Sharp log-Sobolev inequalities in [formula omitted] spaces with applications

Abstract

Given p,N>1, we prove the sharp Lp-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0,N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0,N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf–Lax semigroup in CD(0,N) spaces. The proof of this result uses Hamilton–Jacobi inequality and Sobolev regularity properties of the Hopf–Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0,N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.