Hypoellipticity and Higher Order Gaussian Bounds

Let $(\mathfrak{M},\rho ,\mu )$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\mathrm{\infty })$, and $T(t):L^{p_0}(\mathfrak{M},\mu )\to L^{p_0}(\mathfrak{M},\mu )$, $t\ge 0$, a strongly continuous semi-group. We provide sufficient conditions under which $T(t)$ is given by integration against an integral kernel satisfying higher-order Gaussian bounds of the form $$\mid K_t(x,y)\mid \le C\exp (-c{\left(\frac{\rho (x,y{)}^{2\kappa }}{t}\right)}^{\frac{1}{2\kappa -1}})\mu {\left(B_{\rho }(x,\rho (x,y)+t^{1\mathrm{/}2\kappa })\right)}^{-1},$$ where $B_{\rho }$ denotes the metric ball. We also provide conditions for similar bounds on ``derivatives’’ of $K_t(x,y)$ and our results are localizable. If $A$ is the generator of $T(t)$ the main hypothesis is that ${\mathrm{\partial }}_t-A$ and ${\mathrm{\partial }}_t-A^{\ast }$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs.