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High Energy plurisubharmonic classes

Vincent Guedj, Ahmed Zeriahi

TL;DR

The paper develops a comprehensive framework of high-energy plurisubharmonic classes ${\mathcal E}_{\chi}(\Omega)$ using Orlicz-space technology to handle fast-growing weights. It proves foundational properties including energy continuity, the subextension principle, and capacity-based characterizations, and establishes Moser-Trudinger-type integrability within these classes. A detailed analysis of the Monge-Ampère operator on ${\mathcal E}_{\chi}(\Omega)$ yields compactness and both qualitative and quantitative integrability results, culminating in a conjectural integral characterization of the operator’s range and a formulation of Bedford’s problem in this setting. The work links energy methods, capacity theory, and nonlinear potential theory to propose a path toward an integral description of the image of bounded plurisubharmonic functions under the Monge-Ampère operator, with broad implications for pluripotential theory and complex geometry.

Abstract

Let $Ω\Subset \C^n$ be a bounded strongly pseudoconvex domain. For any concave increasing weight $χ: \R^- \longrightarrow \R^-$ such that $χ(0) = 0$, we introduce and study finite energy classes $\mathcal E_χ(Ω)$ of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge-Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of Pluripotential Theory more than forty years ago.

High Energy plurisubharmonic classes

TL;DR

The paper develops a comprehensive framework of high-energy plurisubharmonic classes using Orlicz-space technology to handle fast-growing weights. It proves foundational properties including energy continuity, the subextension principle, and capacity-based characterizations, and establishes Moser-Trudinger-type integrability within these classes. A detailed analysis of the Monge-Ampère operator on yields compactness and both qualitative and quantitative integrability results, culminating in a conjectural integral characterization of the operator’s range and a formulation of Bedford’s problem in this setting. The work links energy methods, capacity theory, and nonlinear potential theory to propose a path toward an integral description of the image of bounded plurisubharmonic functions under the Monge-Ampère operator, with broad implications for pluripotential theory and complex geometry.

Abstract

Let be a bounded strongly pseudoconvex domain. For any concave increasing weight such that , we introduce and study finite energy classes of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge-Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of Pluripotential Theory more than forty years ago.
Paper Structure (25 sections, 19 theorems, 98 equations)

This paper contains 25 sections, 19 theorems, 98 equations.

Table of Contents

  1. Preliminaries
  2. Orlicz Spaces
  3. The space of weights ${\mathcal{W}}^+$
  4. Extra growth
  5. Range of the Monge-Ampère operator
  6. Bedford-Taylor theory
  7. Dirichlet problem
  8. Maximum principles
  9. Monge-Ampère capacity
  10. High energy classes
  11. Basic properties
  12. Continuity of the weighted energy
  13. The subextension property
  14. Capacity characterization
  15. Choquet integrals
  16. ...and 10 more sections

Key Result

Lemma 1.4

Let $f:\Omega \rightarrow \mathbb{R}^+$ be a Lebesgue measurable function. Let $h$ be a Young function such that $\int_{\Omega} h \circ f d\mu<+\infty$. For all $\kappa>1$, there exists a Young function $\hat{h}$ such that $\hat{h}(t)/h(t) \rightarrow +\infty$ and

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 41 more