Classical Balayage of Charges and Measures

Bulat N. Khabibullin; Enzhe Menshikova

arxiv: 1907.09608 · v1 · pith:CKUDJ5ASnew · submitted 2019-07-22 · 🧮 math.CV · math.FA

Classical Balayage of Charges and Measures

Bulat N. Khabibullin , Enzhe Menshikova This is my paper

Pith reviewed 2026-05-24 17:21 UTC · model grok-4.3

classification 🧮 math.CV math.FA

keywords balayagesubharmonic functionschargesmeasuresdomain geometrypotential theoryEuclidean spaceopen sets

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The pith

Balayage of charges and measures for subclasses of subharmonic functions relates to the geometry of the domain in Euclidean space.

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

The paper investigates properties of the balayage operation applied to charges and measures within specific subclasses of subharmonic functions. It explores how these properties connect to the geometric features of the domain or open set in finite-dimensional Euclidean space. A reader might care because such relationships could reveal how potential-theoretic operations depend on the shape and structure of the underlying space.

Core claim

We investigate some properties of balayage of charges and measures for subclasses of subharmonic functions and their relationship to the geometry of domain or open set in finite-dimensional Euclidean space where this balayage is considered.

What carries the argument

The balayage operation on charges and measures for subclasses of subharmonic functions, which maps them in a manner connected to domain geometry.

If this is right

Balayage preserves or alters certain subharmonic inequalities depending on domain geometry.
The operation yields distinct results in domains with different boundary or interior structures.
Subclasses of subharmonic functions admit balayage behaviors that reflect the domain's geometric constraints.

Where Pith is reading between the lines

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

The findings could support explicit computations of balayage in domains with known symmetries or irregularities.
Similar relationships might appear when extending the study to related classes like superharmonic functions or to higher-dimensional settings.

Load-bearing premise

The balayage operation is well-defined and admits meaningful study for the chosen subclasses of subharmonic functions within the considered domains.

What would settle it

A concrete counterexample in a specific domain where balayage properties for the chosen subclasses show no dependence on geometric features of the domain.

read the original abstract

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 investigates balayage properties for subclasses of subharmonic functions and their geometric ties but states no concrete new theorem or result.

read the letter

The main takeaway is that the work stays inside classical potential theory, examining balayage of charges and measures for chosen subclasses of subharmonic functions and how those properties relate to the geometry of the domain in Euclidean space. The abstract frames it as an investigation rather than a derivation of a new formula or reduction of earlier results to special cases. That keeps the scope narrow and incremental. What the paper does is keep the setting classical and finite-dimensional, which avoids unnecessary abstraction and lets the focus stay on how balayage interacts with domain shape for the selected function classes. That direction is reasonable for the field. The soft spot is the complete absence of any specific property, equation, or example. The abstract gives no indication of what was actually shown or how it improves on existing literature, so novelty and soundness cannot be judged. Without those details the claim reduces to “we looked at this,” which does not supply evidence of progress. The work would mainly interest a small group already active in balayage and subharmonic functions who might want to see extensions to particular subclasses. Most readers in complex analysis or potential theory will not find usable results or tools here. I would not bring it to a reading group, would not cite it, and would not send it to peer review on the basis of the abstract alone.

Referee Report

0 major / 1 minor

Summary. The manuscript investigates properties of the classical balayage operation applied to charges and measures associated with subclasses of subharmonic functions, with emphasis on how these properties relate to the geometry of the underlying domain or open set in finite-dimensional Euclidean space.

Significance. If the claimed properties are established with rigorous proofs, the work would contribute to potential theory by refining the classical balayage construction for restricted classes of subharmonic functions and clarifying its geometric dependence on the domain. Such results could be useful for applications in complex analysis where domain geometry constrains harmonic or subharmonic behavior. The abstract, however, states only the scope of the investigation without identifying specific theorems, so the actual significance cannot be evaluated from the provided information.

minor comments (1)

The abstract is purely descriptive and does not state any concrete results, theorems, or examples; this makes it difficult for a reader to gauge the paper's contribution without reading the full text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The report notes that the abstract describes the scope of the investigation but does not identify specific theorems, which prevents evaluation of significance. We address this point below. No other major comments were provided in the report.

read point-by-point responses

Referee: The abstract, however, states only the scope of the investigation without identifying specific theorems, so the actual significance cannot be evaluated from the provided information.

