An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910032757522432 |
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| author | Huang, Yi C. Ozawa, Tohru Sun, Chenmin Takeuchi, Taiki |
| author_facet | Huang, Yi C. Ozawa, Tohru Sun, Chenmin Takeuchi, Taiki |
| contents | We give a certain $L^{\infty}(\mathbb{R}^2)$-estimate for the heat semigroup $\{e^{tΔ}\}_{t \ge 0}$ that is closely related to the fact $H^1(\mathbb{R}^2) \not\subset L^{\infty}(\mathbb{R}^2)$, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallouët inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as $t \to 0^+$ is indeed optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21665 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding Huang, Yi C. Ozawa, Tohru Sun, Chenmin Takeuchi, Taiki Functional Analysis We give a certain $L^{\infty}(\mathbb{R}^2)$-estimate for the heat semigroup $\{e^{tΔ}\}_{t \ge 0}$ that is closely related to the fact $H^1(\mathbb{R}^2) \not\subset L^{\infty}(\mathbb{R}^2)$, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallouët inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as $t \to 0^+$ is indeed optimal. |
| title | An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2602.21665 |