Displaying similar documents to “Maximal subextensions of plurisubharmonic functions”

A class of maximal plurisubharmonic functions

Azimbay Sadullaev (2012)

Annales Polonici Mathematici

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We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator ( d d c e u ) .

Subextension of plurisubharmonic functions without changing the Monge-Ampère measures and applications

Le Mau Hai, Nguyen Xuan Hong (2014)

Annales Polonici Mathematici

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The aim of the paper is to investigate subextensions with boundary values of certain plurisubharmonic functions without changing the Monge-Ampère measures. From the results obtained, we deduce that if a given sequence is convergent in C n - 1 -capacity then the sequence of the Monge-Ampère measures of subextensions is weakly*-convergent. As an application, we investigate the Dirichlet problem for a nonnegative measure μ in the class ℱ(Ω,g) without the assumption that μ vanishes on all pluripolar...

On a Monge-Ampère type equation in the Cegrell class χ

Rafał Czyż (2010)

Annales Polonici Mathematici

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Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation - χ ( u ) ( d d c u ) = d μ . Under some additional assumption the solution u is uniquely determined.

On the Dirichlet problem in the Cegrell classes

Rafał Czyż, Per Åhag (2004)

Annales Polonici Mathematici

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Let μ be a non-negative measure with finite mass given by φ ( d d c ψ ) , where ψ is a bounded plurisubharmonic function with zero boundary values and φ L q ( ( d d c ψ ) ) , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

Fundamental solutions of the complex Monge-Ampère equation

Halil Ibrahim Celik, Evgeny A. Poletsky (1997)

Annales Polonici Mathematici

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We prove that any positive function on ℂℙ¹ which is constant outside a countable G δ -set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.

Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

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We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is 1 , 1 and the right hand side has a density in L p ( Ω ) for some p > 1, and prove the Hölder continuity of the solution.

The Monge problem for strictly convex norms in d

Thierry Champion, Luigi De Pascale (2010)

Journal of the European Mathematical Society

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We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of d under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.

Potentials with respect to the pluricomplex Green function

Urban Cegrell (2012)

Annales Polonici Mathematici

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For μ a positive measure, we estimate the pluricomplex potential of μ, P μ ( x ) = Ω g ( x , y ) d μ ( y ) , where g(x,y) is the pluricomplex Green function (relative to Ω) with pole at y.

Pseudo-Riemannian and Hessian geometry related to Monge-Ampère structures

S. Hronek, R. Suchánek (2022)

Archivum Mathematicum

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We study properties of pseudo-Riemannian metrics corresponding to Monge-Ampère structures on four dimensional T * M . We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Plücker embedding equation. We also focus on pullbacks of the pseudo-metrics on two dimensional M , and describe the corresponding Hessian structures.

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

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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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 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 χ ( X , ω ) . This is done by...

The complex Monge-Ampère equation for complex homogeneous functions in ℂⁿ

Rafał Czyż (2001)

Annales Polonici Mathematici

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We prove some existence results for the complex Monge-Ampère equation ( d d c u ) = g d λ in ℂⁿ in a certain class of homogeneous functions in ℂⁿ, i.e. we show that for some nonnegative complex homogeneous functions g there exists a plurisubharmonic complex homogeneous solution u of the complex Monge-Ampère equation.

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

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Let ( X , ω ) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L p right hand side, p > 1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range ( X , ω ) of the complex Monge-Ampère operator acting on ω -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with L p -density belong to ( X , ω ) and proving that...

A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the Monge-Ampère-type equation det ( A + λ g ) = const . , where A is the Schouten tensor of a conformally related metric and λ > 0 is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique. ...

Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. Berman, Bo Berndtsson (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P . Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties ( X , Δ ) saying that ( X , Δ ) admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new...

Monge-Ampère measures and Poletsky-Stessin Hardy spaces on bounded hyperconvex domains

Sibel Şahin (2015)

Banach Center Publications

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Poletsky-Stessin Hardy (PS-Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain Ω, the PS-Hardy space H u p ( Ω ) is generated by a continuous, negative, plurisubharmonic exhaustion function u of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces H u p ( Ω ) where the Monge-Ampère measure ( d d c u ) has compact support for the associated...

Convergence in capacity

Pham Hoang Hiep (2008)

Annales Polonici Mathematici

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We prove that if ( Ω ) u j u ( Ω ) in Cₙ-capacity then l i m i n f j ( d d c u j ) n 1 u > - ( d d c u ) n . This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.

On the maximal function for rotation invariant measures in n

Ana Vargas (1994)

Studia Mathematica

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Given a positive measure μ in n , there is a natural variant of the noncentered Hardy-Littlewood maximal operator M μ f ( x ) = s u p x B 1 / μ ( B ) ʃ B | f | d μ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in n . We give some necessary and sufficient conditions for M μ to be bounded from L 1 ( d μ ) to L 1 , ( d μ ) .

On subextension and approximation of plurisubharmonic functions with given boundary values

Hichame Amal (2014)

Annales Polonici Mathematici

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Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in χ ( Ω , H ) , the class of functions with finite χ-energy and given boundary values. We show that, under certain conditions, one can approximate any function in χ ( Ω , H ) by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to L 1 , bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an L p regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an L p ...

Maximal functions and capacities

Lennart Carleson (1965)

Annales de l'institut Fourier

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Pour les fonctions f ( x ) dont les coefficients de Fourier c n satisfont à Σ | c n | 2 λ n < , la capacité est évaluée pour l’ensemble où la fonction maximale satisfait à f * ( x ) λ .