Weighted Hardy inequalities involving supremum for decreasing sequences

Amiran Gogatishvili; Nurgul Kuzeubayeva; Nurzhan Bokayev; Tu\u{g}\c{c}e \"Unver

arxiv: 2606.17558 · v1 · pith:PUENM42Mnew · submitted 2026-06-16 · 🧮 math.FA

Weighted Hardy inequalities involving supremum for decreasing sequences

Tu\u{g}\c{c}e \"Unver , Amiran Gogatishvili , Nurzhan Bokayev , Nurgul Kuzeubayeva This is my paper

Pith reviewed 2026-06-26 22:56 UTC · model grok-4.3

classification 🧮 math.FA

keywords weighted Hardy inequalitiessupremum operatornon-increasing sequencescharacterizationfunctional analysisinequalities

0 comments

The pith

Weighted Hardy inequalities with supremum on non-increasing sequences are fully characterized for all positive parameters by reduction to the non-negative case.

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

The paper establishes a complete characterization of weighted Hardy inequalities that involve the supremum operator when the sequences are restricted to be non-increasing. It achieves this by proving that such inequalities are equivalent to corresponding inequalities stated for arbitrary non-negative sequences. The equivalence widens the available proof methods because the non-negative setting does not impose monotonicity. A reader would care about the result because these inequalities control sums and bounds that appear when working with ordered data in analysis.

Core claim

We provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

What carries the argument

The reduction of the inequalities from the cone of non-increasing sequences to equivalent inequalities on the cone of non-negative sequences.

If this is right

The characterization covers every choice of positive parameters.
Proofs can now be carried out in the wider setting of non-negative sequences rather than only the monotone cone.
The exact conditions on the weights become accessible through the reduced form.

Where Pith is reading between the lines

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

The same reduction idea could be tested on other operators that are commonly restricted to monotone sequences.
The broader non-negative framework may simplify numerical checks of the resulting weight conditions.

Load-bearing premise

The reduction of inequalities restricted to non-increasing sequences to equivalent inequalities on the cone of non-negative sequences is valid and yields a complete characterization for all positive parameters.

What would settle it

A pair of weights and a positive parameter such that the inequality holds for every non-increasing sequence but fails for some non-negative sequence would show the claimed equivalence does not hold.

read the original abstract

In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

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

The paper reduces weighted Hardy inequalities with the supremum on non-increasing sequences to the non-negative case and claims this yields a full characterization for all positive parameters.

read the letter

The core move is reducing the inequality on the cone of non-increasing sequences to an equivalent form on all non-negative sequences. This is presented as the step that delivers the complete characterization across every positive parameter.

The reduction itself is the part that stands out. By dropping the monotonicity restriction they say the problem becomes easier to handle with existing techniques, which widens the set of available proofs. That is a practical step in this corner of sequence-space inequalities.

The limitation is that the abstract gives no derivation or even an outline of how the equivalence is proved. Without that step it is impossible to check whether the constants survive the reduction or whether boundary cases for the weights create extra conditions. The stress-test found no internal contradiction, but that only confirms the claim is consistent on its face.

This is specialized work aimed at people already working on Hardy-type operators and weighted inequalities in sequence spaces. A reader in that subfield could use the reduction if the details hold up. Outside that group the result is too narrow to matter much.

I would not bring it to a general reading group, and I would not cite it in my own papers. It still looks worth sending to referees so someone can verify the equivalence proof and the claimed completeness.

Referee Report

1 major / 0 minor

Summary. The paper claims to provide a complete characterization of weighted Hardy inequalities involving the supremum operator restricted to non-increasing sequences for all positive parameters, achieved by reducing the inequalities to equivalent forms on the larger cone of non-negative sequences.

