Schrödinger semigroups and the Hörmander hypoellipticity condition
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908562311086080 |
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| author | Garofalo, Nicola Lunardi, Alessandra |
| author_facet | Garofalo, Nicola Lunardi, Alessandra |
| contents | We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal T(t)\}_{t\ge 0}$ in $L^2(\Rm)$. Finally, we prove that for $t>0$ the operator $\mathcal T(t)$ satisfies a sharp form of dispersive estimate in $L^p$, for any $1\le p\le 2$, and an uncertainty principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schrödinger semigroups and the Hörmander hypoellipticity condition Garofalo, Nicola Lunardi, Alessandra Analysis of PDEs We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal T(t)\}_{t\ge 0}$ in $L^2(\Rm)$. Finally, we prove that for $t>0$ the operator $\mathcal T(t)$ satisfies a sharp form of dispersive estimate in $L^p$, for any $1\le p\le 2$, and an uncertainty principle. |
| title | Schrödinger semigroups and the Hörmander hypoellipticity condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.04441 |