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

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.

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