arxiv: 2604.07600 · v1 · submitted 2026-04-08 · 🧮 math.CV

Recognition: unknown

New local characterizations of the weighted energy class mathcal{E}_{chi,loc}(Ω)

Hoang Nhat Quy

Pith reviewed 2026-05-10 16:55 UTC · model grok-4.3

classification 🧮 math.CV

keywords weighted energy classplurisubharmonic functionsMonge-Ampère measureslocal characterizationshyperconvex domainscomplex analysis in several variables

0 comments

The pith

Plurisubharmonic functions belong to the local weighted energy class if they have suitable majorants near subdomain boundaries and their Monge-Ampère measures are dominated locally.

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

The paper establishes new criteria for when a negative plurisubharmonic function on a hyperconvex domain in complex space belongs to the local weighted energy class. It shows that the existence of local majorants from the class near the boundaries of compact subdomains ensures that the weighted energy integral stays finite on compact subsets. Moreover, if the function's Monge-Ampère measure is locally dominated by that of a function already in the class, then the function itself is in the class. This provides weaker conditions than before for local membership, extending prior work on the local theory of these functions.

Core claim

If u is a negative plurisubharmonic function on Omega that admits suitable local majorants in E_chi,loc near the boundary of every relatively compact hyperconvex subdomain D, then the weighted energy integral of u over compact sets in D is finite. Furthermore, if additionally the Monge-Ampère measure of u is locally dominated by that of some w in E_chi,loc(D), then u itself belongs to E_chi,loc(D). This is a new local boundedness property and a substantial improvement on the control of the measure.

What carries the argument

The local majorants in the weighted energy class near subdomain boundaries together with the local domination condition on the Monge-Ampère measures.

If this is right

The weighted Monge-Ampère energy remains locally finite for such functions u.
u belongs to the local weighted energy class under the domination condition.
The result holds even without a priori local finiteness of the energy, enlarging the class of admissible functions.
This applies to the unweighted case as a special instance.

Where Pith is reading between the lines

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

This characterization could facilitate the study of singular plurisubharmonic functions with controlled singularities on compact sets.
Similar local conditions might apply to other classes of plurisubharmonic functions or different weights chi.
These results may help in solving local versions of complex Monge-Ampère equations with prescribed measures.

Load-bearing premise

The assumption that u admits suitable local majorants in the weighted energy class near the boundaries of all relatively compact hyperconvex subdomains, plus the local domination of its Monge-Ampère measure.

What would settle it

Construct a negative plurisubharmonic function u on a hyperconvex domain that has local majorants in E_chi,loc near boundaries but for which the integral of chi(u) (dd^c u)^n diverges on some compact set inside a subdomain.

read the original abstract

Let \(\Omega\subset\mathbb{C}^n\) be a hyperconvex domain and let \(\chi:\mathbb{R}^-\to\mathbb{R}^+\) be a decreasing function. This note studies the local weighted energy class \(\mathcal{E}_{\chi,\mathrm{loc}}(\Omega)\) introduced in \cite{HHQ13}. We establish two main results on local membership in this class. First, we prove a new local boundedness property for the weighted Monge--Amp\`ere energy: if \(u\in\mathrm{PSH}^-(\Omega)\) admits suitable local majorants in \(\mathcal{E}_{\chi,\mathrm{loc}}\) near the boundary of every relatively compact hyperconvex subdomain \(D\Subset\Omega\), then the weighted energy \(\int_K \chi(u)(dd^c u)^n\) remains locally finite for every compact set \(K\subset D\). This gives the first explicit local control of the energy functional and is new even in the unweighted setting. Second, we obtain a substantial improvement concerning the local control of the Monge--Amp\`ere measure. We show that if, in addition to the boundary condition, \((dd^c u)^n\) is locally dominated by \((dd^c w)^n\) for some \(w\in\mathcal{E}_{\chi,\mathrm{loc}}(D)\) inside \(D\), then \(u\in\mathcal{E}_{\chi,\mathrm{loc}}(D)\). This domination condition is strictly weaker than the previous requirement of local finiteness of the weighted energy, thereby significantly enlarging the class of admissible functions. Our results extend and refine the local theory developed in \cite{Q24,Q25} and provide a more flexible framework for plurisubharmonic functions with possible singularities on compact subsets.

