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

Zoltán M. Balogh, Alexandru Kristály, Francesca Tripaldi

Published 2022-10-27Version 1

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. Combining the sharp $L^p$-log-Sobolev inequality with the Hamilton-Jacobi inequality, we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. Moreover, a 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.