We prove Sobolev-type and Morrey-type inequalities for Sobolev spaces related to a family of non-smooth vector fields which formally satisfy the Hörmander condition of step 2. The coefficients of the vector fields are not regular enough to define the Carnot-Carathéodory distance. Thus the result is proved by developing a real analysis technique which is based on an approximation procedure of Lipschitz continuous vector fields with a family of left-invariant first order operators on a nilpotent Lie group.
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A. Montanari, Daniele Morbidelli, Sobolev and Morrey Estimates for Non-Smooth Vector Fields of Step Two. Z. Anal. Anwend. 21 (2002), no. 1, pp. 135–157DOI 10.4171/ZAA/1068