Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910108523429888 |
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| author | Bhimani, Divyang G. Dalai, Rupak K. |
| author_facet | Bhimani, Divyang G. Dalai, Rupak K. |
| contents | 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 $-Δ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces Bhimani, Divyang G. Dalai, Rupak K. Analysis of PDEs Functional Analysis 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 $-Δ$ 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. |
| title | Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2507.13220 |