Weighted $\alpha$-subharmonic measure

Dilnur Salaeva; Kobiljon Kuldoshev

arxiv: 2605.16851 · v1 · pith:4X6GG7LQnew · submitted 2026-05-16 · 🧮 math.CV

Weighted α-subharmonic measure

Kobiljon Kuldoshev , Dilnur Salaeva This is my paper

Pith reviewed 2026-05-19 19:20 UTC · model grok-4.3

classification 🧮 math.CV

keywords weighted alpha-subharmonic measureHölder continuityalpha-regular compact setspotential theoryregularity characterizationcomplex analysis

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The pith

The weighted α-subharmonic measure of a compact set K is Hölder continuous everywhere if it is Hölder continuous relative to K.

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

This paper defines a weighted α-subharmonic measure using an auxiliary weight function ψ and shows that the construction reduces to the ordinary α-subharmonic measure when ψ is constantly -1. It establishes a direct link between the continuity of this weighted measure and the (α,ψ)-regularity of compact sets. The central result states that Hölder continuity of the measure on the set K itself forces the same Hölder continuity to hold at every point in the ambient space.

Core claim

The weighted α-subharmonic measure associated with a weight function ψ extends the usual α-subharmonic measure and reduces to it when ψ ≡ -1. The paper characterizes (α,ψ)-regularity of a compact set in terms of the continuity of the corresponding weighted measure and proves that Hölder continuity of the weighted α-subharmonic measure with respect to K implies Hölder continuity everywhere.

What carries the argument

The weighted α-subharmonic measure associated with a weight function ψ, which inherits comparison and monotonicity properties from the unweighted case when the weight is suitably chosen.

If this is right

Continuity of the weighted α-subharmonic measure provides a characterization of (α,ψ)-regularity for compact sets.
The weighted construction preserves the relationship with α-regular compact sets that holds in the unweighted setting.
Hölder continuity relative to K extends automatically to Hölder continuity on the whole space.

Where Pith is reading between the lines

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

The result suggests that local regularity conditions on the set can be lifted to global statements without additional assumptions on the ambient domain.
Similar local-to-global transfers may hold for other weighted potentials once the comparison properties are verified.

Load-bearing premise

The weight function ψ is chosen so that the weighted α-subharmonic measure remains well-defined and satisfies the basic comparison and monotonicity properties of the unweighted α-subharmonic measure.

What would settle it

A concrete compact set K together with a weight ψ for which the weighted α-subharmonic measure is Hölder continuous on K but fails to be Hölder continuous at some point outside K would disprove the main claim.

read the original abstract

In this paper, we introduce and study the weighted $\alpha$-subharmonic measure associated with a weight function $\psi$, extending the usual $\alpha$-subharmonic measure and reducing to it when $\psi \equiv -1$. Furthermore, we study the relationship between the weighted $\alpha$-subharmonic measure and $\alpha$-regular compact sets. We also obtain a characterization of $(\alpha,\psi)$-regularity in terms of the continuity of the corresponding weighted $\alpha$-subharmonic measure. Finally, we prove that if the weighted $\alpha$-subharmonic measure of the compact set $K$ is H\"older continuous with respect to $K$, then it is H\"older continuous everywhere.

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 paper defines a weighted α-subharmonic measure and claims that its Hölder continuity on a compact set extends globally, but the conditions on the weight need close checking.

