The Li-Yau inequality and applications under a curvature-dimension condition

TL;DR

This paper establishes a new, stronger Li-Yau inequality for Markov semigroups under curvature-dimension conditions, leading to improved heat kernel bounds and a new parabolic Harnack inequality applicable in various curvature settings.

Contribution

It introduces a novel, more powerful Li-Yau inequality that enhances existing bounds and applies to general Markov semigroups and Riemannian manifolds.

Findings

Stronger Li-Yau inequality under curvature-dimension condition

Equivalent parabolic Harnack inequality in various curvature regimes

Ultracontractive bounds achieved in positive curvature settings

Abstract

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.