Convergence in capacity of the pluricomplex Green function.
An inequality for the beta function with application to pluripotential theory.
The complex Monge-Ampère operator in the Cegrell classes
The complex Monge-Ampère operator is a useful tool not only within pluripotential theory, but also in algebraic geometry, dynamical systems and Kähler geometry. In this self-contained survey we present a unified theory of Cegrell's framework for the complex Monge-Ampère operator.
On a Monge-Ampère type equation in the Cegrell class
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 . Under some additional assumption the solution u is uniquely determined.
Pluriharmonic extension in proper image domains
Let be a bounded hyperconvex domain in and set , j=1,...,s, s ≥ 3. Also let be the image of D under the proper holomorphic map π. We characterize those continuous functions that can be extended to a real-valued pluriharmonic function in .
The complex Monge-Ampère equation for complex homogeneous functions in ℂⁿ
We prove some existence results for the complex Monge-Ampère equation 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.
Plurisubharmonic extension in non-degenerate analytic polyhedra.
Continuous pluriharmonic boundary values
Let be a bounded hyperconvex domain in and set , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.
Subextension of plurisubharmonic functions without increasing the total Monge-Ampère mass
We prove that subextension of certain plurisubharmonic functions is always possible without increasing the total Monge-Ampère mass.
On the Dirichlet problem in the Cegrell classes
Let μ be a non-negative measure with finite mass given by , where ψ is a bounded plurisubharmonic function with zero boundary values and , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.
Plurisubharmonic functions on compact sets
Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.
Radially symmetric plurisubharmonic functions
In this note we consider radially symmetric plurisubharmonic functions and the complex Monge-Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge-Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge-Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally...
Concerning the energy class for 0 < p < 1
The energy class is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of and its pluricomplex p-energy is proved.
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