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 \({{\mathbb {P}}}_t\nu =e^{-t\sqrt{L}}\nu \) , 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 \({{\mathbb {P}}}_t\nu \) 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.
