Hypoellipticity and Higher Order Gaussian Bounds

TL;DR

This paper establishes higher-order Gaussian bounds for kernels of semigroups on metric measure spaces under hypoelliptic conditions, with applications to subelliptic PDEs.

Contribution

It provides new sufficient conditions for Gaussian bounds and their derivatives for semigroup kernels in metric measure spaces, generalizing previous results.

Findings

Kernel bounds of the form xp(-c( ho(x,y)^{2\u03ba}/t)^{1/(2-1)}) established

Conditions for bounds on derivatives of kernels provided

Results are localizable and apply to subelliptic PDEs

Abstract

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