Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals

TL;DR

This paper investigates the relationship between Lebesgue points of measures and the boundary behavior of Poisson-Hermite integrals, establishing conditions for non-tangential convergence related to measure differentiability.

Contribution

It characterizes Lebesgue points of measures via boundary convergence of Poisson-Hermite integrals and introduces a new notion of convergence stronger than non-tangential, extending classical results.

Findings

Lebesgue point characterization via boundary convergence

Non-tangential convergence at sigma-points of measures

Extension of Fatou's theorem to Hermite setting

Abstract

We study differentiability conditions on a complex measure $\nu$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_t\nu=e^{-t\sqrt L}\nu$, where $L=-\Delta+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $\nu$ iff a slightly stronger notion than non-tangential convergence holds for $P_t\nu$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $\sigma$-point of $\nu$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition.