Wolff-potential estimates and doubling of subelliptic -harmonic measures

Abstract

Abstract. Let X = {X 1 , ..., Xm} be a system of C ∞ vector fields in R n satisfying Hörmander's finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of ∂Ω, to the equationGiven p, 1 < p < ∞, fixed, we impose conditions on the function A = (A 1 , ..., Am) : R n × R m → R m , which imply that the equation is a quasi-linear partial differential equations of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p = 2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.2000 Mathematics Subject Classification.

Boundary behavior of solutions to the parabolic p-Laplace equation

“…Based on this the lower bound established on µ can be interpreted as a non-degeneracy estimate, close to the boundary, of the solution. Our proof of the lower bound for the measure µ is a modification of the elliptic proof, see for example [5,21]. However, our proof is genuinely non-linear, it applies to much more general operators of p-parabolic type, and the result seems to be new already in the case p = 2.…”