Maximal subextension of $m$-subharmonic functions

Ayoub El-Gasmi; Hichame Amal; Sa\"id Asserda

REVIEW 3 major objections 5 minor 25 references

Maximal subextension of $m$-subharmonic functions

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A quasi-m-hyperconvex domain inside a compact Kähler manifold admits, for every function in the weighted Hessian energy class with bounded mass, a maximal m-subharmonic subextension to the whole manifold that preserves the weighted energy a

desk verdict A plausible m-subharmonic analogue of CKZ2 that is worth refereeing, but the energy-preservation step in Theorem 3.4 rests on an unstated convergence theorem and needs a rewritten proof. read the letter →

arxiv 2607.19132 v1 pith:APKKPI53 submitted 2026-07-21 math.CV

Hichame Amal , Saïd Asserda , Ayoub El-Gasmi This is my paper · ORCID

classification math.CV MSC 32U1532Q1532W20

keywords m-subharmonicfunctionsmaximalsubextensionweightedenergyclasscomplexHessianoperatorquasi-m-hyperconvexdomaincompactKählermanifoldFubini-Studyformm-Lelong

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the subextension problem for m-subharmonic functions—a family interpolating between subharmonic and plurisubharmonic functions—has a positive answer on compact Kähler manifolds. Given a domain Ω with a mild convexity property and a function φ in the weighted energy class whose total Hessian mass does not exceed the total volume of the manifold, there is a maximal m-subharmonic function φ̃ on the whole manifold that lies below φ on Ω, belongs to the same weighted energy class, and does not increase the weighted energy. The construction also controls the complex Hessian measure: inside Ω its trace is no larger than that of φ, and the Hessian measure of φ̃ is carried by the set where φ̃ equals φ together with the boundary of Ω. If correct, this unifies and extends known subextension results from the plurisubharmonic setting to m-subharmonic functions and provides a tool for prescribing complex Hessian measures on projective space.

What carries the argument

The central object is the complex Hessian operator H_m(u) = (ω+dd^c u)^m ∧ ω^{n−m} acting on ω-m-subharmonic functions, together with the weighted energy class E_χ^m(Ω,ω) defined by finiteness of ∫ −χ(u)H_m(u). The proof is carried by a three-step mechanism: (1) solve, on the whole manifold, degenerate complex Hessian equations H_m(u_j) = 1_Ω H_m(φ_j) + ε_j ω^n with fixed total mass equal to ∫X ω^n, using a known existence theorem for measures that do not charge m-polar sets; (2) a comparison principle forces φ_j ≥ u_j on Ω, so the upper envelopes of the approximating subextensions are well-defined subextensions; (3) monotonicity of the Hessian measure along decreasing sequences (established

What would settle it

Find a quasi-m-hyperconvex domain Ω ⊂ P^n and φ ∈ E_χ^m(Ω, ω_FS) with ∫_Ω H_m(φ) ≤ ∫_{P^n} ω_FS^n for which the maximal subextension φ̃ satisfies ∫_{P^n} −χ(φ̃)H_m(φ̃) > ∫_Ω −χ(φ)H_m(φ); Theorem 3.4(i) would be false. More directly, exhibit a measure μ on P^n that does not charge m-polar sets, with μ(P^n) = ∫ ω_FS^n, such that the degenerate Hessian equation H_m(u) = μ has no solution u with sup u = −1; this would break the key existence step.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.4 (Main Theorem 1.1): if Ω is a quasi-m-hyperconvex domain in a compact Kähler manifold (X,ω) with ∫_Ω ω^n < ∫_X ω^n, χ is a convex weight, and φ ∈ E_χ^m(Ω,ω) satisfies ∫_Ω H_m(φ) ≤ ∫_X ω^n, then the upper envelope φ̃ := sup{ψ ∈ SH_m(X,ω) : ψ ≤ φ on Ω} lies in E_χ^m(X,ω). It satisfies ∫_X −χ(φ̃)H_m(φ̃) ≤ ∫_Ω −χ(φ)H_m(φ), the Hessian control 1_Ω H_m(φ̃) ≤ 1_Ω H_m(φ) as measures, and supp H_m(φ̃) ⊂ {φ̃ = φ} ∪ ∂Ω. The proof approximates φ by bounded functions in E_0^m, uses solutions of degenerate complex Hessian equations on X to build approximating subextensions, and passes to the limit via monotone convergence of Hessian measures.

