I Background.- 1 Prelude.- 1.1 The Unit Disc.- 1.2 Spaces of Homogeneous Type.- 1.3 Euclidean Half-Spaces.- 1.4 Maximal Operators and Convergence.- 2 Preliminary Results.- 2.1 Approach Regions.- 2.2 The Nagel-Stein Approach Regions.- 2.3 Goals, Problems and Results.- 3 The Geometric Contexts.- 3.1 NTA Domains in ?n.- 3.2 Domains in ?n.- 3.3 Trees.- II Exotic Approach Regions.- 4 Approach Regions for Trees.- 4.1 The Dyadic Tree.- 4.2 The General Tree.- 5 Embedded Trees.- 5.1 The Unit Disc.- 5.2 Quasi-Dyadic Decompositions.- 5.3 The Maximal Decomposition of a Ball.- 5.4 Admissible Embeddings.- 6 Applications.- 6.1 Euclidean Half-Spaces.- 6.2 NTA Domains in ?n.- 6.3 Finite-Type Domains in ?2.- 6.4 Strongly Pseudoconvex Domains in ?n.- Notes.- List of Figures.- Guide to Notation.
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.
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