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.