Boundedness of spectral multipliers on locally compact groups and applications
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917276881518592 |
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| author | Cobos, Santiago Gómez Restrepo, Joel E. Ruzhansky, Michael |
| author_facet | Cobos, Santiago Gómez Restrepo, Joel E. Ruzhansky, Michael |
| contents | We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $φ(|\mathscr{L}|)$ can be estimated if the modulus of the Borel function $φ$ is bounded by a continuous positive monotonically decreasing function that vanishes at infinity $ψ$. As a consequence, we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group $G$ by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on $G$. We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundedness of spectral multipliers on locally compact groups and applications Cobos, Santiago Gómez Restrepo, Joel E. Ruzhansky, Michael Analysis of PDEs Functional Analysis Group Theory 43A15 (Primary) 43A85, 45K05, 35Q41 (Secondary) We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $φ(|\mathscr{L}|)$ can be estimated if the modulus of the Borel function $φ$ is bounded by a continuous positive monotonically decreasing function that vanishes at infinity $ψ$. As a consequence, we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group $G$ by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on $G$. We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups. |
| title | Boundedness of spectral multipliers on locally compact groups and applications |
| topic | Analysis of PDEs Functional Analysis Group Theory 43A15 (Primary) 43A85, 45K05, 35Q41 (Secondary) |
| url | https://arxiv.org/abs/2302.00721 |