Boundary representations of hyperbolic groups: the log-Sobolev case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917747477184512 |
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| author | Boucher, Kevin Špakula, Ján |
| author_facet | Boucher, Kevin Špakula, Ján |
| contents | We study boundary representations of hyperbolic groups $Γ$ on the (compactly embedded) function space $W^{\log,2}(\partialΓ)\subset L^2(\partialΓ)$, the domain of the logarithmic Laplacian on $\partialΓ$. We show that they are not uniformly bounded, and establish their exact growth (up a multiplicative constant): they grow with the square root of the length of $g\inΓ$. We also obtain $L^p$--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary representations of hyperbolic groups: the log-Sobolev case Boucher, Kevin Špakula, Ján Group Theory Functional Analysis 20F67, 43A65, 46E36 We study boundary representations of hyperbolic groups $Γ$ on the (compactly embedded) function space $W^{\log,2}(\partialΓ)\subset L^2(\partialΓ)$, the domain of the logarithmic Laplacian on $\partialΓ$. We show that they are not uniformly bounded, and establish their exact growth (up a multiplicative constant): they grow with the square root of the length of $g\inΓ$. We also obtain $L^p$--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces. |
| title | Boundary representations of hyperbolic groups: the log-Sobolev case |
| topic | Group Theory Functional Analysis 20F67, 43A65, 46E36 |
| url | https://arxiv.org/abs/2408.06446 |