Stratified Lie Groups and Potential Theory for their Sub-Laplacians
The existence, for every sub-Laplacian, of a homogeneous fundamental solution smooth out of the origin, plays a crucial role in the book. This makes it possible to develop an exhaustive Potential Theory, almost completely parallel to that of the classical
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A. Bonfiglioli • E. Lanconelli • F. Uguzzoni
Stratified Lie Groups and Potential Theory for their Sub-Laplacians
A. Bonfiglioli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]
F. Uguzzoni Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]
E. Lanconelli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]
Library of Congress Control Number: 2007929114
Mathematics Subject Classification (2000): 43A80, 35J70, 35H20, 35A08, 31C05, 31C15, 35B50, 22E60
ISSN 1439-7382 ISBN-10 3-540-71896-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71896-3 Springer Berlin Heidelberg New York 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 microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, 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. Typesetting by the author and VTEX using a Springer LATEX macro package Cover design: WMXDesign, Heidelberg, Germany Printed on acid-free paper
SPIN: 12029525
VA 41/3100/VTEX - 5 4 3 2 1 0
To Professor Bruno Pini and to our Families
Preface
With this book we aim to present an introduction to the stratified Lie groups and to their Lie algebras of the left-invariant vector fields, starting from basic and elementary facts from linear algebra and differential calculus for functions of several real variables. The second aim of this book is to perform a potential theory analysis of the sub-Laplacian operators m Xj2 , L= j =1
where the Xj ’s are vector fields, i.e. linear first order partial differential operators, generating the Lie algebra of a stratified Lie group. In recent years, these operators have received considerable attention in literature, mainly due to their basic rôle in the theory of subelliptic second order partial differential equations with semidefinite characteristic form.
1. Some Historical Overviews General second order partial differential equations with non-negative and degenerate characteristic form have appeared in literature since the early 1900s. They were first studied by M. Picone, who called them elliptic-parabolic equations and proved the celebrated weak max
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