Fatou Type Theorems Maximal Functions and Approach Regions
A basic principle governing the boundary behaviour of holomorphic func tions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions ad mit a boundary limit, if we approach
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Fausto Di Biase
Fatou Type Theorems Maximal Functions and Approach Regions
Birkhtiuser Boston • Basel • Berlin
Department of Mathematics Princeton University Princeton, NJ 08540
University Roma-Tor Vergata Dip. Matematica 00133 Rome, Italy
Library of Congress Cataloging-in-Publication Data Di Biase, Fausto, 1962Fatou type theorems: maximal functions and approach regions / Fausto Di Biase. p. cm. -- (Progress in mathematics ; v. 147) Includes bibliographical references (p. - ) and index. ISBN-13: 978-1-4612-7496-4 DOl: 10.1007/978-1-4612-2310-8
e-ISBN-13: 978-1-4612-2310-8
1. Holomorphic functions. 2. Fatou theorems. 3. Functions of several complex variables. I. Title. II. Series: Progress in mathematics (Boston, Mass.) ; vol. 147 QA331.D5581997 95-30694 S15--dc21 CIP
AMS Classification Codes: 3IB25 and 32A40.
Printed on acid-free paper
© 1998 Birkhiiuser
Birkhiiuser
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Typesetting by the author in LATEX
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Contents Preface
I
vii
Background
1
1 Prelude 1.1 1.2 1.3 1.4
The Unit Disc . Spaces of Homogeneous Type Euclidean Half-Spaces . . . . Maximal Operators and Convergence
2 Preliminary Results 2.1 2.2 2.3
Approach Regions . . . . . . . . . . The Nagel-Stein Approach Regions Goals, Problems and Results .
3 The Geometric Contexts 3.1 3.2 3.3
II
NTA Domains in ]Rn Domains in Trees . . . . . . . . .
en
Exotic Approach Regions
4 Approach Regions for Trees 4.1 4.2
The Dyadic Tree The General Tree
5 Embedded Trees 5.1 5.2 5.3
The Unit Disc Quasi-Dyadic Decompositions . . . . . The Maximal Decomposition of a Ball
3 3 16
18 24
27 27
43 51
55 55
59 73
85 87 87 88 99
99 104
108
CONTENTS
vi 5.4
Admissible Embeddings
6 Applications 6.1 Euclidean Half-Spaces 6.2 NTA Domains in IRn . 6.3 Finite-Type Domains in 1(:2 6.4 Strongly Pseudo convex Domains in
. 110
123
en
124 124 125 128
Notes
129
List of Figures
131
Guide to Notation
133
Bibliography
135
Index
149
Preface A basic principle governing the boundary behaviour of holomorphic functions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions admit a boundary
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