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Log-Sobolev inequalities: different roles of Ric and Hess. (English) Zbl 1187.60061

From the author’s abstract: Let \(P_t\) be the diffusion semigroup generated by \(L:=\Delta +\nabla\Delta\) on a complete connected Riemannian manifold with \(\mathrm{Ric}\geq -(\sigma^2\rho_0^2+c)\) for some constants \(\sigma, c>0\) and \(\rho_0\) the Riemannian distance to a fixed point. It is shown that \(P_t\) is hypercontractive, or the log–Sobolev inequality holds for the associated Dirichlet form, provided \(-\mathrm{Hess}_V\geq \delta\) holds outside of a compact set for some constant \(\delta>(1+\sqrt 2)\sigma\sqrt{d-1}\). This indicates that \(\mathrm{Ric}\) and \(-\mathrm{Hess}_V\) play quite different roles for the log–Sobolev inequality to hold.

MSC:

60J60 Diffusion processes
58J32 Boundary value problems on manifolds
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