Load-bearing premise

The proof relies on a black-box existence theorem for degenerate complex Hessian equations on compact Kähler manifolds: any measure that does not charge m-polar sets and has the right total mass must be the Hessian measure of some m-subharmonic function normalized to have supremum −1; if that theorem needs extra integrability, the approximation scheme producing the subextension collapses.

Editorial extensions

If this is right

For any φ in the weighted energy class with total Hessian mass no larger than the manifold's volume, there is a canonical extension φ̃ that is maximal: the largest m-subharmonic minorant of φ on X.
The weighted energy inequality transfers a variational principle from the domain to the ambient manifold: extensions never increase the weighted Hessian energy, so energy minimizers behave well under restriction of the domain.
The support containment means the extension's Hessian measure is zero off the contact set and the boundary; any extra mass appears only as a boundary jump of size ∫X ω^n − ∫Ω H_m(φ), as the paper notes in Remark 3.5.
In the projective case, the correspondence between m-subharmonic functions of logarithmic growth on C^n and ω_FS-m-subharmonic functions on P^n yields global subextensions with prescribed Hessian measure normalized to mass one (Theorem 4.4).
The results provide the m-subharmonic analogue of the classical maximal subextension theory for plurisubharmonic functions, so methods that used that theory can now be attempted for m-subharmonic functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test whether the volume inequality ∫_Ω ω^n < ∫_X ω^n can be weakened to ≤, allowing a zero boundary jump; the proof's reliance on the existence theorem might still hold under a matching condition.
The energy inequality could be sharpened: when the volume deficit is zero, one might ask whether ∫_X −χ(φ̃)H_m(φ̃) equals ∫_Ω −χ(φ)H_m(φ), recovering a conservation law for weighted energy.
On P^1 with m=1, the construction should reproduce the known maximal subextension; checking whether the Hessian measure jumps exactly at ∂Ω by the volume deficit would validate the boundary-charge mechanism.
The support containment suggests defining a fine boundary measure of φ with respect to X—the residual measure ν in the decomposition H_m(φ̃) = f·1_Ω H_m(φ) + ν from Remark 3.5—which could be studied as a new invariant of the domain embedding.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Main takeaway: this is a genuine extension of the Cegrell–Kołodziej–Zeriahi maximal subextension result to weighted energy classes for m-subharmonic functions on compact Kähler manifolds. The main theorem is not in the cited literature, and the paper does the expected work of adapting the CKZ2 architecture: maximal subextension defined by upper envelope, approximation by bounded functions, Hessian mass estimates, support on the contact set. The Hessian-measure comparison in Lemma 3.1 and the capacity estimate in Lemma 2.11 are standard but carefully written. Credit where due: the mass estimates in Section 3 are plausible, and the authors correctly identify the degenerate complex Hessian equation [CN] as the black box. This is a technical advance, not a breakthrough, but a real one.

The soft spots are concentrated in Theorem 3.4, and they are load-bearing. Part (i) invokes "the convergence theorem in [CN]" to claim that φ̃_j ↘ φ̃ implies φ̃ ∈ E_χ^m(X,ω) and that the weighted energy inequality survives the limit. No such statement is included, and the paper's own Theorem 2.25 is proved for domains and for approximating sequences in E_0^m(Ω,ω), not for the φ̃_j on X. That is exactly where the energy preservation lives; as written it is not verified. The stress-test note has this right. Part (iii) uses truncations φ_s, φ_t that are never defined, and the limit passage again relies on the same unstated convergence theorem. Lemma 3.1's Step 1 also suppresses why the set D is open and why balayage gives H_m(φ̃)=0 there. That one is probably fixable with standard arguments, but it needs to be written down.

