The general definition of the complex Monge-Ampère operator

Cegrell, Urban

doi:10.5802/aif.2014

Cited by 180 publications

(143 citation statements)

References 9 publications

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“…Taking r j := 3t j and ε := η/3, we obtain the theorem since the family of pseudo-balls (ω z j (r j )) covers E.◮ As a consequence of the last lemma we obtain a minimum principle for the Cegrell's class F(B) ( [12]) on the unit ball in terms of the pseudo-distance on the unit ball.…”

Section: A Lower Bound For Plurisubharmonic Functionsmentioning

confidence: 88%

A minimum principle for plurisubharmonic functions

Zériahi¹

2007

Indiana Univ. Math. J.

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Abstract:The main goal of this work is to give new and precise generalizations to various classes of plurisubharmonic functions of the classical minimum modulus principle for holomorphic functions of one complex variable, in the spirit of the famous lemma of Cartan-Boutroux. As an application we obtain precise estimates on the size of "plurisubharmonic lemniscates" in terms of appropriate Hausdorff contents.

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Section: A Lower Bound For Plurisubharmonic Functionsmentioning

confidence: 88%

A minimum principle for plurisubharmonic functions

Zériahi¹

2007

Indiana Univ. Math. J.

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“…It follows from [Ce2] thatũ ∈ F(Ω). The following theorem provides a description of the Monge-Ampère measure of the maximal subextension.…”

Section: Monge-ampère Measure Of Maximal Subextensionsmentioning

confidence: 99%

“…Observe that F(D, ω) is a convex set and P 0 (D, ω) ⊂ F(D, ω). The class F(D, ω) is the counterpart of the class defined by Cegrell in [Ce2]. Let D be a hyperconvex domain where the form ω has a plurisubharmonic potential q on D with boundary values 0 and let F(D) the class defined in [Ce2].…”

Section: Weighted Monge-ampère Energy Classesmentioning

confidence: 99%

Maximal subextensions of plurisubharmonic functions

Cegrell¹,

Kołodziej²,

Zériahi³

2011

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact Kähler manifold. We prove that a precise bound on the complex Monge-Ampère mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampère measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampère measure and the right bound on its Monge-Ampère mass on the unit ball in C n for which the maximal subextension to the complex projective space P n does not have a globally well defined complex Monge-Ampère measure.

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“…The following basic properties of E 0 and E 0 which will be used later, and we refer the readers to [6,7] for detailed proofs.…”

Section: A General Characterisationmentioning

confidence: 99%

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

Nguyen

2017

Calc. Var.

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We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Ampère equation admits Hölder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Ampère total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902-930, 2016).

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The general definition of the complex Monge-Ampère operator

Cited by 180 publications

References 9 publications

A minimum principle for plurisubharmonic functions

A minimum principle for plurisubharmonic functions

Maximal subextensions of plurisubharmonic functions

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

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