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A characterization of bounded plurisubharmonic functions

Pham Hoang Hiep (2005)

Annales Polonici Mathematici

We give a characterization for boundedness of plurisubharmonic functions in the Cegrell class ℱ.

A counterexample to the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2005)

Annales Polonici Mathematici

We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.

A decomposition of complex Monge-Ampère measures

Yang Xing (2007)

Annales Polonici Mathematici

We prove a decomposition theorem for complex Monge-Ampère measures of plurisubharmonic functions in connection with their pluripolar sets.

A general Dirichlet problem for the complex Monge-Ampère operator

Urban Cegrell (2008)

Annales Polonici Mathematici

We study a general Dirichlet problem for the complex Monge-Ampère operator, with maximal plurisubharmonic functions as boundary data.

A Hilbert Lemniscate Theorem in $ℂ^{2}$

Thomas Bloom, Norman Levenberg, Yu. Lyubarskii (2008)

Annales de l’institut Fourier

For a regular, compact, polynomially convex circled set $K$ in $C^{2}$ , we construct a sequence of pairs ${P_{n}, Q_{n}}$ of homogeneous polynomials in two variables with $\deg P_{n} =$ $\deg Q_{n} ...$

A note on the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2010) Annales Polonici Mathematici

We prove the almost

^{1, 1}

regularity of the degenerate complex Monge-Ampère equation in a special case.

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

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 regularity theorem for the complex Monge-Ampère equation in ℂPⁿ

Sławomir Kołodziej (2003)

Annales Polonici Mathematici

$C^{1, 1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in $C^{1, 1}$ is proved.

A result on the comparison principle for the log canonical threshold of plurisubharmonic functions

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2014)

Annales Polonici Mathematici

We prove a comparison principle for the log canonical threshold of plurisubharmonic functions under an assumption on complex Monge-Ampère measures.

A viscosity approach to degenerate complex Monge-Ampère equations

Ahmed Zeriahi (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows...

About the Calabi problem: a finite-dimensional approach

H.-D. Cao, J. Keller (2013)

Journal of the European Mathematical Society

Let us consider a projective manifold $M^{n}$ and a smooth volume form $Ω$ on $M$ . We define the gradient flow associated to the problem of $Ω$ -balanced metrics in the quantum formalism, the $Ω$ -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the $Ω$ -balancing flow converges towards a natural flow in Kähler geometry, the $Ω$ -Kähler flow. We also prove the long time existence of the $Ω$ -Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...

An eigenvalue problem for the complex Monge-Ampère operator in pseudoconvex domains

N. D. Kutev, I. P. Ramadanov (1990)

Annales de l'I.H.P. Analyse non linéaire

An energy estimate for the complex Monge-Ampère operator

Urban Cegrell, Leif Persson (1997)

Annales Polonici Mathematici

We prove an energy estimate for the complex Monge-Ampère operator, and a comparison theorem for the corresponding capacity and energy. The results are pluricomplex counterparts to results in classical potential theory.

Approximation in $ℂ^{N}$ .

Levenberg, Norm (2006)

Surveys in Approximation Theory (SAT)[electronic only]

Approximation of Plurisubharmonic Functions and the Dirichlet Problem for the Complex Monge-Ampère Operator.

U. Cegrell, Azim Sadullaev (1992)

Mathematica Scandinavica

Automorphisms of certain Stein mainfolds.

Róbert Szöke (1995)

Mathematische Zeitschrift

Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves

M. Baran, W. Pleśniak (1997)

Studia Mathematica

We show that in the class of compact, piecewise $C^{1}$ curves K in $ℝ^{n}$ , the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.

Boundary values of functions in Cegrell’s class $_{ψ}$

Pham Hoang Hiep (2007)

Annales Polonici Mathematici

We study boundary values of functions in Cegrell’s class $_{ψ}$ .

Calculus inequalities derived from holomorphic Morse inequalities.

Yum-Tong Siu (1990)

Mathematische Annalen

Capacity associated to a positive closed current. (Capacité associée à un courant positif fermé.)

Khalifa, Dabbek, Fredj, Elkhadhra (2006)

Documenta Mathematica

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