Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces

We characterize weighted modulation spaces (data space) for which the heat semigroup $e^{-tL}f$ converges pointwise to the initial data $f$ as time $t$ tends to zero. Here $L$ stands for the standard Laplacian $-\Delta$ or Hermite operator $H=-\Delta +|x|^2$ on the Euclidean space. This is the first result on pointwise convergence with data in a weighted modulation spaces (which do not coincide with weighted Lebesgue spaces). We also prove that the Hardy-Littlewood maximal operator operates on certain modulation spaces. This may be of independent interest. We have highlighted several open questions that arise naturally from our findings.