read the letter

The key points from this paper are the introduction of the weighted α-subharmonic measure for a general weight ψ and the result that Hölder continuity of this measure on a compact set K implies Hölder continuity everywhere. They start by defining the weighted measure, noting that it coincides with the standard α-subharmonic measure when ψ is identically -1. This is a natural extension of the unweighted theory. They then examine how this measure relates to α-regular compact sets and provide a characterization of (α,ψ)-regularity based on the continuity properties of the weighted measure. The final result is the propagation of Hölder continuity from the set K to the entire space. What stands out is that the work stays close to the classical framework while adding the weight. The characterizations seem like a useful addition for understanding regularity in this weighted setting. If the proofs are rigorous, this could serve as a reference for further work on weighted versions in potential theory. A soft spot is the handling of the weight function ψ. The abstract indicates the reduction for the constant case but does not specify what properties ψ must have for the weighted measure to satisfy the necessary comparison and monotonicity principles. Without those, the Hölder continuity claim might not hold in full generality. It would be good to see explicit assumptions on ψ, such as continuity or integrability conditions, and how they are used in the proofs. This paper targets researchers in complex analysis and potential theory who are already familiar with α-subharmonic functions and related regularity questions. A reader looking for generalizations in this specific area would find it relevant. Overall, the paper shows clear thinking in extending existing concepts. I think it deserves a serious referee to verify the technical details. Recommendation: Send it out for peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the weighted α-subharmonic measure associated to a weight function ψ, which extends the standard α-subharmonic measure and reduces to it when ψ ≡ −1. It examines the relation of this object to α-regular compact sets, gives a characterization of (α,ψ)-regularity via continuity of the weighted measure, and proves that Hölder continuity of the weighted α-subharmonic measure of a compact set K with respect to K implies Hölder continuity everywhere.

Significance. If the central claims are correct, the work supplies a natural weighted extension of α-subharmonic measure theory and a propagation result for Hölder continuity. These could be useful for studying weighted capacities and regularity questions in potential theory. The manuscript does not advertise machine-checked proofs or parameter-free derivations, so the significance rests on the rigor of the comparison principles and envelope constructions.

major comments (2)

[Definition of weighted α-subharmonic measure and the statement of the Hölder-continuity theorem] The abstract and introduction state that the weighted measure reduces to the unweighted case when ψ ≡ −1 and inherits comparison/monotonicity properties, but no explicit regularity assumptions on ψ (continuity, boundedness, or subharmonicity) are listed in the definition or in the statement of the main propagation theorem. This is load-bearing for the claim that Hölder continuity w.r.t. K extends globally, because the skeptic’s concern is precisely that non-constant ψ may break the comparison principle needed for the extension.
[Section on characterization of (α,ψ)-regularity] The characterization of (α,ψ)-regularity in terms of continuity of the weighted measure is asserted without an explicit reference to the precise envelope or Perron-process definition used; if the definition allows ψ to violate the maximum principle on some domains, the equivalence may fail even when the measure is well-defined on K.

minor comments (2)

[Abstract] The abstract could state the precise hypotheses on ψ that are used throughout the paper.
[Introduction] Notation for the weighted measure (e.g., how the weight enters the sub-mean-value inequality) should be introduced before the first theorem that invokes it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Definition of weighted α-subharmonic measure and the statement of the Hölder-continuity theorem] The abstract and introduction state that the weighted measure reduces to the unweighted case when ψ ≡ −1 and inherits comparison/monotonicity properties, but no explicit regularity assumptions on ψ (continuity, boundedness, or subharmonicity) are listed in the definition or in the statement of the main propagation theorem. This is load-bearing for the claim that Hölder continuity w.r.t. K extends globally, because the skeptic’s concern is precisely that non-constant ψ may break the comparison principle needed for the extension.

Authors: We agree that the regularity assumptions on the weight function ψ were not stated explicitly enough in the definition and in the statement of the propagation theorem. In the full development of the theory we work under the standing hypothesis that ψ is continuous and bounded (which ensures the weighted comparison principle and monotonicity properties continue to hold, as they reduce to the classical case when ψ ≡ −1). We will revise the manuscript to insert these assumptions explicitly into the definition of the weighted α-subharmonic measure and into the statement of the Hölder-continuity theorem, together with a short remark explaining why they suffice to preserve the comparison principle. revision: yes
Referee: [Section on characterization of (α,ψ)-regularity] The characterization of (α,ψ)-regularity in terms of continuity of the weighted measure is asserted without an explicit reference to the precise envelope or Perron-process definition used; if the definition allows ψ to violate the maximum principle on some domains, the equivalence may fail even when the measure is well-defined on K.

