Observations on gaussian upper bounds for Neumann heat kernels

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