On distributional point values and boundary values of analytic functions

TL;DR

This paper extends Fatou's theorem to distributions as boundary values of analytic functions, establishing conditions under which distributional limits and point values coincide, with implications for boundary behavior analysis.

Contribution

It introduces a version of Fatou's theorem for distributions, linking distributional boundary limits with point values and non-tangential limits of analytic functions.

Findings

Distributional boundary limits imply pointwise limits under boundedness conditions.

The ojasiewicz point value exists when distributional limits are bounded.

Analytic functions approach boundary values non-tangentially under specified conditions.

Abstract

We give the following version of Fatou's theorem for distributions that are boundary values of analytic functions. We prove that if $f\in\mathcal{D}^{\prime}(a,b) $ is the distributional limit of the analytic function $F$ defined in a region of the form $(a,b) \times(0,R),$ if the one sided distributional limit exists, $f(x_{0}+0) =\gamma,$ and if $f$ is distributionally bounded at $x=x_{0}$, then the \L ojasiewicz point value exists, $f(x_{0})=\gamma$ distributionally, and in particular $F(z)\to \gamma$ as $z\to x_{0}$ in a non-tangential fashion.