HARNACK INEQUALITIES FOR WEIGHTED SUBELLIPTIC P-LAPLACE EQUATIONS CONSTRUCTED BY HOORMANDER VECTOR FIELDS

This paper deals with the following weighted subelliptic p-Laplace equation L(p)u := div(X) (< A(x)Xu(x), Xu(x)>(p-2/2) A(x)Xu(x)) = g(x), where the system X = (X-1; ...; X-m) satisfies Hormander's condition, u is an element of W-1,W-p (Omega; w); 1 < p < Q; A (x) is a bounded measurable and m x m symmetric matrix satisfying lambda(-1) w (x)(2/p)vertical bar xi vertical bar(2) <= < A (x) xi, xi > <= lambda w (x)(2/p)vertical bar xi vertical bar(2); for xi is an element of R-m, with w = w (x) being an A(p) function. We first establish a maximum principle for weak solutions to the equation L(p)u = g with the weighted Sobolev inequality and the extension of Moser iteration technique to the weight case. Next, the local boundedness and the Harnack inequality for nonnegative weak solutions to L(p)u = g are proved. As an application, the Holder continuity for nonnegative weak solutions is given. Unlike in many papers, we do not impose any restriction in advance for measures of metric balls.