Ebook: Fatou Type Theorems: Maximal Functions and Approach Regions
Author: Fausto Di Biase (auth.)
- Tags: Functions of a Complex Variable, Analysis, Several Complex Variables and Analytic Spaces
- Series: Progress in Mathematics 147
- Year: 1997
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Prelude....Pages 3-25
Preliminary Results....Pages 27-53
The Geometric Contexts....Pages 55-83
Front Matter....Pages 85-85
Approach Regions for Trees....Pages 87-97
Embedded Trees....Pages 99-122
Applications....Pages 123-128
Back Matter....Pages 129-154
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Prelude....Pages 3-25
Preliminary Results....Pages 27-53
The Geometric Contexts....Pages 55-83
Front Matter....Pages 85-85
Approach Regions for Trees....Pages 87-97
Embedded Trees....Pages 99-122
Applications....Pages 123-128
Back Matter....Pages 129-154
....