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 or Hermite operator H " `|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.