Hessian matrix estimates of heat-type equations via Bismut-Stroock Hessian formula

TL;DR

This paper introduces a new method for estimating the Hessian matrix of heat-type equations on Riemannian manifolds, providing explicit formulas and applications like inequalities and eigenfunction estimates.

Contribution

It develops a novel global Hessian estimate using a Bismut-Stroock formula, with explicit coefficients and growth functions, advancing analysis of heat equations on manifolds.

Findings

Established a new Hessian estimate with explicit coefficients

Derived a backward weak Harnack inequality

Provided a pointwise Hessian estimate for eigenfunctions

Abstract

In this paper, we establish a new global Hessian matrix estimate for heat-type equations on Riemannian manifolds using a Bismut-type Hessian formula. Our results feature fully explicit coefficients as well as delay / growth rate functions. These estimates yield two key applications: a novel backward weak Harnack inequality and a precise pointwise Hessian estimate for eigenfunctions.