Sharp log-Sobolev inequalities in CD(0, N) spaces with applications

Given p, N > 1, we prove the sharp L-p-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 L-2-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 Rieman nian manifolds will be a simple consequence of our sharp log-Sobolev inequality.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

The article investigates the sharp L-p-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0, N) condition, and proves it using isoperimetric inequality, symmetrization, and scaling argument. It also establishes hypercontractivity estimate for the Hopf-Lax semigroup and obtains Gaussian-type L-2-log-Sobolev inequality and hypercontractivity estimate in RCD(0, N) spaces.