Boundary representations of hyperbolic groups: the log-Sobolev case

Abstract

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.