Authors: We agree that a more informative abstract would help. The manuscript establishes several specific results, including properties of balayage for restricted classes of subharmonic functions and their dependence on the geometry of the domain in Euclidean space. In the revised version we will update the abstract to explicitly mention the main theorems and their geometric implications. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states only that it investigates properties of balayage for subclasses of subharmonic functions and their geometric relationships in Euclidean domains. No equations, derivations, predictions, or first-principles results are presented in the provided abstract or description. The work is descriptive and exploratory within standard potential theory; the well-definedness of classical balayage is presupposed as background rather than derived internally. No self-citations, fitted inputs, or ansatzes are visible that could reduce claims to inputs by construction. This is the expected honest non-finding for an investigation-style manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms invoked, or new entities postulated; ledger is empty by default.

pith-pipeline@v0.9.0 · 5547 in / 1014 out tokens · 27447 ms · 2026-05-24T17:21:18.372353+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Yu., Talipova G

Bayguskarov T. Yu., Talipova G. R., Khabibullin B. N. Subsequences of zeros for classes of entire functions of exponential type, allocated by restrictions on their growth // Algebra i Analiz. 2016.28:2, 1–33; English transl. in St. Petersburg Math. J. 2017. 28:2, 127–151

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Cole B. J., Ransford Th. J. Jensen measures and harmonic measures // J. reine angew. Math. 2001,541, 29–53

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Doob J. L. Classical Potential Theory and Its Probabilistic Counterpart. Springer- Verlag, 1984

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Gamelin T. W. Uniform Algebras and Jensen Measures. Cambridge Univ. Press, 1978

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Gardiner S. J. Harmonic Approximation. Cambridge Univ. Press, 1995

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Gauthier P. M. Uniform approximation // In book: Complex Potential Theory. Nether- lands: Kluwer Academic Publisher. 1994. 235–271

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Potential theory and approximation: highlights from the scientiﬁc work of Stephen Gardiner // Analysis and Mathematical Physics

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The Uncertainty Principle in Harmonic Analysis

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Khabibullin B. N. Criteria for (sub-)harmonicity and continuation of (sub-)harmonic functions // Sibirsk. Mat. Zh. 2003,44:4, 905–925; Siberian Math. J. 2003,44:4, 713– 728

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N., Rozit A

Khabibullin B. N., Rozit A. P., Khabibullina E. B. Order versions of the Hahn–Banach theoremandenvelopes.II.Applicationstothefunctiontheory, ComplexAnalysis.Math- ematical Physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 162, 11 VINITI, Moscow, 2019, 93–135; translated in English and are published by Springer in Journal of Mathematica...

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N., Talipova G

Khabibullin B. N., Talipova G. R., Khabibullin F. B. Zero subsequences for Bernstein’s spaces and the completeness of exponential systems in spaces of functions on an interval //AlgebraiAnaliz.2014. 26:2, 185–215; Englishtransl.inSt.PetersburgMath.J.2015. 26:2, 319–340

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Le¸ cons sur le th´ eor` eme Beurling et Malliavin

Koosis P. Le¸ cons sur le th´ eor` eme Beurling et Malliavin. Montr´ eal: Les Publications CRM, 1996

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Landkoﬀ N. S. Foundations of modern potential theory. Springer-Verlag, 1972

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B., Khabibullin B

Menshikova E. B., Khabibullin B. N. On the Distribution of Zero Sets of Holomorphic Functions. II // Funktsional. Anal. i Prilozhen. 2019,53:1, 84-87; English transl. in Funct. Anal. Appl. 2019,53:1

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The logarithmic integral

Koosis P. The logarithmic integral. II. Cambridge Univ. Press, 1992

work page 1992
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Potential Theory in the Complex Plane

Ransford Th. Potential Theory in the Complex Plane. Cambridge University Press, 1995

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Approximation of Jensen measures by image measures under holomorphic functions and applications // Trans

Bu S., Schachermayer W. Approximation of Jensen measures by image measures under holomorphic functions and applications // Trans. Amer. Math. Soc. 1992.331:2, 585– 608

work page 1992
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Hermann, 1967

Analyse Math´ ematique. Hermann, 1967. 12

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[1] [1]

Yu., Talipova G

Bayguskarov T. Yu., Talipova G. R., Khabibullin B. N. Subsequences of zeros for classes of entire functions of exponential type, allocated by restrictions on their growth // Algebra i Analiz. 2016.28:2, 1–33; English transl. in St. Petersburg Math. J. 2017. 28:2, 127–151

work page 2016

[2] [2]

