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

Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed

Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 1, p. 81-96

Abstract

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.

MR 2413673 | Zbl 1150.32011 | 3 citations in Numdam

Classification: 32W20, 32Q25, 32U05

@article{ASNSP_2008_5_7_1_81_0,
     author = {Benelkourchi, Slimane and Guedj, Vincent and Zeriahi, Ahmed},
     title = {A priori estimates for weak solutions of complex Monge-Amp\`ere equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 7},
     number = {1},
     year = {2008},
     pages = {81-96},
     zbl = {1150.32011},
     mrnumber = {2413673},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2008_5_7_1_81_0}
}

Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed. A priori estimates for weak solutions of complex Monge-Ampère equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 1, pp. 81-96. http://www.numdam.org/item/ASNSP_2008_5_7_1_81_0/

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