Pointwise convergence to initial data of heat and Hermite-heat equations in modulation spaces

Abstract

We characterize weighted modulation spaces (data space) for which the heat semigroup 𝑒−𝑡⁢𝐿⁢𝑓 converges pointwise to the initial data f as time t tends to zero. Here L stands for the standard Laplacian −Δ or Hermite operator 𝐻 =−Δ +|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.