None of this looks fatal. The architecture is sound and the gaps look repairable: state the convergence theorem, verify its hypotheses (capacity bounds for the φ̃_j, or approximation by functions already in E_χ^m(X,ω)), define the truncations, and expand the support argument. The comparison principle and the mass estimates are coherent. I would not desk-reject this; it deserves a serious referee, ideally someone who knows [CN] and CKZ2 and can check whether the quoted black box actually implies the convergence used.

Who gets value: pluripotential theory people working on complex Hessian equations and subextension. The P^n/Lelong-class section is a natural application, though it inherits the same gaps. I would send it to peer review, and I would probably cite the main theorem once the proof is cleaned up, but I would not rely on the current proof of Theorem 3.4.

Referee Report

3 major / 5 minor

Summary. The paper claims that if Ω is a quasi-m-hyperconvex domain in a compact Kähler manifold (X,ω) with ∫_Ω ω^n < ∫_X ω^n, and χ is a convex weight function, then every φ in the weighted Hessian energy class E_χ^m(Ω,ω) with ∫_Ω H_m(φ) ≤ ∫_X ω^n admits a maximal ω-m-subharmonic subextension φ̃ to X. The main theorem asserts that φ̃ ∈ E_χ^m(X,ω), the weighted energy does not increase, the local Hessian domination 1_Ω H_m(φ̃) ≤ 1_Ω H_m(φ) holds, and H_m(φ̃) is supported on {φ̃ = φ} ∪ ∂Ω. The proof follows the CKZ2 strategy: approximate φ by functions in E_0^m, construct approximate subextensions via the solvability of degenerate complex Hessian equations on X, and pass to decreasing limits. Section 4 adapts the result to C^n via the Fubini–Study correspondence and introduces an m-Lelong class.

Significance. If the proof is completed, this would be a natural and useful extension of the CKZ2 maximal-subextension theorem to m-subharmonic functions and weighted energy classes, with potential applications to complex Hessian operators on compact Kähler manifolds. The paper contains coherent comparison principles, mass estimates, and approximation arguments, and the statement of the main theorem is precise and falsifiable. The main weakness is that the decisive limit passage in Theorem 3.4 is delegated to an unnamed convergence theorem, and the support argument uses undefined approximations; these are load-bearing verification gaps rather than stylistic issues.

major comments (3)

[Theorem 3.4(i)] The proof states: "Since φ̃_j ↘ φ̃ on X, the convergence theorem in [CN] implies φ̃ ∈ E_χ^m(X,ω)" and gives the weighted measure inequality. No such theorem is stated, and the paper's own Theorem 2.25 is proved only for domains Ω and for approximating sequences in E_0^m(Ω,ω); the φ̃_j are bounded ω-m-subharmonic functions on X without vanishing boundary data. To conclude the energy-preservation claim, one needs a precise convergence theorem for weighted Hessian measures along decreasing sequences on X, including convergence of −χ(φ̃_j)H_m(φ̃_j) to −χ(φ̃)H_m(φ̃), convergence of −χ(φ_j)1_ΩH_m(φ_j), and preservation of the measure inequality. Please state the theorem from [CN], verify its hypotheses for the present sequence, or supply a proof. This is the central claim (i) of the main theorem.
[Theorem 3.4(iii)] The support statement uses several undefined objects. The proof reuses the symbol φ_j for the truncations max{φ,−j}, conflicting with the approximating sequence φ_j ∈ E_0^m(Ω,ω) used earlier. Later "for fixed s and t" the functions φ_s and φ̃_s are not defined. The passage 1_{φ>−j}H_m(φ_j) ↗ 1_{φ>−∞}H_m(φ) is asserted by reference to [CN] without stating hypotheses, and Lemma 2.11 is invoked without checking the required uniform capacity control. Since claim (iii) is part of the main theorem, these gaps must be fixed.
[Theorem 3.2] The construction of the approximate subextensions rests on the black-box solvability statement [CN, Theorem 1.3], used to produce u_j with H_m(u_j) = 1_Ω H_m(φ_j) + ε_j ω^n and sup_X u_j = −1. The exact hypotheses of that theorem are not stated. In particular, the paper should verify that 1_Ω H_m(φ_j) + ε_j ω^n does not charge m-polar sets and satisfies whatever integrability or regularity conditions [CN, Theorem 1.3] requires. If the theorem needs additional hypotheses beyond "does not charge m-polar sets", the approximation scheme collapses and the existence of φ̃ is not established.

