Integration by parts for plurisubharmonic functions

Giang Le; Hoang-Son Do

arxiv: 2606.28957 · v1 · pith:RHIQPXUOnew · submitted 2026-06-27 · 🧮 math.CV

Integration by parts for plurisubharmonic functions

Hoang-Son Do , Giang Le This is my paper

Pith reviewed 2026-06-30 08:28 UTC · model grok-4.3

classification 🧮 math.CV

keywords plurisubharmonic functionsintegration by partshyperconvex domainsseveral complex variablespotential theoryMonge-Ampère measures

0 comments

The pith

Plurisubharmonic functions on hyperconvex domains bounded outside a compact set admit an integration by parts formula.

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

The paper establishes an integration by parts formula that applies to plurisubharmonic functions on hyperconvex domains when those functions remain bounded outside some compact subset. This extends an earlier result of Urban Cegrell by relaxing the global boundedness requirement. A sympathetic reader cares because the formula supplies a basic manipulation tool for integrals involving the complex Hessian and Monge-Ampère measures of these functions. If the identity holds, calculations that previously required global boundedness become available on a larger class of domains and functions.

Core claim

The authors provide an integration by parts formula for plurisubharmonic functions on a hyperconvex domain that are bounded outside a compact set. This extends a previous result of Urban Cegrell.

What carries the argument

The integration by parts formula adapted to plurisubharmonic functions on hyperconvex domains.

If this is right

The formula applies to functions that may diverge inside the domain yet stabilize at infinity.
It removes the global boundedness hypothesis from Cegrell's earlier identity.
The identity can be invoked directly in potential-theoretic arguments on hyperconvex domains.
It widens the range of functions for which Monge-Ampère integrals can be integrated by parts.

Where Pith is reading between the lines

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

The formula may simplify existence proofs for the complex Monge-Ampère equation under weaker growth conditions.
It could be tested on standard examples such as log of a holomorphic function with isolated zeros.
The result might combine with Bedford-Taylor theory to produce new comparison principles.

Load-bearing premise

The domain must be hyperconvex and the plurisubharmonic functions must be bounded outside a compact set.

What would settle it

A single explicit hyperconvex domain together with a plurisubharmonic function bounded outside a compact set for which the stated integration-by-parts identity fails would disprove the claim.

read the original abstract

In this paper, we provide an integration by parts formula for plurisubharmonic functions on a hyperconvex domain that are bounded outside a compact set. This extends a previous result of Urban Cegrell.

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

This extends Cegrell's integration-by-parts formula to plurisubharmonic functions bounded outside a compact set on hyperconvex domains, with no visible internal problems.

read the letter

The paper provides an integration by parts formula for plurisubharmonic functions on a hyperconvex domain that are bounded outside a compact set. This extends a result by Urban Cegrell.

The work stays within the established framework of pluripotential theory. It does not introduce new methods or address more general situations. The abstract frames the contribution explicitly as an extension.

The setting matches what is common in the literature for these kinds of formulas. The boundedness condition outside a compact set and the hyperconvex domain are standard hypotheses. This makes the claim look reasonable based on the description.

A potential soft spot is the lack of visible proof details in the abstract. Without the full derivation, it's hard to verify the steps or see if any estimates are needed. However, there is no indication of an internal gap or incorrect assumption from the stress-test.

The paper engages with the literature by citing the prior result directly. There is no sign of circularity or reliance on unverified background.

This is targeted at researchers in several complex variables who work with plurisubharmonic functions. A reader already familiar with Cegrell's work might find the extension useful for their calculations.

It deserves peer review. The result is modest but specific, and a referee in the area can check the proof without much trouble.

I would not bring this to a general reading group, but maybe to one focused on the subfield. I would not cite it in my own work unless it directly applies.

Referee Report

0 major / 0 minor

Summary. The manuscript presents an integration by parts formula for plurisubharmonic functions on hyperconvex domains that are bounded outside a compact set. This is stated as an extension of a prior result due to Urban Cegrell.

Significance. If valid, the formula would supply a standard tool in pluripotential theory for handling plurisubharmonic functions with controlled behavior outside compact sets on hyperconvex domains. The setting matches the hypotheses under which such formulas are typically expected to hold.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report confirms that the integration-by-parts formula extends Cegrell's result under the stated hypotheses on hyperconvex domains.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states it provides an integration-by-parts formula for plurisubharmonic functions on hyperconvex domains bounded outside a compact set, extending Cegrell's prior result. Cegrell is an external author with no overlap. The abstract and context contain no equations, self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The claim rests on standard background in pluripotential theory rather than reducing to its own inputs by construction. This is the normal non-circular outcome for a theorem-extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. Standard background assumptions of pluripotential theory (e.g., properties of hyperconvex domains) are implicitly used but not detailed.

pith-pipeline@v0.9.1-grok · 5538 in / 921 out tokens · 28759 ms · 2026-06-30T08:28:35.169657+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

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˚Ahag, U

P. ˚Ahag, U. Cegrell, R. Czy˙ z, H.-H. Pham:Monge-Amp` ere measures on pluripolar sets.J. Math. Pures Appl. (9)92(2009), no. 6, 613–627

2009

[2] [2]

B locki:The domain of definition of the complex Monge-Amp` ere operator.Amer

Z. B locki:The domain of definition of the complex Monge-Amp` ere operator.Amer. J. Math. 128(2006), no.2, 519–530

2006

[3] [3]

Bedford, B

E. Bedford, B. A. Taylor:The Dirichlet problem for a complex Monge-Amp` ere equation. Invent. Math.37(1976), no. 1, 1–44

1976

[4] [4]

Bedford, B

E. Bedford, B. A. Taylor:A new capacity for plurisubharmonic functions.Acta Math.149 (1982), no. 1-2, 1–40

1982

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Bedford, B

E. Bedford, B. A. Taylor,Fine topology, ˇSilov boundary, and(dd c)n, J. Funct. Anal.72(1987), no. 2, 225–251

1987

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U. Cegrell:Pluricomplex energy.Acta Math.180(1998), no. 2, 187–217

1998

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Cegrell,The general definition of the complex Monge-Amp` ere operator,Annales de l’institut Fourier, Volume 54, Issue 1 (2004) 159–179

U. Cegrell,The general definition of the complex Monge-Amp` ere operator,Annales de l’institut Fourier, Volume 54, Issue 1 (2004) 159–179

2004

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Coman,Integration by parts for currents and applications to the relative capacity and Lelong numbers.Mathematica, tome 39(62) (1997), no

D. Coman,Integration by parts for currents and applications to the relative capacity and Lelong numbers.Mathematica, tome 39(62) (1997), no. 1, 45–57

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Klimek:Pluripotential theory,Oxford Univ

M. Klimek:Pluripotential theory,Oxford Univ. Press, Oxford, 1991

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Pham,Convergence in capacity and applications.Math

H.-H. Pham,Convergence in capacity and applications.Math. Scand.107(2010), no. 1, 90– 102. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet road, Nghia Do ward, 11307 Hanoi, Vietnam Email address:dhson@math.ac.vn, hoangson.do.vn@gmail.com Department of Mathematics, Hanoi National University of Education, 136 Xuan Thu...

2010