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

Two local refinements to weighted energy classes: boundedness under boundary majorants and membership via MA-measure domination instead of energy finiteness.

read the letter

The paper's main contributions are a local boundedness result for the weighted energy integral and a weaker membership test for the class E_chi,loc. If a negative plurisubharmonic u has suitable majorants from the class near the boundary of every compactly contained hyperconvex subdomain, then the integral of chi(u) times the Monge-Ampere measure stays finite on compacts inside those subdomains. Adding the condition that the Monge-Ampere measure of u is dominated by that of some w already in the class then puts u itself in the class. Both claims are presented as improvements over the local theory in HHQ13, Q24, and Q25, and the second one is explicitly weaker than requiring local finiteness of the energy itself. This is new even without the weight chi. The abstract is clear about what is being relaxed and why it enlarges the admissible functions. The stress-test note finds no obvious circularity in the majorant conditions or internal contradictions, which matches what the abstract states. The work stays inside the standard framework of hyperconvex domains and decreasing weights, so the technical steps appear to be incremental but useful refinements rather than a wholesale change. The only real soft spot is that we have only the abstract and the reader's summary; without the actual proofs it is impossible to check whether the majorants are constructed in a way that avoids hidden assumptions or whether the domination argument really goes through without extra regularity. That said, nothing in the stated claims looks load-bearing or self-referential. This note is for people already working on local pluripotential theory, singular metrics, or Monge-Ampere equations on hyperconvex domains. It is not broad enough to interest a general complex-analysis audience, but it supplies concrete local control that those specialists can use. I would send it to peer review so the proofs can be examined in detail; the claims are modest enough that a referee can decide quickly whether they hold.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes two new local results for the weighted energy class E_chi,loc(Omega) of plurisubharmonic functions on hyperconvex domains Omega in C^n, with chi a decreasing weight. First, if u in PSH^-(Omega) admits suitable local majorants from E_chi,loc near the boundary of every relatively compact hyperconvex D subset Omega, then the weighted energy integral chi(u)(dd^c u)^n is finite on every compact K subset D. Second, if in addition (dd^c u)^n is locally dominated by (dd^c w)^n for some w in E_chi,loc(D), then u belongs to E_chi,loc(D). These are presented as strictly weaker than prior conditions and as extensions of the local theory in HHQ13, Q24, and Q25, with the first result claimed to be new even in the unweighted case.

Significance. If the results hold, they supply the first explicit local control on the weighted energy functional and weaken the hypotheses needed for local membership in E_chi,loc, thereby enlarging the class of admissible singular plurisubharmonic functions. This refines the local theory and may prove useful in applications involving Monge-Ampere measures with compactly supported singularities.

minor comments (2)

The abstract refers to 'suitable local majorants' without an immediate definition; this notion should be stated explicitly in the introduction or in the section recalling the definition of E_chi,loc.
The claim that the domination condition is 'strictly weaker' than local finiteness of the weighted energy should be accompanied by a brief comparison (perhaps in the introduction) with the hypotheses of the cited works Q24 and Q25.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The referee's summary accurately describes the two main results in the manuscript. As no explicit major comments were raised in the report, we have nothing further to address point by point at this stage. We will make the appropriate minor revisions to the manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes two new local results for membership in the weighted energy class E_chi,loc(Omega): a local finiteness property for the weighted energy integral under the assumption of suitable local majorants near boundaries of compact subdomains, and full membership when the Monge-Ampere measure is additionally dominated by that of a function already in the class. These claims are presented as extensions of prior local theory, with the domination condition explicitly noted as strictly weaker than previous requirements. No step in the stated claims reduces by construction to a fitted parameter, self-definition of the class, or a load-bearing self-citation whose content is unverified; the citations to HHQ13, Q24 and Q25 supply background definitions and earlier results but do not substitute for the new proofs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the work builds on standard pluripotential theory on hyperconvex domains.

pith-pipeline@v0.9.0 · 5620 in / 1094 out tokens · 46598 ms · 2026-05-10T16:55:42.779870+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

[1]

˚Ahag,A Dirichlet problem for the complex Monge-Amp` ere operator inF(f), Michigan Math

P. ˚Ahag,A Dirichlet problem for the complex Monge-Amp` ere operator inF(f), Michigan Math. J,55(2007), 123–138

2007
[2]

˚Ahag, U

P. ˚Ahag, U. Cegrell , R. Czyz and Pham Hoang Hiep,Monge-Amp` ere measures on pluripolar sets, J. Math. Pures Appl., 92 (2009), 613–627

2009
[3]

Bedford and B

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

1982
[4]

Math,37(1976), 1–44

E.Bedford and B.A.Taylor,The Dirichlet problem for a complex Monge-Amp` ere equation, Invent. Math,37(1976), 1–44

1976
[5]

Benelkourchi,Weighted Pluricomplex Energy, Potential Analysis,31(2009), 1– 20

S. Benelkourchi,Weighted Pluricomplex Energy, Potential Analysis,31(2009), 1– 20

2009
[6]

