Hypoellipticity and Higher Order Gaussian Bounds

Let $(\mathfrak{M},ρ,μ)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},μ)\rightarrow L^{p_0}(\mathfrak{M},μ)$, $t\geq 0$, 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 $\left| K_t(x,y) \right| \leq C \exp\left( -c \left( \frac{ρ(x,y)^{2κ}}{t} \right)^{\frac{1}{2κ-1}} \right) μ\left( B_ρ\left(x,ρ(x,y)+t^{1/2κ}\right) \right)^{-1},$ where $B_ρ$ 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 $\partial_t -A$ and $\partial_t-A^{*}$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs.

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