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

2025/07/17 by Bhimani, Divyang G., Dalai, Rupak K.
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.13220

Abstract

We characterize weighted modulation spaces (data space) for which the heat semigroup e^-tLf 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.

Citations

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Related

• A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces
• A note on the almost everywhere convergence to initial data for some evolution equations
• Pointwise convergence to initial data for some evolution equations on symmetric spaces
• Phase space analysis of the Hermite semigroup and applications to\n nonlinear global well-posedness
• Convergence of Hermite expansions in modulation spaces