Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear
Universitext Editorial Board (North America): S Axler F.W Gehring K.A Ribet Springer Science+Business Media, LLC U niversitext Editors (North America): S Axler, F.W Gehring, and K.A Ribet AksoylKhamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BachmanINariciIBeckenstein: Fourier and Wavelet Analysis BalakrishnanlRanganathan: A Textbook ofGraph Theory Balser: Fonnal Power Series and Linear Systems ofMeromorphic Ordinary Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals ofReal Analysis BoossIBleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course BiittcherlSilbermann: Introduction to Large Truncated Toeplitz Matrices CarlesonlGamelin: Complex Dynamics Cecil: Lie Sphere Geometry: With Applications to Submanifolds Chae: Lebesgue Integration (2nd ed.) Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear Algebra Curtis: Matrix Groups DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology ofHypersurfaces Edwards: A Formal Background to Mathematics I alb Edwards: A Formal Background to Mathematics II alb Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles Fuhrmann: A Polynomial Approach to Linear Algebra Gardiner: A First Course in Group Theory GardingITambour: Algebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonlRao: Numerical Range: The Field ofValues ofLinear Operators and Matrices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Heinonen: Lectures on Analysis on Metric Spaces Holmgren: A First Course in Discrete Dynamical Systems Howe!fan: Non-Abelian Harmonic Analysis: Applications ofSL(2, R) Howes: Modem Analysis and Topology Hsieh/Sibuya: Basic Theory ofOrdinary Differential Equations HumiIMiller: Second Course in Ordinary Differential Equations HurwitzlKritikos: Lectures on Number Theory Jennings: Modem Geometry with Applications JonesIMorrisIPearson: Abstract Algebra and Famous Impossibilities KannanIKrueger: Advanced Analysis KeIlylMatthews: The Non-Euclidean Hyperbolic Plane Kostrikin: Introduction to Algebra LueckingIRubel: Complex Analysis: A Functional Analysis Approach MacLaneIMoerdijk: Sheaves in Geometry and Logic Marcus: Number Fields (continued after index) Juha Heinonen Lectures on Analysis on Metric Spaces i Springer Juha Heinonen F.W Gehring Mathematics Department Mathematics Department East HaU East Hali University of Michigan University of Michigan Ann Arbor, MI 48109-1109 Ann Arbor, MI 48109-1109 USA USA Editorial Board (North America): S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 28Axx, 43A85, 46Exx Library of Congress Cataloging-in-Publication Data Heinonen, Juha Lectures on analysis on metric spaces / Juha Heinonen p cm - (Universitext) Inc1udes bibliographical references and index ISBN 978-1-4612-6525-2 ISBN 978-1-4613-0131-8 (eBook) DOI 10.1007/978-1-4613-0131-8 1 Metric spaces 2 Mathematical analysis 1 Title II Series QA611.28.H44 2001 514'.32 dc21 00-056266 Printed on acid-free paper © 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 2001 Softcover reprint of the hardcover lst edition 2001 All rights reserved This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive narnes, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Michael Koy; manufacturing supervised by Jerome Basma Typeset by TechBooks, Fairfax, VA 987654 32 1 ISBN 978-1-4612-6525-2 To my teacher Professor Olli Martio Preface Analysis in spaces with no a priori smooth structure has progressed to include con- cepts from first-order calculus In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial related articles The book is a written version of lectures from a graduate course I taught at the University of Michigan in Fall 1996 I have added remarks and references to works that have appeared thereafter The material can roughly be divided into three different types: classical, standard but sometimes with a new twist, and recent For instance, the treatment of covering theorems and their applications is classical, with little novelty in the presentation On the other hand, the ensuing discussion on Sobolev spaces is more modern, emphasizing principles that are valid in larger contexts Finally, most ofthe material on quasisymmetric maps is relatively recent and appears for the first time in book format; one can even find a few previously unpublished results The bibliography is extensive but by no means exhaustive The reader is assumed to be familiar with basic analysis such as Lebesgue's theory of integration and the elementary Banach space theory Some knowledge of complex analysis is helpful but not necessary in the chapters that deal with quasisymmetric maps Occasionally, some elementary facts from other fields are quoted, mostly in examples and remarks The students in the course, many colleagues, and friends made a number of useful suggestions on the earlier versions of the manuscript and found many errors I thank them all Special thanks go to four people: Bruce Hanson carefully read viii Preface the entire manuscript and detected more errors than anyone else; years later, use of Kari Hag's handwritten class notes preserved the spirit of the class; Fred Gehring insisted that I prepare these notes for publication; and Jeremy Tyson provided crucial help in preparing the final version of the text Finally, I am grateful for having an inspiring audience for my lectures at Michigan Juha Heinonen Ann Arbor, Michigan July 1999 Contents Preface vii 1 Covering Theorems 1 2 Maximal Functions 10 3 Sobolev Spaces 14 4 Poincare Inequality 27 5 Sobolev Spaces on Metric Spaces 34 6 Lipschitz Functions 43 7 Modulus of a Curve Family, Capacity, and Upper Gradients 49 8 Loewner Spaces 59 9 Loewner Spaces and Poincare Inequalities 68 10 Quasisymmetric Maps: Basic Theory I 78 11 Quasisymmetric Maps: Basic Theory II 88 12 Quasisymmetric Embeddings of Metric Spaces in Euclidean Space 98 13 Existence of Doubling Measures 103 x Contents 14 Doubling Measures and Quasisymmetric Maps 109 IS Conformal Gauges 119 References 127 Index 137 1 Covering Theorems All covering theorems are based on the same idea: from an arbitrary cover of a set in a metric space, one tries to select a subcover that is, in a certain sense, as disjointed as possible In order to have such a result, one needs to assume that the covering sets are somehow nice, usually balls In applications, the metric space normally comes with a measure /-L, so that if F = {B} is a covering of a set A by balls, then always /-L(A)::: L/-L(B) F (with proper interpretation of the sum if the collection F is not countable) What we often would like to have, for instance, is an inequality in the other direction, /-L(A) 2: C L /-L(B), F' for some subcollection F' c F that still covers A and for some positive constant C that is independent of A and the covering F There are many versions of this theme In the following, we describe three covering theorems, which we call the basic covering theorem, the Vitali covering theorem, and the Besicovitch-Federer cov- ering theorem (The terms may not be standard.) Each of these three theorems has a slightly different purpose, as well as different domains of validity, and I have chosen them as representatives of the many existing covering theorems We will not strive for the greatest generality here 1.1 Convention Let X be a metric space When we speak of a ball B in X, it is understood that it comes with a fixed center and radius, although these in general J Heinonen, Lectures on Analysis on Metric Spaces © Springer Science+Business Media New York 2001