Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients

We study interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. In particular, we show that solutions of large classes of sub elliptic equations with bounded measurable coefficients are Holder continuous. We present two types of results dealing with such equations. The first type generalizes the celebrated Fefferman-Phong geometric characterization of sub ellipticity in the smooth case. We introduce a notion of $L^q$-sub ellipticity for the rough case and develop an axiomatic method which provides a near characterization of the notion of $L^q$-sub ellipticity.The second type deals with generalizing a case of Hormanders' celebrated algebraic characterization of sub ellipticity for sums of squares of real analytic vector fields. In this case, we introduce a 'flag condition' as a substitute for the Hormander commutator condition which turns out to be equivalent to it in the smooth case. The question of regularity for quasilinear sub elliptic equations with smooth coefficients provides motivation for our study, and we briefly indicate some applications in this direction, including degenerate Monge-Ampere equations.