Fatou Type Theorems

1. Front Matter

   Pages i-xi

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2. Background

   1. Front Matter

      Pages 1-1

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   2. Prelude

      • Fausto Di Biase

      Pages 3-25

   3. Preliminary Results

      • Fausto Di Biase

      Pages 27-53

   4. The Geometric Contexts

      • Fausto Di Biase

      Pages 55-83

3. Exotic Approach Regions

   1. Front Matter

      Pages 85-85

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   2. Approach Regions for Trees

      • Fausto Di Biase

      Pages 87-97

   3. Embedded Trees

      • Fausto Di Biase

      Pages 99-122

   4. Applications

      • Fausto Di Biase

      Pages 123-128

4. Back Matter

   Pages 129-154

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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.

• Fatou Type
• Finite
• Pseudoconvexity
• function
• mathematics
• maximum
• theorem

• Department of Mathematics, Princeton University, Princeton, USA

  Fausto Biase

• Dip. Matematica, University Roma-Tor Vergata, Rome, Italy

  Fausto Biase

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