Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups

Abstract

Let $\mathcal{L}$ be a non-negative self-adjoint $N\times N$ matrix-valued operator of order $a\le Q$ on a Carnot group $\mathbb{G}$. Here $Q$ is the homogeneous dimension of $\mathbb{G}$. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp $L^p$ estimates in appropriate Sobolev spaces), $L^p$-maximal hypoellipticity (i.e. sharp $L^p$estimates in appropriate Sobolev spaces for {$1<p<\infty$}), and what we call maximal subellipticity of $\mathcal{L}$ (which is basically a sharp higher order energy estimate).