Observations on gaussian upper bounds for Neumann heat kernels

TL;DR

This paper establishes Gaussian upper bounds for Neumann heat kernels on domains within Riemannian manifolds, leading to results on semigroup analyticity and spectral multipliers.

Contribution

It proves new Gaussian upper bounds for Neumann heat kernels under geometric conditions, and derives consequences for semigroup analyticity and spectral multipliers.

Findings

Heat kernel satisfies Gaussian upper bound under certain conditions.

Semigroup $e^{-tA}$ is analytic on $L^p( ext{Omega})$ for all } p ext{ in } [1, ext{infinity}).

Spectral multiplier results are obtained from the bounds.

Abstract

Given a domain $\Omega$ of a complete Riemannian manifold $\mathcal{M}$ and define $\mathcal{A}$ to be the Laplacian with Neumann boundary condition on $\Omega$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$ h(t,x,y)\leq \frac{C}{\left[V\_\Omega(x,\sqrt{t})V\_\Omega (y,\sqrt{t})\right]^{1/2}}\left( 1+\frac{d^2(x,y)}{4t}\right)^{\delta}e^{-\frac{d^2(x,y)}{4t}},\;\; t\textgreater{}0,\; x,y\in \Omega . $$ Here $d$ is the geodesic distance on $\mathcal{M}$, $V\_\Omega (x,r)$ is the Riemannian volume of $B(x,r)\cap \Omega$, where $B(x,r)$ is the geodesic ball of center $x$ and radius $r$, and $\delta$ is a constant related to the doubling property of $\Omega$. As a consequence we obtain analyticity of the semigroup $e^{-t {\mathcal A}}$ on $L^p(\Omega)$ for all $p \in [1, \infty)$ as well as a spectral multiplier result.