A Harnack inequality and Holder continuity for weak solutions to parabolic operators with Hormander vector fields

This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hormander vector fields. Adapting the iteration scheme of Jurgen Moser for elliptic and parabolic equations in Rn we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form ut = sum Xi(aijXju ) we use this to show Holder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parabolic equations of the form ut = - X*i Aj(Xju). With this we show both a Harnack inequality and Holder continuity of weak solutions.