# Fatou Type Theorems

## Maximal Functions and Approach Regions

• Fausto Di Biase
Book

Part of the Progress in Mathematics book series (PM, volume 147)

1. Front Matter
Pages i-xi
2. ### Background

1. Front Matter
Pages 1-1
2. Fausto Di Biase
Pages 3-25
3. Fausto Di Biase
Pages 27-53
4. Fausto Di Biase
Pages 55-83
3. ### Exotic Approach Regions

1. Front Matter
Pages 85-85
2. Fausto Di Biase
Pages 87-97
3. Fausto Di Biase
Pages 99-122
4. Fausto Di Biase
Pages 123-128
4. Back Matter
Pages 129-154

### Introduction

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 the bounda-ry point within certain approach regions. For example, for bounded harmonic functions in the open unit disc, the natural approach regions are nontangential triangles with one vertex in the boundary point, and entirely contained in the disc [Fat06]. In fact, these natural approach regions are optimal, in the sense that convergence will fail if we approach the boundary inside larger regions, having a higher order of contact with the boundary. The first theorem of this sort is due to J. E. Littlewood [Lit27], who proved that if we replace a nontangential region with the rotates of any fixed tangential curve, then convergence fails. In 1984, A. Nagel and E. M. Stein proved that in Euclidean half­ spaces (and the unit disc) there are in effect regions of convergence that are not nontangential: These larger approach regions contain tangential sequences (as opposed to tangential curves). The phenomenon discovered by Nagel and Stein indicates that the boundary behaviour of ho)omor­ phic functions (and harmonic functions), in theorems of Fatou type, is regulated by a second principle, which predicts the existence of regions of convergence that are sequentially larger than the natural ones.

### Keywords

Fatou Type Finite Pseudoconvexity function mathematics maximum theorem

#### Authors and affiliations

• Fausto Di Biase
• 1
• 2
1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
2. 2.Dip. MatematicaUniversity Roma-Tor VergataRomeItaly

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-2310-8
• Copyright Information Birkhäuser Boston 1998
• Publisher Name Birkhäuser Boston
• eBook Packages
• Print ISBN 978-1-4612-7496-4
• Online ISBN 978-1-4612-2310-8
• Series Print ISSN 0743-1643
• Series Online ISSN 2296-505X
• Buy this book on publisher's site