Li-Yau multiplier set and optimal Li-Yau gradient estimate on hyperbolic spaces

TL;DR

This paper introduces the Li-Yau multiplier set, computes it via heat kernels, and derives an optimal gradient estimate on hyperbolic spaces, leading to sharp Harnack inequalities for heat equations.

Contribution

It presents a novel Li-Yau multiplier set concept and applies heat kernel recurrence relations to establish optimal gradient estimates on hyperbolic spaces.

Findings

Computed Li-Yau multiplier set using heat kernel

Derived optimal Li-Yau gradient estimate on hyperbolic spaces

Established sharp Harnack inequalities for heat equations

Abstract

In this paper, motivated by finding sharp Li-Yau type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an optimal Li-Yau type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Finally, as an application, we obtain sharp Harnack inequalities on hyperbolic spaces.