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

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.