Asian Journal of Mathematics

Volume 11 (2007)

Number 2

The Boundary Behavior of Holomorphic Functions: Global and Local Results

Pages: 179 – 200

DOI: http://dx.doi.org/10.4310/AJM.2007.v11.n2.a2

Author

Steven G. Krantz

Abstract

We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions.

As a result of this methodology, theorems of Calderón type about local boundary behavior on a set of positive measure may be proved in a new and more natural way.

We also study the question of nontangential boundedness (on a set of positive measure) versus admissible boundedness. Under suitable hypotheses, these two conditions are shown to be equivalent.

Keywords

Fatou theorem; admissible convergence; Calderón theorem; boundary limits

2010 Mathematics Subject Classification

Primary 32A40. Secondary 32A35, 32A50.

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