Estimates of harmonic measures associated with degenerate Laplacian on strictly pseudoconvex domains
Let D be a smooth, bounded, strictly pseudoconvex domain in \({\mathbb{C}}^ n\) and suppose that D is given by \(D=\{z\in U:\lambda (z)<0\}\), where U is an open neighbourhood of the closure of D and \(\lambda\) is an infinitely differentiable, strictly plurisubharmonic function on U such that \(d\lambda\neq 0\) on the boundary \(\partial D\) of D. The author announces results on harmonic measures associated with the Laplace- Beltrami operator L of the complete Kähler metric \(-\partial {\bar \partial} \log (-\lambda)\) on D. Theorem 1 states that the L-harmonic measure \(d\omega\) relative to a fixed point z of D and the induced Euclidean measure \(d\sigma\) on \(\partial D\) are mutually absolutely continuous; moreover there is a function \(k_ z\) on \(\partial D\) such that \(k_ z,k_ z^{-1}\in L^{\infty}(d\sigma)\) and \(d\omega =k_ z d\sigma\). Theorem 2, which is obtained from Theorem 1 and some other results, concerns the existence of admissible limits of L-harmonic functions in D. Indications of proofs are given; details are to appear elsewhere.
scientific article; zbMATH DE number 4159068