Int´ egration, Chapitre 5.´El´ ements de math´ ematique

Nicolas Bourbaki. Int´ egration, Chapitre 5.´El´ ements de math´ ematique. Springer-Verlag, 1967–2007

work page 1967

[3] [3]

J., Ransford Th

Cole B. J., Ransford Th. J. Subharmonicity without Upper Semicontinuity // J. Funct. Anal. 1997.147. 420–442

work page 1997

[4] [4]

J., Ransford Th

Cole B. J., Ransford Th. J. Jensen measures and harmonic measures // J. reine angew. Math. 2001,541, 29–53

work page 2001

[5] [5]

Doob J. L. Classical Potential Theory and Its Probabilistic Counterpart. Springer- Verlag, 1984

work page 1984

[6] [6]

Gamelin T. W. Uniform Algebras and Jensen Measures. Cambridge Univ. Press, 1978

work page 1978

[7] [7]

Gardiner S. J. Harmonic Approximation. Cambridge Univ. Press, 1995

work page 1995

[8] [8]

Gauthier P. M. Uniform approximation // In book: Complex Potential Theory. Nether- lands: Kluwer Academic Publisher. 1994. 235–271

work page 1994

[9] [9]

Gautier P. M. Subharmonic extensions and approximations // Can. Math. Bull. 1994. 37:5, 46–53

work page 1994

[10] [10]

Potential theory and approximation: highlights from the scientiﬁc work of Stephen Gardiner // Analysis and Mathematical Physics

Ghergu m., Manolaki m., Netuka i., Render H. Potential theory and approximation: highlights from the scientiﬁc work of Stephen Gardiner // Analysis and Mathematical Physics. 2019.9:2, 679–709

work page 2019

[11] [11]

The Uncertainty Principle in Harmonic Analysis

Havin V., J¨ oriicke B. The Uncertainty Principle in Harmonic Analysis. Springer-Verlag, 1994

work page 1994

[12] [12]

Helms, L. L. Introduction to Potential Theory, Wiley-Interscience, 1969

work page 1969

[13] [13]

Khabibullin B. N. Criteria for (sub-)harmonicity and continuation of (sub-)harmonic functions // Sibirsk. Mat. Zh. 2003,44:4, 905–925; Siberian Math. J. 2003,44:4, 713– 728

work page 2003

[14] [14]

N., Rozit A

Khabibullin B. N., Rozit A. P., Khabibullina E. B. Order versions of the Hahn–Banach theoremandenvelopes.II.Applicationstothefunctiontheory, ComplexAnalysis.Math- ematical Physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 162, 11 VINITI, Moscow, 2019, 93–135; translated in English and are published by Springer in Journal of Mathematica...

work page 2019

[15] [15]

N., Talipova G

Khabibullin B. N., Talipova G. R., Khabibullin F. B. Zero subsequences for Bernstein’s spaces and the completeness of exponential systems in spaces of functions on an interval //AlgebraiAnaliz.2014. 26:2, 185–215; Englishtransl.inSt.PetersburgMath.J.2015. 26:2, 319–340

work page 2014

[16] [16]

Le¸ cons sur le th´ eor` eme Beurling et Malliavin

Koosis P. Le¸ cons sur le th´ eor` eme Beurling et Malliavin. Montr´ eal: Les Publications CRM, 1996

work page 1996

[17] [17]

Landkoﬀ N. S. Foundations of modern potential theory. Springer-Verlag, 1972

work page 1972

[18] [18]

B., Khabibullin B

Menshikova E. B., Khabibullin B. N. On the Distribution of Zero Sets of Holomorphic Functions. II // Funktsional. Anal. i Prilozhen. 2019,53:1, 84-87; English transl. in Funct. Anal. Appl. 2019,53:1

work page 2019

[19] [19]

The logarithmic integral

Koosis P. The logarithmic integral. II. Cambridge Univ. Press, 1992

work page 1992

[20] [20]

Potential Theory in the Complex Plane

Ransford Th. Potential Theory in the Complex Plane. Cambridge University Press, 1995

work page 1995

[21] [21]

Approximation of Jensen measures by image measures under holomorphic functions and applications // Trans

Bu S., Schachermayer W. Approximation of Jensen measures by image measures under holomorphic functions and applications // Trans. Amer. Math. Soc. 1992.331:2, 585– 608

work page 1992

[22] [22]

Hermann, 1967

Analyse Math´ ematique. Hermann, 1967. 12

work page 1967