Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

TL;DR

This paper establishes new heat kernel bounds, $L^p$ estimates, and second order Davies-Gaffney estimates for covariant derivatives of heat semigroups on differential forms on Riemannian manifolds, with implications for Riesz transforms.

Contribution

It provides novel bounds and estimates for the covariant derivative of the heat semigroup on differential forms, extending understanding of geometric analysis on manifolds.

Findings

Li-Yau type heat kernel bounds for $ abla e^{-toldsymbol{ riangle}_j}$

Exponential weighted $L^p$ bounds for the heat kernel of $ abla e^{-toldsymbol{ riangle}_j}$

Boundedness of $ abla e^{-toldsymbol{ riangle}_j}$ in $L^p$ for all $1 extless p extless \infty$

Abstract

With $\vec{\Delta}_j\geq 0$ is the uniquely determined self-adjoint realization of the Laplace operator acting on $j$-forms on a geodesically complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant derivative, we prove amongst other things a Li-Yau type heat kernel bound for $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, an exponentially weighted $L^p$ bound for the heat kernel of $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$, if the curvature tensor of $M$ and its covariant derivative are bounded, that $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$ is bounded in $L^p$ for all $1\leq p<\infty$, if the curvature tensor of $M$ and its covariant derivative are bounded, and a second order Davies-Gaffney estimate (in terms of $\nabla$ and $\vec{\Delta}_j$) for $\mathrm{e}^{ -t\vec{\Delta}_j }$ for small times, if the $j$-th degree Bochner-Lichnerowicz potential $V_j=\vec{\Delta}_j-\nabla^{\dagger}\nabla$ of $M$ is bounded from below (where $V_1=\mathrm{Ric}$), which is shown to fail for large times if $V_j$ is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $\nabla (\vec{\Delta}_j+\kappa)^{-1/2}$ in $L^p$ for all $1\leq p<\infty$ (which we prove for $1\leq p\leq 2$), and explain its implications to geometric analysis, such as the $L^p$-Calder\'on-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$.