Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Abstract Let L = ∑ j = 1 m X j 2 be a Hormander sum of squares of vector fields in space R n , where any X j is homogeneous of degree 1 with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for L in the X-Sobolev spaces W X k , p ( R n ) , where X = { X 1 , … , X m } . In our approach, we combine local results for general Hormander sums of squares, the homogeneity property of the X j 's, plus a global lifting technique for homogeneous vector fields.