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

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.