Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals
Guillermo Flores, Gustavo Garrigós, Beatriz Viviani
TL;DR
The paper investigates boundary behavior of Poisson-Hermite integrals ${\mathbb P}_t=e^{-t\sqrt L}$ for measures with respect to the Hermite operator $L=-\Delta+|x|^2$. It shows that a point $x_0$ is a Lebesgue point for a complex measure $\nu$ precisely when a strengthened non-tangential convergence of ${\mathbb P}_t\nu$ occurs at $x_0$, with the non-tangential limit equal to the symmetric derivative $D\nu(x_0)$; it also introduces $\sigma$-points as a weaker condition ensuring non-tangential convergence in low dimensions and under additional density bounds in higher dimensions. The results rely on sharp kernel decompositions for the Hermite Poisson kernel, including a radial vs non-radial split and a polar-coordinate treatment of measures, to control near-field and far-field contributions in a non-convolution setting. The findings extend Fatou-type boundary results to Poisson-Hermite integrals and provide dimension-dependent criteria for convergence at singular points of the initial measure, with special cases (e.g., $x_0=0$) offering improved hypotheses. Overall, the work advances the understanding of when Poisson-Hermite boundary limits reflect local differentiability properties of measures and highlights new techniques for non-convolution kernels in harmonic analysis.
Abstract
We study differentiability conditions on a complex measure $ν$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_tν=e^{-t\sqrt L}ν$, where $L=-Δ+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $ν$ iff a slightly stronger notion than non-tangential convergence holds for $P_tν$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $σ$-point of $ν$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition.