Comparison principles for Monge-Ampère measures on pluripolar sets

Abstract: In this paper, we introduce a notion of singularity comparison for plurisubharmonic functions based on the Bedford--Taylor capacity. We establish comparison principles for the complex Monge--Ampère operator on pluripolar sets in the Cegrell classes. As applications, we obtain a characterization of this relation via auxiliary functions in the energy class and prove a corresponding uniqueness result for the Monge--Ampère equation.

• The paper introduces a capacity-based preorder to compare singularities of plurisubharmonic functions in Cegrell classes.
• The paper establishes monotonicity and uniqueness principles for Monge-Ampère measures on pluripolar sets within bounded hyperconvex domains.
• The paper links capacity singularity ordering with negligible perturbations, enhancing analytic tools for studying degenerate complex Monge-Ampère equations.

Comparison Principles for Monge-Ampère Measures on Pluripolar Sets

Introduction

The paper "Comparison principles for Monge-Ampère measures on pluripolar sets" (2604.19347) develops a refined framework for the comparison of singularities of plurisubharmonic (psh) functions and the associated complex Monge-Ampère measures, specifically targeting the non-trivial context of pluripolar sets within Cegrell classes. In pluripotential theory, understanding the behavior of Monge-Ampère measures on pluripolar sets informs both fine aspects of singularities and the structure of solutions to degenerate complex Monge-Ampère equations.

Preliminaries: Cegrell Classes, Monge-Ampère Capacity, and Singularity Notions

The study is conducted on a bounded hyperconvex domain $\Omega\subset\mathbb{C}^n$ and leverages Cegrell's hierarchy of classes for negative psh functions: $\mathcal{E}_0(\Omega)$, $\mathcal{F}(\Omega)$, and $\mathcal{E}(\Omega)$. These classes ensure the well-definedness of the Monge-Ampère operator in non-smooth contexts.

Central to the analysis is the Bedford-Taylor Monge-Ampère capacity, denoted $\operatorname{Cap}_\Omega(\cdot)$, which quantifies the "size" of sets via testing with negative psh functions. The authors introduce a new preorder for singularity comparison, called singularity comparison in capacity: for $\varphi,\psi : \Omega \to [-\infty,+\infty]$, $\varphi \preceq_{\operatorname{Cap}} \psi$ if

$$\varliminf_{t \to +\infty} t^n\, \operatorname{Cap}_\Omega\big(\{z \in D: \varphi(z) < \psi(z) - t\}\big) = 0$$

for all relatively compact $D \Subset \Omega$. The resulting preorder is strictly weaker than the local singularity order, which requires pointwise control up to additive constants on compacts.

Main Results: Comparison Principles on Pluripolar Sets

The authors establish several strong comparison principles for Monge-Ampère measures associated with psh functions in Cegrell classes, when restricted to pluripolar Borel sets. The essential results are as follows:

Comparison Principle for Pluripolar Sets

Let $\varphi_1,\ldots,\varphi_n$ and $\mathcal{E}_0(\Omega)$0 with $\mathcal{E}_0(\Omega)$1 for all $\mathcal{E}_0(\Omega)$2. For every pluripolar Borel set $\mathcal{E}_0(\Omega)$3,

$\mathcal{E}_0(\Omega)$4

This provides a precise monotonicity principle for mixed Monge-Ampère masses supported on pluripolar subsets, generalizing previous principles established only in more restrictive settings.

Strengthened Comparison and Uniqueness Principles

The ordering $\mathcal{E}_0(\Omega)$5 enables extended comparison principles under boundary or majorant hypotheses. If $\mathcal{E}_0(\Omega)$6 and either (i) $\mathcal{E}_0(\Omega)$7 for all $\mathcal{E}_0(\Omega)$8, or (ii) $\mathcal{E}_0(\Omega)$9 with $\mathcal{F}(\Omega)$0 for some $\mathcal{F}(\Omega)$1, then discrete mass and energy-type inequalities hold between their Monge-Ampère measures on sublevel sets and at their singular loci. This leads to a strong uniqueness result: if two such functions yield equal Monge-Ampère measures and are locally comparable in capacity order, then the functions must coincide.

Characterization of Capacity Singularity Ordering

A structural characterization is achieved: for $\mathcal{F}(\Omega)$2 and $\mathcal{F}(\Omega)$3, $\mathcal{F}(\Omega)$4 if and only if there exists $\mathcal{F}(\Omega)$5 (i.e., a negative psh function with zero Monge-Ampère mass on pluripolar Borel sets) such that $\mathcal{F}(\Omega)$6. This links the capacity order to augmentation by negligible singularities.

Technical Developments

The proofs rely on several deep tools:

• Decompositions of psh functions into maximal and negligible components in Cegrell classes.
• Refined Stokes-type arguments and stability properties of the Monge-Ampère operator under decreasing limits.
• Quantitative estimates relating Monge-Ampère masses on sublevel sets to capacity, with precise control over error terms by adjusting $\mathcal{F}(\Omega)$7-perturbations and truncations.
• Careful reduction of the pluripolar set case to technical statements for functions in $\mathcal{F}(\Omega)$8, drawing upon earlier work on extremal functions and subextensions.

The authors demonstrate, with explicit examples, that their preorder is strictly weaker than classical local comparison, hence admitting strictly larger equivalence classes relevant for pluripotential applications.

Implications and Prospects

The established comparison principles provide new analytic tools for the qualitative and quantitative study of singularities for psh functions and complex Monge-Ampère equations. In particular:

• They extend uniqueness and comparison theory for degenerate Monge-Ampère equations to a broader class of functions and to more singular supports.
• The characterization via negligible Monge-Ampère mass perturbations supports the robustness of the associated PDE theory under pluripolar singularities.
• The results have significant implications for the study of fine properties of plurisubharmonic singularities (e.g., type, order, and envelope regularity) and for the construction and comparison of solutions to Dirichlet-type problems beyond the smooth or non-singular context.
• The methods and capacity techniques are anticipated to transfer to related domains such as degenerate potential theory, field extensions in Kähler geometry, and variational methods in complex dynamical systems.
• The framework paves the way for further exploration of envelope constructions and the role of pluripolar sets in pluripotential geodesics and rooftop envelopes.

Conclusion

This work establishes comprehensive comparison principles for Monge-Ampère measures associated with functions in Cegrell's classes on pluripolar sets. The framework based on capacity singularity order enlarges the class of admissible singularities, leading to new uniqueness, monotonicity, and structural results which refine the understanding of the Monge-Ampère operator in singularity-dominated regimes. The techniques and results will inform further advances in pluripotential theory and the analysis of degenerate complex Monge-Ampère equations (2604.19347).