Quantitative $C^1$-estimates by Bismut formulae

TL;DR

This paper develops probabilistic Bismut formulae to obtain quantitative $C^1$-estimates for functions under elliptic operators, providing new bounds and conditions related to the zero-mean property, with extensions to differential forms.

Contribution

It introduces a probabilistic approach to derive quantitative derivative estimates for elliptic PDEs and extends these results to differential forms, offering new tools for analysis.

Findings

Derived explicit bounds for $du$ in terms of $u$ and $Lu$

Established a condition for the zero-mean value property of $ riangle u$

Extended estimates to differential forms

Abstract

For a $C^2$ function $u$ and an elliptic operator $L$, we prove a quantitative estimate for the derivative $du$ in terms of local bounds on $u$ and $Lu$. An integral version of this estimate is then used to derive a condition for the zero-mean value property of $\Delta u$. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings.