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Local Boundary Behavior of Bounded Holomorphic Functions

Published online by Cambridge University Press:  20 November 2018

Alexander Nagel  and
Walter Rudin
Alexander Nagel
Affiliation:
University of Wisconsin, Madison, Wisconsin
Walter Rudin
Affiliation:
University of Wisconsin, Madison, Wisconsin
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Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 583 - 592
Copyright
Copyright © Canadian Mathematical Society 1978

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