Two-sided estimates of heat kernels on metric measure spaces (Q428149)

Let \((M,\mu,d)\) be a locally compact separable measure metric space with fully supported Radon measure \(\mu\) and geodesic metric \(d\). Let \(F(r)\) be a given function with a certain regularity, which includes the functions \(r^\beta\) (\(\beta>1\)) as special cases. Denote by \(V(x,r)\) the measure of the open ball of radius \(r\) centered at \(x\). It is shown that if the doubling volume property, the elliptic Harnack inequality and the estimate of the mean exit time are satisfied and all metric balls are precompact, then the heat kernel \(p_t(x,y)\) exists, is Hölder continuous in \(x\), \(y\), and satisfies the upper estimate NEWLINE\[NEWLINEp_t(x,y)\leq \frac{C}{V(x,\mathcal{R}(t))}\exp \left(-\frac{1}{2}t\Phi\left(c\frac{d(x,y)}{t}\right)\right),NEWLINE\]NEWLINE where \(\mathcal{R}=F^{-1}\) and NEWLINE\[NEWLINE\Phi(s):=\sup_{r>0}\left\{\frac{s}{r}-\frac{1}{F(r)}\right\},NEWLINE\]NEWLINE and the near-diagonal lower estimate NEWLINE\[NEWLINEp_t(x,y)\geq \frac{c}{V(x,\mathcal{R}(t))}\quad \mathrm{if} \;d(x,y)\leq \eta \mathcal{R}(t),NEWLINE\]NEWLINE where \(\eta>0\) is a small enough constant.