Boundary representations of hyperbolic groups: the log-Sobolev case
Boundary representations of hyperbolic groups: the log-Sobolev case
We study boundary representations of hyperbolic groups Γ on the (compactly embedded) function space Wlog,2(∂Γ)⊂L2(∂Γ), the domain of the logarithmic Laplacian on ∂Γ. 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∈Γ. We also obtain Lp--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces.
Boucher, Kevin L
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Špakula, Ján
c43164e4-36a7-4372-9ce2-9bfbba775d77
Boucher, Kevin L
2d93ec11-c486-4af3-a513-58d84eec4258
Špakula, Ján
c43164e4-36a7-4372-9ce2-9bfbba775d77
Boucher, Kevin L and Špakula, Ján
(2026)
Boundary representations of hyperbolic groups: the log-Sobolev case.
Potential Analysis, 65, [6 (2026)].
(doi:10.48550/arXiv.2408.06446).
Abstract
We study boundary representations of hyperbolic groups Γ on the (compactly embedded) function space Wlog,2(∂Γ)⊂L2(∂Γ), the domain of the logarithmic Laplacian on ∂Γ. 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∈Γ. We also obtain Lp--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces.
Text
log-sobolev-3
- Accepted Manuscript
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Accepted/In Press date: 12 August 2024
e-pub ahead of print date: 6 May 2026
Identifiers
Local EPrints ID: 494939
URI: http://eprints.soton.ac.uk/id/eprint/494939
ISSN: 1572-929X
PURE UUID: 413b06b2-68fa-49b8-80d5-2143177a7707
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Date deposited: 23 Oct 2024 16:54
Last modified: 27 May 2026 01:46
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Author:
Kevin L Boucher
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