A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi; Vincent Guedj; Ahmed Zeriahi

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 1, page 81-96
  • ISSN: 0391-173X

Abstract

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Let X be a compact Kähler manifold and ω be a smooth closed form of bidegree ( 1 , 1 ) which is nonnegative and big. We study the classes ℰ χ ( X , ω ) of ω -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight χ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ℰ χ ( X , ω ) . This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori 𝒞 0 -estimate.

How to cite

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Benelkourchi, Slimane, Guedj, Vincent, and Zeriahi, Ahmed. "A priori estimates for weak solutions of complex Monge-Ampère equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.1 (2008): 81-96. <http://eudml.org/doc/272282>.

@article{Benelkourchi2008,
abstract = {Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes $\{\mathcal \{E\}\}_\{\chi \}(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class $\{\mathcal \{E\}\}_\{\chi \}(X,\omega )$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori $\{\mathcal \{C\}\}^0$-estimate.},
author = {Benelkourchi, Slimane, Guedj, Vincent, Zeriahi, Ahmed},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {81-96},
publisher = {Scuola Normale Superiore, Pisa},
title = {A priori estimates for weak solutions of complex Monge-Ampère equations},
url = {http://eudml.org/doc/272282},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Benelkourchi, Slimane
AU - Guedj, Vincent
AU - Zeriahi, Ahmed
TI - A priori estimates for weak solutions of complex Monge-Ampère equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 1
SP - 81
EP - 96
AB - Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal {E}}_{\chi }(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal {E}}_{\chi }(X,\omega )$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori ${\mathcal {C}}^0$-estimate.
LA - eng
UR - http://eudml.org/doc/272282
ER -

References

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