Benelkourchi, V.Guedj and A.Zeriahi,Plurisubharmonic functions with weak sin- gularities, In: Proceedings from the Kiselmanfest, Uppsala University, V¨ astra Aros (2009), 57–74

S. Benelkourchi, V.Guedj and A.Zeriahi,Plurisubharmonic functions with weak sin- gularities, In: Proceedings from the Kiselmanfest, Uppsala University, V¨ astra Aros (2009), 57–74

2009
[7]

Blocki,On the definition of the Monge-Amp` ere operator inC 2, Math

Z. Blocki,On the definition of the Monge-Amp` ere operator inC 2, Math. Ann,328 (2004), 415–423

2004
[8]

Blocki,The domain of definition of the complex Monge-Amp` ere operator, Amer

Z. Blocki,The domain of definition of the complex Monge-Amp` ere operator, Amer. J. Math,128(2006), 519–530

2006
[9]

Cegrell,Pluricomplex energy,Acta

U. Cegrell,Pluricomplex energy,Acta. Math,180(1998), 187–217

1998
[10]

Cegrell,The general definition of the complex Monge-Amp` ere operator,Ann

U. Cegrell,The general definition of the complex Monge-Amp` ere operator,Ann. Inst. Fourier (Grenoble),54(2004), 159–179

2004
[11]

Cegrell,A general Dirichlet problem for the complex Monge-Amp` ere operator, Ann

U. Cegrell,A general Dirichlet problem for the complex Monge-Amp` ere operator, Ann. Polon. Math,94(2008), 131–147

2008
[12]

Cegrell, A

U. Cegrell, A. ZeriahiSubextension of plurisubharmonic functions with bounded Monge - Amp´ ere mass, C. R. Acad. Sci. Paris, Ser. I 336 (2003), 305–308. 11

2003
[13]

Demailly,Monge-Amp` ere operators, Lelong numbers and intersection theory, Complex analysis and geometry, Univ

J-P. Demailly,Monge-Amp` ere operators, Lelong numbers and intersection theory, Complex analysis and geometry, Univ. Ser. Math, Plenium, New York (1993), 115– 193

1993
[14]

Guedj and A

V. Guedj and A. Zeriahi,The weighted Monge-Amp` ere energy of quasiplurisubhar- monic functions, J. Funct. Anal,250(2007), 442–482

2007
[15]

Le Mau Hai and Pham Hoang Hiep,Some Weighted Energy Classes of Plurisubhar- monic Functions, Potential Analysis,34(2011), 43–56

2011
[16]

Le Mau Hai, Pham Hoang Hiep, Hoang Nhat Quy,Local property of the classE χ,loc, J. Math. Anal. Appli.,402(2013), 440–445

2013
[17]

and Elliptic Equations,53(2008), 675–684

Pham Hoang Hiep,Pluripolar sets and the subextension in Cegrell’s classes, Com- plex Var. and Elliptic Equations,53(2008), 675–684

2008
[18]

Kiselman,Sur la d´ efinition de l’op´ erateur de Monge-Amp` ere complexe, Com- plex Analysis (Toulouse, 1983), 139 - 150, Lectures Notes in Math

C.O. Kiselman,Sur la d´ efinition de l’op´ erateur de Monge-Amp` ere complexe, Com- plex Analysis (Toulouse, 1983), 139 - 150, Lectures Notes in Math. 1094, Springer, Berlin, 1984

1983
[19]

Ko lodziej,The range of the complex Monge-Amp` ere operator, II, Indiana Univ

S. Ko lodziej,The range of the complex Monge-Amp` ere operator, II, Indiana Univ. Math. J,44(1995), 765–782

1995
[20]

Ko lodziej,The Monge-Amp` ere equation, Acta Math.,180(1998), 69–117

S. Ko lodziej,The Monge-Amp` ere equation, Acta Math.,180(1998), 69–117

1998
[21]

Hoang Nhat Quy,The topology on the spaceδE χ, Univ. Iagel. Acta. Math.51(2014), 61 –73

2014
[22]

of Pure and Appl.,54(4)(2024), 419 –424

Hoang Nhat Quy,The local properties of some subclasses of plurisubharmonic func- tions, Indian J. of Pure and Appl.,54(4)(2024), 419 –424

2024
[23]

Hoang Nhat Quy,The complex Monge-Amp` ere equation in the Cegrell’s classes, Results in Math.,80(14)(2025), https://doi.org/10.1007/s00025-024-02335-9

work page doi:10.1007/s00025-024-02335-9 2025
[24]

J. B. Walsh, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18(1968), 143–148

1968