Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators.pdf

a r X i v : 0 8 0 3 .0 7 8 4 v 1 [ m a t h .A P ] 6 M a r 2 0 0 8 MUTUAL ABSOLUTE CONTINUITY OF HARMONIC AND SURFACE MEASURES FOR HO?RMANDER TYPE OPERATORS LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU Dedicated to Professor Maz’ya, on his 70th birthday 1. Introduction In this paper we study the Dirichlet problem for the sub-Laplacian associated with a system X = {X1, ...,Xm} of C∞ real vector fields in Rn satisfying Ho?rmander’s finite rank condition (1.1) rank Lie[X1, ...,Xm] ≡ n. Throughout this paper n ≥ 3, and X?j denotes the formal adjoint of Xj . The sub-Laplacian associated with X is defined by (1.2) Lu = m∑ j=1 X?jXju . A distributional solution of Lu = 0 is called L-harmonic. Ho?rmander’s hypoellipticity theorem [H] guarantees that every L-harmonic function is C∞, hence it is a classical solution of Lu = 0. We consider a bounded open set D ? Rn, and study the Dirichlet problem (1.3) { Lu = 0 in D , u = φ on ?D . Using Bony’s maximum principle [B] one can show that for any φ ∈ C(?D) there exists a unique Perron-Wiener-Brelot solution HDφ to (1.3). We focus on the boundary regularity of the solution. In particular, we identify a class of domains, which are referred to as ADPX domains (admissible for the Dirichlet problem), for which we prove the mutual absolute continuity of the L-harmonic measure dωx and of the so-called horizontal perimeter measure dσX = PX(D; ·) on ?D. The latter constitutes the appropriate replacement for the standard surface measure on ?D and plays a central role in sub-Riemannian geometry. Moreover, we show that a reverse Ho?lder inequality holds for a suitable Poisson kernel which is naturally associated with the system X. As a consequence of such reverse Ho?lder inequality we then derive the solvability of (1.3) for boundary data φ ∈ Lp(?D, dσX), for 1 p ≤ ∞. If instead the domain D belongs to the smaller class σ?ADPX introduced in Definition 8.10 below, we prove that L-harmonic measure is mutually absolutely continuous