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

TL;DR

This paper characterizes when the heat semigroup converges pointwise to initial data in weighted modulation spaces, introducing new results for these spaces and analyzing the Hardy-Littlewood maximal operator.

Contribution

It provides the first characterization of pointwise convergence in weighted modulation spaces for heat equations and explores the boundedness of the Hardy-Littlewood maximal operator on these spaces.

Findings

Identifies conditions for pointwise convergence in weighted modulation spaces.

Proves the Hardy-Littlewood maximal operator acts on certain modulation spaces.

Highlights open questions related to these findings.

Abstract

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.