Boundary representations of hyperbolic groups: the log-Sobolev case

Abstract

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log ,2}(\partial \Gamma )\subset L^2(\partial \Gamma )$, the domain of the logarithmic Laplacian on $\partial \Gamma$. 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 $g\in \Gamma$. 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.

arXiv comment19 pages