Significance. If the asserted reduction is equivalence-preserving and yields an explicit complete characterization, the result would expand the analytical framework for supremum-based Hardy inequalities and allow a wider range of proof techniques; the manuscript's explicit assertion of equivalence is a strength in this regard.

major comments (1)

[Abstract] The central reduction from the cone of non-increasing sequences to non-negative sequences is asserted in the abstract to be equivalence-preserving and to yield a complete characterization for all positive parameters, but no explicit statement of the resulting characterization (e.g., the form of the weight conditions or the equivalent inequality) is supplied in the provided text, preventing verification that the reduction does not introduce or hide parameter restrictions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to address the concern raised. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] The central reduction from the cone of non-increasing sequences to non-negative sequences is asserted in the abstract to be equivalence-preserving and to yield a complete characterization for all positive parameters, but no explicit statement of the resulting characterization (e.g., the form of the weight conditions or the equivalent inequality) is supplied in the provided text, preventing verification that the reduction does not introduce or hide parameter restrictions.

Authors: The full manuscript establishes the equivalence-preserving reduction and derives the explicit characterization (including the resulting weight conditions and equivalent inequality on non-negative sequences) in the main theorems. The abstract is a concise summary of this achievement. We agree that an explicit reference to the form of the characterization in the abstract would facilitate immediate verification that no parameter restrictions are introduced or hidden. We will revise the abstract to include a brief statement of the equivalent inequality obtained after reduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper asserts a reduction of the weighted Hardy inequalities with supremum operator from the cone of non-increasing sequences to equivalent inequalities on the cone of non-negative sequences, claiming this yields a complete characterization for all positive parameters. This reduction is presented as an equivalence-preserving analytical step that expands the proof framework, with no equations or steps shown to reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The abstract and described structure contain no self-referential loops, renamed empirical patterns, or uniqueness theorems imported from prior author work; the central claim remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract contains no information about free parameters, axioms or invented entities.

pith-pipeline@v0.9.1-grok · 5591 in / 863 out tokens · 33215 ms · 2026-06-26T22:56:34.373089+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references

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N. Bokayev, A. Gogatishvili, G. Karshygina, N. Kuzeubayeva, and T. ¨Unver, Reduction theorems for the discrete Hardy operator on the cones of monotone sequences.J. Math. Sci.291(2025), no. 2, 217–223

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Gogatishvili, and L

A. Gogatishvili, and L. Pick, A reduction theorem for supremum operators.J. Comput. Appl. Math.208 (2007), no. 1, 270–279

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Gogatishvili, L

A. Gogatishvili, L. Pick, and T. ¨Unver, Weighted inequalities for discrete iterated kernel operators.Math. Nachr.295(2022), no. 11, 1–26

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A. Gogatishvili, and V.D. Stepanov, Reduction theorems for operators on the cones of monotone functions. J. Math. Anal. Appl.405(2013), no. 1, 156–172

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Gogatishvili, and V.D

A. Gogatishvili, and V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions.Russ. Math. Surv.64(2013), no. 4, 597–664

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Grosse-Erdmann,The blocking technique, weighted mean operators and Hardy’s inequality.Lecture Notes in Mathematics, no

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Oinarov, and S.Kh

R. Oinarov, and S.Kh. Shalginbaeva, Weighted Hardy inequalities on the cone of monotone sequences. Izv. Minister. Nauki Akad. Nauk Resp. Kaz. Ser. Fiz.-Mat.(1998), no. 1, 33–42

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G. Sinnamon, Hardy’s inequality and monotonicity, Function Spaces and Nonlinear Analysis. Mathe- matical Institute of the Academy of Sciences of the Czech Republic, Prague (2005), 292–310https: //www.math.uwo.ca/faculty/sinnamon/pdf/fsdona.pdf T. ¨Unver, Department of Mathematics, Kirikkale University, 71450 Yahsihan, Kirikkale, T¨urkiye Email address:tug...

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[1] [1]

Bennett, and K.-G

G. Bennett, and K.-G. Grosse-Erdmann, On series of positive terms.Houston J. Math.31(2005), no. 2, 541–586

2005

[2] [2]

Bennett, and K.-G

G. Bennett, and K.-G. Grosse-Erdmann, Weighted Hardy inequalities for decreasing sequences and func- tions.Math. Ann.334(2006), no. 3, 489–531

2006

[3] [3]

Bokayev, A

N. Bokayev, A. Gogatishvili, G. Karshygina, N. Kuzeubayeva, and T. ¨Unver, Reduction theorems for the discrete Hardy operator on the cones of monotone sequences.J. Math. Sci.291(2025), no. 2, 217–223

2025

[4] [4]

Cianchi, R

A. Cianchi, R. Kerman, B. Opic, and L. Pick. A sharp rearrangement inequality for fractional maximal operator.Studia Math.138(2000), no. 3, 277–284

2000

[5] [5]

Copson, Note on series of positive terms.J

E.T. Copson, Note on series of positive terms.J. London Math. Soc.2(1927), 9–12

1927

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Gogatishvili, M

A. Gogatishvili, M. Kˇ repela, R. Ol’hava, and L. Pick, Weighted inequalities for discrete iterated Hardy operators.Mediterr. J. Math.17(2022), no. 4, 132

2022

[7] [7]

Gogatishvili, and R

A. Gogatishvili, and R. Mustafayev, Iterated Hardy-type inequalities involving suprema.Math. Inequal. Appl.20(2017), no. 4, 901–927. 18 TU ˘GC ¸ E¨UNVER, AMIRAN GOGATISHVILI, NURZHAN BOKAYEV AND NURGUL KUZEUBAYEVA

2017

[8] [8]

Gogatishvili, B

A. Gogatishvili, B. Opic, and L. Pick, Weighted inequalities for Hardy-type operators involving suprema. Collect. Math.57 (2006), no. 3, 227–255

2006

[9] [9]

Gogatishvili, and L

A. Gogatishvili, and L. Pick, A reduction theorem for supremum operators.J. Comput. Appl. Math.208 (2007), no. 1, 270–279

2007

[10] [10]

Gogatishvili, L

A. Gogatishvili, L. Pick, and T. ¨Unver, Weighted inequalities for discrete iterated kernel operators.Math. Nachr.295(2022), no. 11, 1–26

2022

[11] [11]

Gogatishvili, and V.D

A. Gogatishvili, and V.D. Stepanov, Reduction theorems for operators on the cones of monotone functions. J. Math. Anal. Appl.405(2013), no. 1, 156–172

2013

[12] [12]

Gogatishvili, and V.D

A. Gogatishvili, and V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions.Russ. Math. Surv.64(2013), no. 4, 597–664

2013

[13] [13]

Grosse-Erdmann,The blocking technique, weighted mean operators and Hardy’s inequality.Lecture Notes in Mathematics, no

K.-G. Grosse-Erdmann,The blocking technique, weighted mean operators and Hardy’s inequality.Lecture Notes in Mathematics, no. 1679, Berlin, 1998

1998

[14] [14]

Oinarov, and S.Kh

R. Oinarov, and S.Kh. Shalginbaeva, Weighted Hardy inequalities on the cone of monotone sequences. Izv. Minister. Nauki Akad. Nauk Resp. Kaz. Ser. Fiz.-Mat.(1998), no. 1, 33–42

1998

[15] [15]

Sawyer, Boundedness of classical operators on classical Lorentz spaces.Studia Math.96(1990), no

E. Sawyer, Boundedness of classical operators on classical Lorentz spaces.Studia Math.96(1990), no. 2, 145–158

1990

[16] [16]

Sinnamon, Hardy’s inequality and monotonicity, Function Spaces and Nonlinear Analysis

G. Sinnamon, Hardy’s inequality and monotonicity, Function Spaces and Nonlinear Analysis. Mathe- matical Institute of the Academy of Sciences of the Czech Republic, Prague (2005), 292–310https: //www.math.uwo.ca/faculty/sinnamon/pdf/fsdona.pdf T. ¨Unver, Department of Mathematics, Kirikkale University, 71450 Yahsihan, Kirikkale, T¨urkiye Email address:tug...

2005