Author | Biase, Fausto Di. author |
---|---|

Title | Fatou Type Theorems [electronic resource] : Maximal Functions and Approach Regions / by Fausto Di Biase |

Imprint | Boston, MA : Birkhรคuser Boston, 1998 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-2310-8 |

Descript | XII, 154 p. online resource |

SUMMARY

A basic principle governing the boundary behaviour of holomorphic funcยญ tions (and harmonic functions) is this: Under certain growth conditions, for almost every point in the boundary of the domain, these functions adยญ 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 halfยญ 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)omorยญ 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

I Background -- 1 Prelude -- 2 Preliminary Results -- 3 The Geometric Contexts -- II Exotic Approach Regions -- 4 Approach Regions for Trees -- 5 Embedded Trees -- 6 Applications -- Notes -- List of Figures -- Guide to Notation

Mathematics
Mathematical analysis
Analysis (Mathematics)
Functions of complex variables
Mathematics
Functions of a Complex Variable
Analysis
Several Complex Variables and Analytic Spaces