Observations on gaussian upper bounds for Neumann heat kernels

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

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