointwise Convergence of Bochner–Riesz Means in Sobolev Spaces ....135 Leonardo Colzani and Sara Volpi Sub-Finsler Geometry and Finite Propagation Speed ............147 Michael G. Cowling and Alessio Martini On the Boundary Behavior of Holomorphic and Harmonic Functions . 207 Fausto Di Biase xi xii Contents Constructing Laplacians on Limit Spaces of Self-similar Groups .....245 Alfredo Donno Some Remarks on Generalized Gaussian Noise ...............277 Saverio Giulini Eigenvalues of the Vertex Set Hecke Algebra of an Affine Building ....291 Anna Maria Mantero and Anna Zappa A Liouville Type Theorem for Carnot Groups: A Case Study .......371 Alessandro Ottazzi and Ben Warhurst Stochastic Properties of Riemannian Manifolds and Applications to PDE’s ..................................381 Gregorio Pacelli Bessa, Stefano Pigola, and Alberto G. Setti Characterization of Carleson Measures for Besov Spaces on Homogeneous Trees ..........................399 Maria Rosaria Tupputi Atomic and Maximal Hardy Spaces on a Lie Group of Exponential Growth ..................................409 Maria Vallarino The Maximal Singular Integral: Estimates in Terms of the Singular Integral ..................................425 Joan Verdera The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees Jean-Philippe Anker, Pierre Martinot, Emmanuel Pedon, and Alberto G. Setti Abstract We solve explicitly the shifted wave equation on Damek–Ricci spaces, using the inverse dual Abel transform and Ásgeirsson’s theorem. As an application, we investigate Huygens’ principle. A similar analysis is carried out in the discrete setting of homogeneous trees. Keywords Abel transform · Damek–Ricci space · Homogeneous tree · Huygens’ principle · Hyperbolic space · Wave equation · Wave propagation Mathematics Subject Classification (2010) Primary 35L05 · 43A85 · Secondary 20F67 · 22E30 · 22E35 · 33C80 · 43A80 · 58J45 J.-P. Anker (B) Fédération Denis Poisson (FR 2964) and Laboratoire MAPMO (UMR 7349), Bâtiment de Mathématiques, Université d’Orléans and CNRS, B.P. 6759, 45067 Orléans Cedex 2, France e-mail: [email protected] P. Martinot 2 rue de la Maladrie, 54630 Flavigny-sur-Moselle, France e-mail: [email protected] E. Pedon Fédération ARC (FR 3399) and Laboratoire de Mathématiques (EA 4535), Université de Reims (Champagne-Ardenne), Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France e-mail: [email protected] A.G. Setti Sezione di Matematica, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy e-mail: [email protected] M.A. Picardello (ed.), Trends in Harmonic Analysis, Springer INdAM Series 3, 1 DOI 10.1007/978-88-470-2853-1_1, © Springer-Verlag Italia 2013 2 J.-P. Anker et al. 1 Introduction In the book [17] Helgason uses Ásgeirsson’s mean value theorem (see Theorem II.5.28) to solve the wave equation ∂2u(x, t) = Δ u(x, t), t x (1) u(x, 0) = f(x), ∂t |t=0 u(x, t) = g(x), on Euclidean spaces Rd (see [17, Exercise II.F.1, p. 342] and its solution at pp. 574– 575) and the shifted wave equation ∂2u(x, t) ={Δ + (d − 1)2/4}u(x, t), t x (2) u(x, 0) = f(x), ∂t |t=0 u(x, t) = g(x), on real hyperbolic spaces H d (R) (see [17, Exercise II.F.2, p. 343] and its solution at pp. 575–577). In this work we extend this approach both to Damek–Ricci spaces and to homogeneous trees. Along the way we clarify the role of the inverse dual Abel transform in solving the shifted wave equation. Recall that Damek–Ricci spaces are Riemannian manifolds, which contain all hy- perbolic spaces H d (R), H d (C), H d (H), H 2(O) as a small subclass and share nev- ertheless several features with these spaces.
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