In this paper we mainly prove weighted Poincaré inequalities for vector fields satisfying Hörmander'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.
Cite this article
Guozhen Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Rev. Mat. Iberoam. 8 (1992), no. 3, pp. 367–439DOI 10.4171/RMI/129