Pluricomplex Green Functions and the Dirichlet Problem for the Complex Monge-Ampere Operator

摘要

Let us recall some important notions which will be used here. Let D be an open subset in C~n, and denote by PSH(D) the cone of plurisubharmonic functions u: D → [-∞, +∞[ on D not identically equal to -∞ on any component of D. Let u ∈ PSH(D). For a ∈ D and 0 < r < d_a : = dist(a; C~nD), we set M_u(a,r): = ∫_(|ξ|=1) u(a + rξ)dσ(ξ), (0.1) where dσ(ξ) is the normalized area measure on the unit Euclidean sphere in C~n. It is well known that the function r | → M_u(a, r) is increasing and convex in log r.