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

Summary, in English

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.

• Author: Magnus Carlehed; Urban Cegrell; Frank Wikström
• Publishing year: 1999
• Language: English
• Pages: 87-103
• Publication/Series: Annales Polonici Mathematici
• Volume: 71
• Issue: 1
• Document type: Journal article
• Publisher: Institute of Mathematics, Polish Academy of Sciences
• Topic: Mathematics; Mathematical Analysis
• Status: Published
• ISBN/ISSN/Other: ISSN: 1730-6272