Title: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications
Abstract: In this paper we mainly prove weighted Poincare inequalities for vector fields satisfying Hormander's condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the local regularity of weak solutions for certain classes of strongly degenerate differential operators formed by vector fields.