Sharp log-Sobolev inequalities in ${\sf CD}(0,N)$ spaces with applications

Abstract: Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ 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. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ 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 (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.