Boundary Representations of Hyperbolic Groups: The Log–Sobolev Case
Abstract
Abstract We study boundary representations of hyperbolic groups $$\Gamma $$ Γ on the (compactly embedded) function space $$W^{\log ,2}(\partial \Gamma )\subset L^2(\partial \Gamma )$$ W log , 2 ( ∂ Γ ) ⊂ L 2 ( ∂ Γ ) , the domain of the logarithmic Laplacian on $$\partial \Gamma $$ ∂ 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 $$g\in \Gamma $$ g ∈ Γ . We also obtain $$L^p$$ L p –analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors–David regular metric measure spaces.
Fields of Study & Subjects
Geometric and Algebraic Topology
Quantum chaos and dynamical systems
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Potential Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 65
- Issue
- 1
- Publication date
- 6 May 2026
- Language
- EN