minor comments (5)

[Throughout] There are numerous typos and notation slips: Theorem 3.2 says "quasi-hyperconvex" instead of "quasi-m-hyperconvex"; Proposition 2.17(3) has ω^{m−n} where it should be ω^{n−m}; the abstract says "a good control properties"; Proposition 3.3 uses "eφ" and "eφ_j" for φ̃ and φ̃_j.
[Theorem 2.9] The proof begins "Since P(f)^* ≤ f, then P(f) = P(f)^*", but P(f) is not known to be upper semicontinuous before the proof. This step needs justification or rewording.
[Lemma 3.1] The symbol \hat φ appears before the subextension φ̃ is introduced, and the argument with the characteristic function χ_K should specify that K is compact (hence closed) and explain the approximation by continuous functions more carefully.
[Theorem 4.3] The hypothesis says H_m(u) "charges no pluripolar sets", while elsewhere the relevant notion is m-polar sets. Please align the terminology and clarify whether the stronger condition is needed in the C^n-to-P^n transfer.
[Remark 3.5] The decomposition H_m(φ̃) = f · 1_Ω H_m(φ) + ν with ν supported on ∂Ω is asserted without saying with respect to which reference measure the density f is taken. A short explanation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central derivation rests on external theorems and an independent citation, not on its own conclusions.

full rationale

The paper's main theorem is an existence and energy-inequality result for a maximal subextension, where the subextension is defined independently as the supremum of all admissible subextensions, not by the target inequality. The Hessian measure control (ii) is derived in Lemma 3.1 through balayage, comparison, and weak convergence, then upgraded in Theorem 3.4 using bounded weighted-energy estimates; no fitted parameter is introduced and no quantity is renamed as a prediction. The weighted-energy preservation step in Theorem 3.4(i) cites an external convergence theorem from [CN] after establishing the uniform bounds ∫X −χ(φ̃j)H_m(φ̃j) ≤ C; [CN] is a separate published work by Lu and Nguyen, not the present authors' earlier result, so this is an external dependency rather than a self-citation chain. The only self-citation, [AE], appears in Theorem 4.5 as '[AE, Lemma 3.2] or [MV]', with [MV] an independent source providing the same estimate, so the self-citation is not load-bearing. There are genuine verification/correctness gaps: the precise hypotheses and conclusions of [CN, Theorem 1.3] and the invoked convergence theorem are not restated or checked, and the passage of weighted Hessian measures to the limit is asserted rather than proved from stated assumptions. These are gaps of support and rigor, but they do not make the derivation equivalent to its inputs by construction. No circular step satisfying the required evidentiary standard is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants; auxiliary numbers such as ε_j and γ are determined by the given data, not free parameters. The load-bearing inputs are a stack of prior theorems on complex Hessian equations, Perron envelopes, capacity estimates, and weighted energy theory, plus one lemma from the authors' own previous paper [AE] that is also supported by [MV]. No new physical or geometric entities are postulated; the m-Lelong class in Section 4 is a definition, not an independent entity.

