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).
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Bruno Franchi, Annalisa Baldi, Maria Carla Tesi, Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups. J. Eur. Math. Soc. 11 (2009), no. 4, pp. 777–798DOI 10.4171/JEMS/166