Log-Sobolev inequalities: Different roles of Ric and Hess

TL;DR

This paper investigates how Ricci curvature and the Hessian of the potential function differently influence the validity of log-Sobolev inequalities on Riemannian manifolds, revealing their distinct roles in geometric analysis.

Contribution

It establishes conditions under which the log-Sobolev inequality holds, highlighting the different impacts of Ricci curvature and Hessian of the potential.

Findings

Log-Sobolev inequality holds if $- ext{Hess}_V$ is sufficiently negative outside a compact set.

Ricci curvature and Hessian of the potential function have different roles in the inequality.

Conditions for supercontractivity and ultracontractivity are also derived.

Abstract

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$ for some constants $\sigma, c>0$ and $\rho_o$ 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 $-\operatorname {Hess}_V\ge\delta$ 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 $\operatorname {Ric}$ and $-\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.