Boundary representations of hyperbolic groups: the log-Sobolev case

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