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

Abstract

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.

Jensen measures and boundary values of plurisubharmonic functions

“…Therefore, we have limr If we assume is addition that the domain is hyperconvex, then the situation is more satisfactory, since we do not require an extension of the boundary function. To prove this, we will require (a part of) a theorem from [2]. Proof.…”

Estimates for k-Hessian operator and some applications

“…The above two propositions for the complex Monge-Ampère operator were proved by R. Czyż [5], M. Carlehed, U. Cegrell and F. Wikström [2], respectively. Although we use methods from pluripotential theory, our results are completely new for the kHessian operator and some of them are even stronger.…”

“…Now we proceed with the proof of Proposition 1.2. Our method is due to M. Carlehed, U. Cegrell and F. Wikström [2]. We need the following lemma.…”

Pluriharmonic extension in proper image domains

“…If Ω is a hyperconvex domain, then supp µ ⊂ ∂Ω for all z ∈ ∂Ω and all µ ∈ J c z (see [3]). Moreover, for a bounded hyperconvex domain Ω ⊆ C n it was proved in [11] that Ω has the approximation property if, and only if,…”