A note on $\sigma$-point and nontangential convergence

TL;DR

This paper extends Shapiro's theorem on nontangential convergence of Poisson integrals, introducing $\sigma$-points and analyzing their relation to strong derivatives, with results applicable to a broad class of convolution kernels.

Contribution

It introduces the concept of $\sigma$-points for measures and explores their connection to nontangential limits and strong derivatives, broadening the understanding of convergence behavior.

Findings

Convolution integrals have nontangential limits at $\sigma$-points.

In one dimension, $\sigma$-points coincide with strong derivative points.

The results apply to a wide class of convolution kernels.

Abstract

In this article, we generalize a theorem of Victor L. Shapiro concerning nontangential convergence of the Poisson integral of a $L^p$-function. We introduce the notion of $\sigma$-points of a locally finite measure and consider a wide class of convolution kernels. We show that convolution integrals of a measure have nontangential limits at $\sigma$-points of the measure. We also investigate the relationship between $\sigma$-point and the notion of the strong derivative introduced by Ramey and Ullrich. In one dimension, these two notions are the same.