Abstract. We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in B-regular domain. This theorem implies that the two classes of Jensen measures coincide in B regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.

Abstract. We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs triangle. Finally, we study the behaviour of the pluricomplex Green function g(z, w) as the pole w tends to a boundary point.

Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function [Formula: see text], we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on [Formula: see text].

Abstract. We consider the pluricomplex Green function with multiple poles as introduced by Lelong. We give a partial solution to a question concerning the set where the multipole Green function coincides with the sum of the corresponding single pole Green functions.

Abstract. Let V be an analytic variety in a domain Ω ⊂ C n and let K ⊂⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V , we prove that the relative extremal function ω K is continuous on V if Ω is hyperconvex and K is regular.1. Introduction. Let P (D) be a linear partial differential operator with constant coefficients. Hörmander [9] gave a characterization of when P (D) is surjective on the space A(Ω) of real-analytic functions when Ω is a convex domain in R n . This characterization is stated in terms of Phragmén-Lindelöf type estimates for plurisubharmonic functions on the zero variety of the symbol of P (D). Hörmander's result has been the main inspiration for trying to find various kinds of geometric or algebraic criteria for recognizing varieties satisfying such Phragmén-Lindelöf estimates, and there are a large number of papers attacking this problem. One of the tools that have been used is the relative extremal function ω K . (See for example [1,2,3].) The main goal of this paper is to prove that ω K is continuous if K is regular. To state the result more precisely, we will require some preliminary definitions.there is an open neighborhood U of x such that V ∩ U is irreducible, we say that V is locally irreducible at x. We denote by V irr the set of locally irreducible points in V . If x ∈ V irr , we say that V is reducible at x; we denote the set of reducible points by V red . Note that V irr is an open dense subset of V containing all regular points of V .