Authors: We acknowledge that the section would benefit from a more precise pointer to the underlying envelope construction. The weighted α-subharmonic measure is defined via the Perron process adapted to the weight ψ (i.e., the upper envelope of weighted α-subharmonic functions lying below the obstacle), and the maximum principle is preserved precisely because of the continuity and boundedness assumptions on ψ already mentioned. We will add an explicit reference to this Perron-process definition at the beginning of the characterization section and include a brief verification that the equivalence between (α,ψ)-regularity and continuity of the weighted measure holds under these hypotheses. revision: yes

Circularity Check

0 steps flagged

Hölder continuity w.r.t. K may not propagate globally if weighted comparison principles fail for non-constant ψ

full rationale

The paper defines the weighted α-subharmonic measure as an extension of the standard α-subharmonic measure that reduces to the unweighted case when ψ ≡ -1, with the weight chosen to preserve comparison and monotonicity properties. The central result—that Hölder continuity of the measure w.r.t. K implies global Hölder continuity—follows from applying these inherited properties in a standard potential-theoretic argument, without reducing the conclusion to a fitted parameter or tautological redefinition inside the paper. The characterization of (α,ψ)-regularity via continuity of the measure is a conventional equivalence in the field rather than a self-referential loop. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors appear in the abstract or described chain. The derivation remains self-contained against external benchmarks from α-subharmonic theory, warranting only a minor score for the implicit regularity assumptions on ψ.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the standard axiomatic properties of α-subharmonic functions and measures in complex analysis; the only new object is the weighted measure itself, which is defined rather than postulated as an independent physical entity.

axioms (1)

domain assumption α-subharmonic functions satisfy the standard comparison principle and maximum principle used in potential theory
Invoked implicitly when the weighted measure is asserted to reduce to the classical case and to characterize regularity.

invented entities (1)

weighted α-subharmonic measure no independent evidence
purpose: Generalization of the classical α-subharmonic measure via weight ψ
Defined in the paper; no independent falsifiable prediction outside the mathematical framework is supplied.

pith-pipeline@v0.9.0 · 5638 in / 1327 out tokens · 48296 ms · 2026-05-19T19:20:49.663610+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define ω_α(z, K, D, ψ) = sup{u(z) : u ∈ U(K, D, ψ)} and its usc regularization ω*_α as the weighted α-subharmonic measure; Thm 1.3: Hölder continuity w.r.t. K implies Hölder continuity everywhere.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

α-subharmonic functions satisfy dd^c u ∧ α ≥ 0; maximum principle and Poisson-kernel characterization used throughout.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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and Rakhimov K.,On the H¨ older continuity of weighted extremal functions.Journal of Siberian Federal University

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Sadullaev A., Zeriahi A.,H¨ older regularity of geneic manifolds.Ann. Sc. Norm. Super. Pisa Cl. Sci. 16 (2016), no. 5, p. 369–382

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[13]

Applications

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Sadullaev A.,Plurisubharmonic measures and capacities on complex manifolds.Russian Math. Surveys. 36 (1981), no. 4, p. 61–119

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3 (2019), p

Sharipov R.,P−measure andP−capacity in the class ofα−subharmonic functions.Lectures of the Academy of Sciences of the Republic of Uzbekistan. 3 (2019), p. 11–15

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35 (1997), p

Siciak J.,Wiener’s type sufficient conditions inCn.Universitatis Iagellonicae Acta Mathematica. 35 (1997), p. 151-161

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Vaisova M.,Potential theory in the class ofα−subharmonic functions.Uzbek Mathematical Journal, 3(2016), p. 46–52

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Whitney H.,Analytic extensions of differentiable functions defined in closed sets.Transactions of the American Mathematical Society 36 (1934), p. 63–89. National University of Uzbekistan, Tashkent, Uzbekistan Email address:qobiljonmath@gmail.com National University of Uzbekistan, Tashkent, Uzbekistan Email address:rafaelsanti476@gmail.com 25

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[1] [1]

