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

Carlehed, Magnus; Cegrell, Urban and Wikström, Frank LU (1999) In Annales Polonici Mathematici 71(1). p.87-103
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... (More)
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)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Annales Polonici Mathematici
volume
71
issue
1
pages
87 - 103
publisher
Institute of Mathematics, Polish Academy of Sciences
ISSN
1730-6272
language
English
LU publication?
no
id
ee3090b7-a73d-425a-bd24-bc09b681879e (old id 1314612)
date added to LUP
2009-05-11 15:00:31
date last changed
2017-01-13 13:29:06
@article{ee3090b7-a73d-425a-bd24-bc09b681879e,
  abstract     = {Let <br/><br>
 be a bounded domain in CN. Let z be a point in <br/><br>
 and let Jz be the set of all Jensen<br/><br>
measures on <br/><br>
 with barycenter at z with respect to the space of functions continuous on <br/><br>
 and<br/><br>
plurisubharmonic in <br/><br>
. The authors prove that <br/><br>
 is hyperconvex if and only if, for every z 2 @<br/><br>
,<br/><br>
measures in Jz are supported by @<br/><br>
. From this they deduce that a pluricomplex Green function<br/><br>
g(z,w) with its pole at w continuously extends to @<br/><br>
 with zero boundary values if and only if <br/><br>
<br/><br>
is hyperconvex.<br/><br>
Then the authors give a criterion for Reinhardt domains to be hyperconvex and explicitly compute<br/><br>
the pluricomplex Green function on the Hartogs triangle.<br/><br>
The last sections are devoted to the boundary behaviour of pluricomplex Green functions. Such<br/><br>
a function has Property (P0) at a point w0 2 @<br/><br>
 if limw!w0 g(z,w) = 0 for every z 2 <br/><br>
. If the<br/><br>
convergence is uniform in z on compact subsets of <br/><br>
r{w0}, then w0 has Property (P0). Several<br/><br>
sufficient conditions for points on the boundary with these properties are given.},
  author       = {Carlehed, Magnus and Cegrell, Urban and Wikström, Frank},
  issn         = {1730-6272},
  language     = {eng},
  number       = {1},
  pages        = {87--103},
  publisher    = {Institute of Mathematics, Polish Academy of Sciences},
  series       = {Annales Polonici Mathematici},
  title        = {Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function},
  volume       = {71},
  year         = {1999},
}