<p>We study boundary representations of hyperbolic groups <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> on the (compactly embedded) function space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$W^{\log ,2}(\partial \Gamma )\subset L^2(\partial \Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mo>log</mo> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the domain of the logarithmic Laplacian on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that they are not uniformly bounded, and establish their exact growth (up a multiplicative constant): they grow with the square root of the length of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g\in \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation>. We also obtain <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>–analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors–David regular metric measure spaces.</p>

Boundary Representations of Hyperbolic Groups: The Log–Sobolev Case

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2026年5月6日

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Kevin Boucher,
Ján Špakula

摘要

We study boundary representations of hyperbolic groups $\Gamma $ $Γ$ on the (compactly embedded) function space $W^{\log ,2}(\partial \Gamma )\subset L^2(\partial \Gamma )$ $W^{log, 2} (\partial Γ) \subset L^{2} (\partial Γ)$ , the domain of the logarithmic Laplacian on $\partial \Gamma $ $\partial Γ$ . 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\in \Gamma $ $g \in Γ$ . We also obtain $L^p$ $L^{p}$ –analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors–David regular metric measure spaces.