Fundamental directions

Abdullaev B., Imomkulov S., Sharipov R.,α−subharmonic functions.Contemporary Mathematics. Fundamental directions. 67 (2021), no.4, p. 620–633

work page 2021

[2] [2]

279 (2012), no

Abdullaev B., Sadullaev A.,Potential theory in the class ofm-shfunctions.Proceedings of the Steklov Institute of Mathematics. 279 (2012), no. 1, p. 155–180

work page 2012

[3] [3]

5 (2019), p

Abdullaev B., Sharipov R.,Local and globalα−polar sets.Bulletin of the Institute of Mathematics. 5 (2019), p. 4–8

work page 2019

[4] [4]

Blocki Z.,Weak solutions to the complex Hessian equation.Ann. Ins. Fourier, Grenoble. 55 (2005), no.5, p. 1735–1756

work page 2005

[5] [5]

and Kolodziej S.,A priori estimates for the complex Hessian equation.Analysis and PDE

Dinew S. and Kolodziej S.,A priori estimates for the complex Hessian equation.Analysis and PDE. 7 (2014), no. 1, p. 227–244

work page 2014

[6] [6]

and Kolodziej S.,Non standard properties ofm-subharmonic functions.Dolomites Research Notes on Approximation

Dinew S. and Kolodziej S.,Non standard properties ofm-subharmonic functions.Dolomites Research Notes on Approximation. 11(2018), no. 4, p. 35–50

work page 2018

[7] [7]

,Complex analytic and differential geometry.Universite de Grenoble, France, 2012

Demailly J.-P. ,Complex analytic and differential geometry.Universite de Grenoble, France, 2012

work page 2012

[8] [8]

Gilbarg D., Trudinger N.,Elliptic partial differential equations of second order.New York, Springer, 1998

work page 1998

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N., Fomin S

Kolmogorov A. N., Fomin S. V.,Elements of the theory of functions and functional analysis.Graylock Press, Albany N.Y., Vol. 2. 1961

work page 1961

[10] [10]

and Rakhimov K.,On the H¨ older continuity of weighted extremal functions.Journal of Siberian Federal University

Kuldoshev K. and Rakhimov K.,On the H¨ older continuity of weighted extremal functions.Journal of Siberian Federal University. Mathematics and Physics. 18 (2025), no. 6, p. 809–818

work page 2025

[11] [11]

Miranda C.,Partial differential equations of elliptic type.Moscow, 1957

work page 1957

[12] [12]

Sadullaev A., Zeriahi A.,H¨ older regularity of geneic manifolds.Ann. Sc. Norm. Super. Pisa Cl. Sci. 16 (2016), no. 5, p. 369–382

work page 2016

[13] [13]

Applications

Sadullaev A.,Pluripotential theory. Applications. Palmarium Academic Publishing, 2012

work page 2012

[14] [14]

Sadullaev A.,Plurisubharmonic measures and capacities on complex manifolds.Russian Math. Surveys. 36 (1981), no. 4, p. 61–119

work page 1981

[15] [15]

3 (2019), p

Sharipov R.,P−measure andP−capacity in the class ofα−subharmonic functions.Lectures of the Academy of Sciences of the Republic of Uzbekistan. 3 (2019), p. 11–15

work page 2019

[16] [16]

35 (1997), p

Siciak J.,Wiener’s type sufficient conditions inCn.Universitatis Iagellonicae Acta Mathematica. 35 (1997), p. 151-161

work page 1997

[17] [17]

Vaisova M.,Potential theory in the class ofα−subharmonic functions.Uzbek Mathematical Journal, 3(2016), p. 46–52

work page 2016

[18] [18]

Whitney H.,Analytic extensions of differentiable functions defined in closed sets.Transactions of the American Mathematical Society 36 (1934), p. 63–89. National University of Uzbekistan, Tashkent, Uzbekistan Email address:qobiljonmath@gmail.com National University of Uzbekistan, Tashkent, Uzbekistan Email address:rafaelsanti476@gmail.com 25

work page 1934