The Poisson kernel for certain degenerate elliptic operators

Let L = 1 2∑_{k = 1}^d V_k² + V₀ be a smooth second order differential operator on Rⁿ written in Hörmander form, and G be a bounded open set with smooth noncharacteristic boundary. Under a global condition that ensures that the Dirichlet problem is well posed for L on G and a nondegeneracy condition at the boundary (precisely: the Lie algebra generated by the vector fields V₀, V₁,..., V_d is of full rank on the boundary) then the harmonic measure for L starting at any point in G has a smooth density with respect to the natural boundary measure. Estimates on the derivatives of this density (the Poisson kernel) similar to the classical estimates for the Poisson kernel for the Laplacian on a half space are given.