Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

Åhag, Per; Cegrell, Urban; Kołodziej, Sławomir; Phạm, Hoàng Hiệp; Zériahi, Ahmed

doi:10.1016/j.aim.2009.07.002

Cited by 36 publications

(46 citation statements)

References 23 publications

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“…It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,17]). For further information on pluripotential theory we refer to [16,19,20].…”

Section: Introductionmentioning

confidence: 99%

On Dirichlet's principle and problem

Åhag¹,

Cegrell²,

Czyż³

2012

MATH. SCAND.

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Abstract. The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampère operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods. IntroductionThroughout this note let Ω ⊆ C n , n ≥ 1, be a bounded, connected, open, and hyperconvex set. By E 0 we denote the family of all bounded plurisubharmonic functions ϕ defined on Ω such that lim z→ξ ϕ(z) = 0 for every ξ ∈ ∂Ω , andwhere (dd c · ) n is the complex Monge-Ampère operator. Next let E p , p > 0, denote the family of plurisubharmonic functions u defined on Ω such that there exists a decreasing sequence {u j }, u j ∈ E 0 , that converges pointwise to u on Ω, as j tends to +∞, and 10,14]). It should be noted that it follows from [10] that the complex Monge-Ampère operator is well-defined on E p . It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,17]). For further information on pluripotential theory we refer to [16,19,20]. The purpose of this paper is to give a new proof of Theorem B below and use Theorem B to prove (2) implies (1) the following theorem:Theorem A (Dirichlet's problem). Let µ be a non-negative Radon measure, then the following conditions are equivalent:(1) there exists a function u ∈ E 1 such that (dd c u) n = µ, (2) there exists a constant B > 0, such that Ω (−ϕ) dµ ≤ B e 1 (ϕ) 1 n+1 for all ϕ ∈ E 1 , (1.1) (3) the class E 1 is contained in L 1 (µ),2000 Mathematics Subject Classification. Primary 35J20; Secondary 32W20.

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Section: Introductionmentioning

confidence: 99%

On Dirichlet's principle and problem

Åhag¹,

Cegrell²,

Czyż³

2012

MATH. SCAND.

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“…Some authors say that K is a cone if it satisfies (1), and a convex cone if it satisfies (1) and (2). Furthermore, there are literature that say that K is a pointed convex cone with vertex at zero if it satisfies (1)-(3).…”

Section: Ordered Vector Spacesmentioning

confidence: 97%

“…It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,18]). …”

Section: Introductionmentioning

confidence: 99%

Modulability and duality of certain cones in pluripotential theory

Åhag

Czyz

2010

Journal of Mathematical Analysis and Applications

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ModulabilityOrdered vector space Quasi-Banach space Topological dual Let p > 0, and let E p denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δE p = E p − E p , with the vector ordering induced by the cone E p is σ -Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone E p in the quasi-normed space δE p is closed, generating, and has empty interior.

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“…To finish the proof of the proposition we use an argument which is inspired by the proofs in [1]. Let K ⊂ be compact.…”

Section: Proposition 29mentioning

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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Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

Abstract: International audienc

Cited by 36 publications

References 23 publications

On Dirichlet's principle and problem

On Dirichlet's principle and problem

Modulability and duality of certain cones in pluripotential theory

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

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