Boundary representations of hyperbolic groups: the log-Sobolev case

2024/08/12 by Boucher, Kevin, Špakula, Ján
#20F67 #43A65 #46E36 #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2408.06446

Abstract

We study boundary representations of hyperbolic groups Γ on the (compactly embedded) function space W^log,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 L^p--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces.

Related

• Sobolev spaces and uniform boundary representations
• Boundary representations of hyperbolic groups
• Boundary Representations of Locally Compact Hyperbolic Groups
• Riesz operators and L^p-boundary representations for hyperbolic groups
• Logarithmic Sobolev inequalities on homogeneous spaces