Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. (English) Zbl 0804.35015

Lu, Guozhen

Rev. Mat. Iberoam. 8, No. 3, 367-439 (1992).

Summary: We 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 show the local regularity of weak solutions for certain classes of strongly degenerate differential operators formed by vector fields.

MSC:

35B45	A priori estimates in context of PDEs
35D10	Regularity of generalized solutions of PDE (MSC2000)
35J70	Degenerate elliptic equations

Keywords:

weighted Poincaré inequalities; Hörmander’s condition; Sobolev type inequalities; degenerate differential operators