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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2312.10824 |
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| _version_ | 1866912275422511104 |
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| author | Gutowski, Michał Kwaśnicki, Mateusz |
| author_facet | Gutowski, Michał Kwaśnicki, Mateusz |
| contents | For an arbitrary regular Dirichlet form $\mathscr{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding Sobolev-Bregman form $\mathscr{E}_p(u) = -\tfrac{1}{p} \frac{d}{d t}\bigr\vert_{t = 0} \|T_t u\|_p^p$, where $p \in (1, \infty)$. We prove a variant of the Beurling-Deny formula for $\mathscr{E}_p$. As an application, we prove the corresponding Hardy-Stein identity. Our results extend previous works in this area, which either required that $\mathscr{E}$ is translation-invariant, or that $u$ is sufficiently regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10824 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Beurling-Deny formula for Sobolev-Bregman forms Gutowski, Michał Kwaśnicki, Mateusz Analysis of PDEs For an arbitrary regular Dirichlet form $\mathscr{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding Sobolev-Bregman form $\mathscr{E}_p(u) = -\tfrac{1}{p} \frac{d}{d t}\bigr\vert_{t = 0} \|T_t u\|_p^p$, where $p \in (1, \infty)$. We prove a variant of the Beurling-Deny formula for $\mathscr{E}_p$. As an application, we prove the corresponding Hardy-Stein identity. Our results extend previous works in this area, which either required that $\mathscr{E}$ is translation-invariant, or that $u$ is sufficiently regular. |
| title | Beurling-Deny formula for Sobolev-Bregman forms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.10824 |