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

Abstract

Let ℒ be a non-negative self-adjoint N × N matrix-valued operator of order a ≤ Q on a Carnot group G. Here Q is the homogeneous dimension of G. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp _L_2 estimates in appropriate Sobolev spaces), Lp-maximal hypoellipticity (i.e. sharp Lp estimates in appropriate Sobolev spaces for 1 < p < ∞), and what we call maximal subellipticity of ℒ (which is basically a sharp higher order energy estimate).