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July 2009 Log-Sobolev inequalities: Different roles of Ric and Hess
Feng-Yu Wang
Ann. Probab. 37(4): 1587-1604 (July 2009). DOI: 10.1214/08-AOP444


Let Pt be the diffusion semigroup generated by L:=Δ+∇V on a complete connected Riemannian manifold with Ric≥−(σ2ρo2+c) for some constants σ, c>0 and ρo the Riemannian distance to a fixed point. It is shown that Pt is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided −HessVδ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}$. This indicates, at least in finite dimensions, that Ric and −HessV play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.


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Feng-Yu Wang. "Log-Sobolev inequalities: Different roles of Ric and Hess." Ann. Probab. 37 (4) 1587 - 1604, July 2009.


Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1187.60061
MathSciNet: MR2546756
Digital Object Identifier: 10.1214/08-AOP444

Primary: 58G32 , 60J60

Keywords: diffusion semigroup , Log-Sobolev inequality , Ricci curvature , Riemannian manifold

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 4 • July 2009
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