Maximal subextensions of plurisubharmonic functions

U. Cegrell[1]; S. Kołodziej[2]; A. Zeriahi[3]

  • [1] Department of Mathematics, University of Umea, S-90187 Umea, Sweden.
  • [2] Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, 30-348 Kraków, Poland.
  • [3] Institut de Mathématiques de Toulouse, UPS, 118 Route de Narbonne, 31062 Toulouse cedex, France.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: S2, page 101-122
  • ISSN: 0240-2963

Abstract

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In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in ℂ n with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function ϕ on a given quasi-hyperconvex domain D ⊂ X of a compact Kähler manifold ( X , ω ) , with well defined Monge-Ampère measure such that ∫ D ( ω + d d c ϕ ) n ≤ ∫ X ω n , we prove that ϕ admits a global quasi-plurisubharmonic subextension ϕ ˜ to the whole manifold X . If moreover ( ω + d d c ϕ ) n puts no mass on pluripolar sets of D , the maximal subextension is shown to have a well defined global Monge-Ampère measure on X . Moreover we give a good control on the weigthed energy of the subextension in terms of the weigthed energy of the original function. Finally we provide an exemple in ℙ 2 which shows that in general the maximal subextension do not have a well defined Monge-Ampère measure on ℙ 2 if the original function concentrates some mass in an analytic disc.

How to cite

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Cegrell, U., Kołodziej, S., and Zeriahi, A.. "Maximal subextensions of plurisubharmonic functions." Annales de la faculté des sciences de Toulouse Mathématiques 20.S2 (2011): 101-122. <http://eudml.org/doc/219735>.

@article{Cegrell2011,
abstract = {In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in $\mathbb\{C\}^ n$ with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function $\varphi $ on a given quasi-hyperconvex domain $D \subset X$ of a compact Kähler manifold $(X,\omega )$, with well defined Monge-Ampère measure such that $\int _D (\omega + dd^c \varphi )^n \le \int _X \omega ^ n$, we prove that $\varphi $ admits a global quasi-plurisubharmonic subextension $\tilde\{\varphi \}$ to the whole manifold $X$. If moreover $(\omega + dd^c \varphi )^n$ puts no mass on pluripolar sets of $D$, the maximal subextension is shown to have a well defined global Monge-Ampère measure on $X$. Moreover we give a good control on the weigthed energy of the subextension in terms of the weigthed energy of the original function. Finally we provide an exemple in $\{\mathbb\{P\}\}^2$ which shows that in general the maximal subextension do not have a well defined Monge-Ampère measure on $\{\{\mathbb\{P\}\}\}^2$ if the original function concentrates some mass in an analytic disc.},
affiliation = {Department of Mathematics, University of Umea, S-90187 Umea, Sweden.; Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, 30-348 Kraków, Poland.; Institut de Mathématiques de Toulouse, UPS, 118 Route de Narbonne, 31062 Toulouse cedex, France.},
author = {Cegrell, U., Kołodziej, S., Zeriahi, A.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {plurisubharmonic function; subextension; Monge-Ampère measure; quasiplurisubharmonic function; hyperconvex domain; Lelong class},
language = {eng},
month = {4},
number = {S2},
pages = {101-122},
publisher = {Université Paul Sabatier, Toulouse},
title = {Maximal subextensions of plurisubharmonic functions},
url = {http://eudml.org/doc/219735},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Cegrell, U.
AU - Kołodziej, S.
AU - Zeriahi, A.
TI - Maximal subextensions of plurisubharmonic functions
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 101
EP - 122
AB - In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in $\mathbb{C}^ n$ with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function $\varphi $ on a given quasi-hyperconvex domain $D \subset X$ of a compact Kähler manifold $(X,\omega )$, with well defined Monge-Ampère measure such that $\int _D (\omega + dd^c \varphi )^n \le \int _X \omega ^ n$, we prove that $\varphi $ admits a global quasi-plurisubharmonic subextension $\tilde{\varphi }$ to the whole manifold $X$. If moreover $(\omega + dd^c \varphi )^n$ puts no mass on pluripolar sets of $D$, the maximal subextension is shown to have a well defined global Monge-Ampère measure on $X$. Moreover we give a good control on the weigthed energy of the subextension in terms of the weigthed energy of the original function. Finally we provide an exemple in ${\mathbb{P}}^2$ which shows that in general the maximal subextension do not have a well defined Monge-Ampère measure on ${{\mathbb{P}}}^2$ if the original function concentrates some mass in an analytic disc.
LA - eng
KW - plurisubharmonic function; subextension; Monge-Ampère measure; quasiplurisubharmonic function; hyperconvex domain; Lelong class
UR - http://eudml.org/doc/219735
ER -

References

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