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The Poisson kernel for certain degenerate elliptic operators. (English) Zbl 0556.35036

Let \(V_ 0,V_ 1,...,V_ d\) be given vector fields and consider the differential operator \(L=\sum^{d}_{k=1}V^ 2_ k+V_ 0\) where the vector fields are interpreted as directional derivatives, and let \(G\) be a bounded open set with smooth non-characteristic boundary. If the Dirichlet problem for \(L\) on \(G\) is well posed then there exists a weakly continuous map \(x\to H(x,\cdot)\) from \(G\) to probability measures on \(\partial G\) such that \(u_ f(x)=\int f(y)H(x,dy),\) \(x\in G\), \(f\in C(\partial G)\) is the solution of the Dirichlet problem, \(Lu_ f=0\) in \(G\), \(u_ f=f\) on \(\partial G\). The present article studies the harmonic measure \(H(x,\cdot)\). It is shown that \(H(x,\cdot)\) admits a smooth density \(h(x,\cdot)\) with respect to the natural boundary measure, provided that certain Hörmander-type nondegeneracy conditions and strong nontangency conditions at the boundary are satisfied. The approach differs from the usual analytic strategy and can be viewed as a generalization of the classical subordination procedure used to study the Poisson kernel in the upper half space.
Reviewer: G. Gudmundsdottir

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
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