Dimension-Free Harnack Inequality and its Applications

Abstract

This paper presents a self-contained account concerning a dimension-free Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical Li-Yau’s Harnack inequality did, but also provides a direct way to describe various dimension-free properties of finite and infinite-dimensional diffusion semigroups. The author starts with a standard weighted Laplace operator on a Riemannian manifold with curvature bounded from below, and then move further to the unbounded below curvature case and its infinite-dimensional settings.

Keywords

Harnack inequality / diffusion semigroup / Riemannian manifold / heat kernel / hypercontractivity / supercontractivity / ultracontractivity / 58J65 / 60H30

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