Fundamental solution for higher order homogeneous hypoelliptic operators structured on H\"{o}rmander vector fields

Abstract

We introduce and study a new class of higher order differential operators defined on $\mathbb{R}^{n}$, which are built with H\"{o}rmander vector fields, homogeneous w.r.t. a family of dilations (but not left invariant w.r.t. any structure of Lie group) and have a structure such that a suitably lifted version of the operator is hypoelliptic. We call these operators ''generalized Rockland operators''. We prove that these operators are themselves hypoelliptic and, under a natural condition on the homogeneity degree, possess a global fundamental solution $\Gamma\left( x,y\right) $ which is jointly homogeneous in $\left( x,y\right) $ and satisfies sharp pointwise estimates. Our theory can be applied also to some higher order heat-type operators and their fundamental solutions.