assumptions (6)

domain assumption The Bedford–Taylor-style complex Hessian operator H_m is well-defined for bounded ω-m-sh functions and is continuous under decreasing sequences (Definitions 2.4–2.6, Remark 2.5).
Standard extension of the classical theory to the ω-m setting; used throughout the approximation arguments.
domain assumption Degenerate complex Hessian equations on compact Kähler manifolds are solvable for measures not charging m-polar sets with total mass ∫X ω^n ([CN, Theorem 1.3]).
Used in Theorem 3.2 and Theorem 4.4 to produce the global approximating functions u_j; this is the main non-elementary black box.
domain assumption The Dirichlet problem for the complex Hessian equation on balls admits smooth solutions ([GN, Lemma 3.16]) with the a priori estimates of [DK, Theorem 2.7].
Used in Theorem 2.9 to prove continuity of the Perron envelope and vanishing of H_m(P(f)) on {P(f) < f}.
domain assumption Weighted Hessian energy theory on compact Kähler manifolds, including convergence properties of E_χ(X, ω), follows [GZ], [CGZ], and the unstated 'convergence theorem in [CN]'.
Needed in Theorem 3.4(i) to conclude φ̃ ∈ E_χ^m(X, ω) from the decreasing sequence of subextensions.
standard math Comparison principle and integration-by-parts identities for bounded ω-m-sh functions (Propositions 2.12, 2.13, Theorem 2.19).
The propositions are asserted to follow from local theory ([BT2], [BT3]); they are load-bearing for every mass comparison in Section 3.
domain assumption The m-subharmonic subextension theorem for F^m classes ([MV, Theorem 1]) and the analogue recorded in [AE, Lemma 3.2].
Used in Section 4 to move from balls to general domains and in Theorem 4.5 to control the Hessian measure of the maximal logarithmic subextension. [AE] overlaps with the present authors, but [MV] provides independent support.

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Pith. "Pith review of Maximal subextension of $m$-subharmonic functions." pith.science (2026). https://pith.science/paper/APKKPI53

@misc{pith2026260719132,
  author       = {Pith},
  title        = {Pith review of: Maximal subextension of $m$-subharmonic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/APKKPI53}},
  note         = {Machine review of arXiv:2607.19132}
}

abstract

In this paper, we prove that given a quasi-$m$-hyperconvex domain $\Omega \subset X$ in a compact K\"ahler manifold $(X, \omega)$, and a function $\varphi $ in the weighted energy class $\mathcal{E}_\chi^m(\Omega, \omega)$ with respect to a convex weight function $\chi : \mathbb{R} \to \mathbb{R}$, then there exists a maximal $\omega$-$m$-subharmonic subextension $\tilde{\varphi}$ to $X$ that preserves the weighted energy and satisfies a good control properties for its Hessian measure $ \mathbf{1}_\Omega H_m(\tilde{\varphi}) \leq \mathbf{1}_\Omega H_m(\varphi) $. In the last part, we study the particular case where $(X,\omega)=(\mathbb{P}^n,\omega_{FS}).$

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Ahag and R

P. Ahag and R. Czyz, On the Cegrell classes, Math. Z. 256 (2007), 243–264

2007

[2] [2]

Subextension and approximation of m -subharmonic functions with given boundary values

AMAL H., El-Gasmi A. Subextension and approximation of m -subharmonic functions with given boundary values. Ukrainian Mathematical Journal, Vol. 77, No. 2, 2025. https://doi.org/10.3842/umzh.v77i2.8215

work page doi:10.3842/umzh.v77i2.8215 2025

[3] [3]

Weighted Green functions for complex Hessian operators

El Aini, Hadhami; Zeriahi, Ahmed. Weighted Green functions for complex Hessian operators. Ann. Polon. Math. 130 (2023), no. 1, 1--32

2023

[4] [4]

Bedford, Eric; Taylor, B. A. The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37 (1976), no. 1, 1--44

1976

[5] [5]

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

1982

[6] [6]

Bedford, Eric; Taylor, B. A. Fine topology, Šilov boundary, and (dd^c)^n . J. Funct. Anal. 72 (1987), no. 2, 225--251

1987

[7] [7]

Benelkourchi, Weighted pluricomplex energy, Potential Anal., 31(2009), 1-20

S. Benelkourchi, Weighted pluricomplex energy, Potential Anal., 31(2009), 1-20

2009

[8] [8]

Pluripolar sets and the subextension in Cegrell’s classes

Hiep PH. Pluripolar sets and the subextension in Cegrell’s classes. Complex Var. Elliptic Equ. 2008;53:675–684

2008