Fundamental solution for higher order homogeneous hypoelliptic operators structured on Hörmander vector fields
Stefano Biagi, Marco Bramanti
TL;DR
The paper develops a global theory for higher-order hypoelliptic operators built from Hörmander vector fields that are homogeneous under a dilations family but not left-invariant on any Lie group. By lifting to an appropriate homogeneous group and applying a saturation argument, the authors prove hypoellipticity of generalized Rockland operators, establish a Liouville-type property, and construct a global, jointly homogeneous fundamental solution $\Gamma(x,y)$ when the homogeneity degree $\nu$ satisfies $\nu<q$, with sharp pointwise estimates. The results extend to heat-type operators $\mathcal{L}\pm\partial_t$, providing a robust framework for higher-order hypoelliptic operators beyond the classical left-invariant homogeneous-group setting. This advances understanding of non-left-invariant, higher-order hypoelliptic structures and their fundamental solutions, with potential applications to global regularity and evolution problems in non-homogeneous geometric contexts.
Abstract
We introduce and study a new class of higher order differential operators defined on $\mathbb{R}^{n}$, which are built with Hörmander 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 $Γ\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.