Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function

Abstract

Let be a bounded domain in CN. Let z be a point in and let Jz be the set of all Jensen measures on with barycenter at z with respect to the space of functions continuous on and plurisubharmonic in . The authors prove that is hyperconvex if and only if, for every z 2 @ , measures in Jz are supported by @ . From this they deduce that a pluricomplex Green function g(z,w) with its pole at w continuously extends to @ with zero boundary values if and only if is hyperconvex. Then the authors give a criterion for Reinhardt domains to be hyperconvex and explicitly compute the pluricomplex Green function on the Hartogs triangle. The last sections are devoted to the boundary behaviour of pluricomplex Green functions. Such a function has Property (P0) at a point w0 2 @ if limw!w0 g(z,w) = 0 for every z 2 . If the convergence is uniform in z on compact subsets of r{w0}, then w0 has Property (P0). Several sufficient conditions for points on the boundary with these properties are given. (Less)