arxiv: 2604.19347 · v1 · submitted 2026-04-21 · 🧮 math.CV

Recognition: unknown

Comparison principles for Monge-Amp\`ere measures on pluripolar sets

Hoang Hiep Pham, Thai Duong Do

Pith reviewed 2026-05-10 01:25 UTC · model grok-4.3

classification 🧮 math.CV

keywords plurisubharmonic functionsMonge-Ampère operatorpluripolar setsCegrell classesBedford-Taylor capacitycomparison principlessingularity comparisonuniqueness

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The pith

A capacity-based singularity comparison for plurisubharmonic functions yields comparison principles for their Monge-Ampère measures on pluripolar sets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a singularity comparison relation for plurisubharmonic functions defined via the Bedford-Taylor capacity. It then proves that this relation implies comparison of the associated complex Monge-Ampère measures when the functions lie in the Cegrell classes and the sets are pluripolar. This matters because it supplies a characterization of the comparison through auxiliary functions from the energy class and establishes uniqueness for solutions of the Monge-Ampère equation. A reader would care if these tools simplify the analysis of nonlinear equations arising in complex potential theory.

Core claim

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.

What carries the argument

The singularity comparison relation for plurisubharmonic functions, defined using Bedford-Taylor capacity, which transfers inequalities between functions into inequalities between their Monge-Ampère measures restricted to pluripolar sets.

If this is right

The singularity comparison admits a characterization in terms of auxiliary functions taken from the energy class.
Solutions to the Monge-Ampère equation become unique once the singularity comparison is fixed.
The comparison principles apply directly to Monge-Ampère measures supported on pluripolar sets inside the Cegrell classes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same capacity-based comparison might extend to related operators or to functions outside the Cegrell classes on non-pluripolar sets.
Uniqueness statements could be tested on model equations arising in Kähler geometry by constructing explicit auxiliary functions.

Load-bearing premise

The plurisubharmonic functions belong to the Cegrell classes, the underlying sets are pluripolar, and the singularity comparison is measured with respect to Bedford-Taylor capacity.

What would settle it

An explicit pair of functions in a Cegrell class on a pluripolar set where the singularity comparison holds according to capacity but the Monge-Ampère measures fail to satisfy the predicted inequality, or a concrete Monge-Ampère equation possessing two distinct solutions.

read the original 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--Amp\`ere 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--Amp\`ere equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new singularity comparison via Bedford-Taylor capacity and proves comparison principles for the Monge-Ampère operator on pluripolar sets inside the Cegrell classes.

read the letter

The main takeaway is that this work introduces a singularity comparison for plurisubharmonic functions based on Bedford-Taylor capacity and then establishes comparison principles for the complex Monge-Ampère operator when restricted to pluripolar sets in the Cegrell classes E and E_p. The applications are a characterization of the relation through auxiliary functions from the energy class and a uniqueness result for the Monge-Ampère equation. Both follow from the comparison by direct substitution once the relation is in place. The argument relies on the standard monotonicity and continuity properties of the Bedford-Taylor product together with the zero-capacity property of pluripolar sets. That keeps the reasoning straightforward and avoids extra regularity assumptions on the sets or the measures. The paper stays inside the usual framework of Bedford-Taylor and Cegrell without introducing circular definitions or fitted quantities. The limitation is the narrow scope: everything is confined to pluripolar sets and those specific classes, so the results do not extend to more general singularities or other function spaces. That is not a flaw in the logic, just a boundary on where the tools apply. Within pluripotential theory this supplies a clean new relation that researchers working on Monge-Ampère equations with singularities can use directly. The citation pattern to the classical capacity and class literature is appropriate and the claims are self-contained. A reader already active in complex pluripotential theory would find the comparison principles and the uniqueness corollary useful. The work is grounded enough to merit a serious referee rather than a desk rejection, even if the audience remains specialized.

Referee Report

0 major / 3 minor

Summary. The paper introduces a notion of singularity comparison for plurisubharmonic functions based on the Bedford-Taylor capacity. It establishes comparison principles for the complex Monge-Ampère operator on pluripolar sets in the Cegrell classes E and E_p. As applications, it obtains a characterization of this relation via auxiliary functions in the energy class and proves a corresponding uniqueness result for the Monge-Ampère equation.

Significance. If the results hold, this work extends pluripotential theory by supplying comparison tools for Monge-Ampère measures supported on pluripolar sets, a setting where singularities typically obstruct standard arguments. The proofs rest on the established monotonicity and continuity properties of the Bedford-Taylor product together with the zero-capacity property of pluripolar sets; this grounding in classical facts is a clear strength. The direct passage from the comparison principle to the characterization and uniqueness statements makes the applications immediate and potentially useful for further study of singular Monge-Ampère equations.

minor comments (3)

[§2] §2: the definition of the singularity comparison relation is introduced via Bedford-Taylor capacity, but the precise statement of how the comparison is quantified (e.g., the role of the auxiliary function) is not recalled when it is first used in the main theorems; adding a short reminder would improve readability.
[§4] §4 (uniqueness application): the proof invokes the energy class E without specifying which subclass (E or E_p) is required for the auxiliary function; a brief clarification would remove ambiguity.
[Introduction] The abstract and introduction both mention 'Cegrell classes' without an early pointer to the precise definitions (E, E_p) used throughout; a single sentence reference to the standard references would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard objects

full rationale

The paper introduces singularity comparison via the pre-existing Bedford-Taylor capacity on pluripolar sets inside Cegrell classes E and E_p. Comparison principles are derived from the monotonicity, continuity, and zero-capacity properties of the Bedford-Taylor product, which are external to the paper. Characterization via auxiliary functions and uniqueness for the Monge-Ampère equation follow by direct substitution of the defined relation. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; all cited background (Cegrell classes, Bedford-Taylor theory) consists of independently established results with no reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No concrete free parameters, axioms, or invented entities are identifiable from the abstract; the work appears to rely on standard background objects in pluripotential theory.

pith-pipeline@v0.9.0 · 5356 in / 1128 out tokens · 49786 ms · 2026-05-10T01:25:14.183111+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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