Boundary Representations of Hyperbolic Groups: The Log–Sobolev Case

Abstract We study boundary representations of hyperbolic groups $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> on the (compactly embedded) function space $$W^{\log ,2}(\partial \Gamma )\subset L^2(\partial \Gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , the domain of the logarithmic Laplacian on $$\partial \Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> . 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 $$g\in \Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> . We also obtain $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> –analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors–David regular metric measure spaces.