An embedding theorem for sobolev spaces related to non-smooth vector fieldsand harnack inequality

In the first part of this paper we study some differentiabilityperties properties (In Sobolev space setting)of functions belonging to thespace generated by the vector fields in aJ J Dx.connected open subset of Rn ; here the λjs are continuous (but in generalnot smooth) and nonnegative. In particular, under suitable geometricalhypotheses involving the integral curves of we get thenfollowing embedding estimate: for all test functions u and for suitable positive constants e and CP depending only on the "order of degeneration" of the vector fields Such a problem arises naturally in study of the pointewise estimates for the weak solutions of some degenerate elliptic equations with measurable coefficients.