Weighted Hardy Inequality in $l_2$

Ivan Gadjev

arxiv: 2606.09057 · v1 · pith:WSR6UVDZnew · submitted 2026-06-08 · 🧮 math.CA

Weighted Hardy Inequality in l₂

Ivan Gadjev This is my paper

Pith reviewed 2026-06-27 14:48 UTC · model grok-4.3

classification 🧮 math.CA

keywords weighted Hardy inequalitybest constantsconvergence rateextremal sequencesfinite sequencesl2 spaces

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The pith

The exact rate of convergence for the smallest constant dn in the weighted Hardy inequality is established, along with an almost extremal sequence.

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

The paper examines the smallest constant dn making the weighted Hardy inequality hold for every real sequence a of length n. It proves the precise asymptotic rate at which dn converges as n tends to infinity. The work also identifies an almost extremal sequence that nearly attains the bound. This sharpens the understanding of how the inequality performs in finite weighted l2 spaces.

Core claim

The infimum dn of constants satisfying the inequality sum (1/k sum_{j=1 to k} a_j)^2 k^epsilon <= dn sum a_k^2 k^epsilon for k=1 to n admits an exact rate of convergence as n grows, and an almost extremal sequence is constructed.

What carries the argument

dn, the infimum of constants C such that the weighted Hardy inequality holds for all real sequences a of length n.

Load-bearing premise

That dn admits an exact asymptotic description via some test sequence or variational method.

What would settle it

A sequence of length n for which the ratio of the left-hand side to the right-hand side of the inequality exceeds the claimed asymptotic expression for dn by more than the stated rate.

read the original abstract

We study the behaviour of the smallest possible constant $d_n$ in weighted Hardy inequality $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2 k^\epsilon\le d(n,\epsilon)\,\sum_{k=1}^{n}{a_k^2}\,k^\epsilon $$ The exact rate of convergence of $d_n$ is established and the ``almost extremal'' sequence is found.

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 claims the exact rate of convergence for the best constant d_n in this finite-n weighted Hardy inequality on l2 plus an almost extremal sequence, but the abstract supplies no method or verification.

read the letter

The main takeaway is that the authors establish the precise asymptotic rate for d_n (the infimum of constants making the weighted inequality hold for sequences of length n) and produce an almost extremal sequence. This is the stated contribution.

What is new is the exact rate together with the explicit sequence in the weighted finite-dimensional l2 setting. The inequality itself is the standard form of the Hardy averaging operator with matching weights k^epsilon on both sides, so the variational definition of d_n as the operator norm is the usual one.

The paper does one thing cleanly: it isolates the finite-n case and claims a sharp rate plus a test sequence that nearly achieves the bound. That kind of concrete information can be useful for people who need to control the constant when n is large but finite.

The soft spot is obvious from the abstract alone: there is no derivation, no sketch of how the rate was obtained, and no indication whether the argument uses a variational characterization, an explicit construction, or something else. Without seeing the steps it is impossible to tell whether the upper and lower bounds actually meet or whether the sequence is shown to be almost extremal in a rigorous way. The stress-test note finds no internal inconsistency in the setup, which is fair, but that does not substitute for a proof.

This is for specialists working on Hardy inequalities and sharp constants in analysis. A reader who already cares about rates of convergence in discrete inequalities might extract something usable if the details hold. It is thin on evidence as presented, but the claim is specific enough that a serious editor should send it to referees rather than desk-reject it so the proof can be checked.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the smallest constant d_n in the weighted Hardy inequality for real sequences a of length n: ∑_{k=1}^n [(1/k) ∑_{j=1}^k a_j]^2 k^ε ≤ d(n,ε) ∑_{k=1}^n a_k^2 k^ε. It claims to establish the exact rate of convergence of d_n (presumably as n→∞) and to identify an almost extremal sequence.

Significance. If the central claims hold with rigorous proofs, the result supplies precise asymptotics for the operator norm of the weighted Hardy averaging operator on the finite-dimensional weighted ℓ² space. This is a natural finite-n counterpart to the classical Hardy inequality and could inform variational characterizations or numerical approximations in weighted sequence spaces.

minor comments (2)

The abstract uses both d_n and d(n,ε) without clarifying whether ε is fixed or the dependence on ε is tracked in the rate; this notation should be unified in the introduction.
No explicit statement appears on whether ε is assumed positive, zero, or in a specific range; the range of validity for the claimed rate should be stated clearly near the definition of d_n.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines d_n variationally as the infimum of constants satisfying the weighted Hardy inequality for sequences of length n, which is the standard operator-norm characterization. It claims to derive the exact asymptotic rate of convergence of this quantity and an almost-extremal sequence. No equations or steps in the abstract reduce a claimed prediction to a fitted input by construction, invoke self-citations as load-bearing uniqueness results, or smuggle ansatzes. The derivation is presented as a direct analysis of the finite-dimensional weighted l2 operator and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; relies on standard properties of real sequences and l2 norm but no specific free parameters or invented entities are mentioned.

axioms (1)

standard math Standard axioms of real numbers and finite sums
Implicit in defining the inequality and the constant d_n.

pith-pipeline@v0.9.1-grok · 5588 in / 875 out tokens · 19682 ms · 2026-06-27T14:48:56.846598+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 1 canonical work pages

[1]

G. H. Hardy,Notes on some points in the integral calculus, LI. On Hilbert’s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals, Messenger Math. 48 (1919), 107–112

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G. H. Hardy,Notes on a theorem of Hilbert, Math. Z. 6 (1920), 314–317

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G. H. Hardy,Notes on some points in the integral calculus, LX. An inequality between inte- gral, Messenger Math. 54 (1925), 150–156

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Landau,Letter to G

E. Landau,Letter to G. H. Hardy, June 21, 1921

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Landau,A note on a theorem concerning series of positive terms: Extract from a letter of Prof

E. Landau,A note on a theorem concerning series of positive terms: Extract from a letter of Prof. E. Landau to Prof. I. Schur, J. London Math. Soc. 1 (1926), 38–39

1926
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Widom, H., On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc., 88(1), 491-522 (1958)

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Widom, H., Extreme eigenvalues of translation kernels, Trans. Amer. Math. Soc., 100(2), 252-262 (1961)

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Wilf, H., On finite sections of the classical inequalities, Nederl

H.S. Wilf, H., On finite sections of the classical inequalities, Nederl. Akad. Wet. Amsterdam Proc. Ser. A, 1962 - core.ac.uk

1962
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Bruijn, N. G. DE and H. S. Wilf, On Hilbert’s Inequality in n dimensions, Bull. Amer. Math. Soc
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ˇCiˇ zmeˇ sija, J

A. ˇCiˇ zmeˇ sija, J. Peˇ cari´ c, Mixed means and Hardy’s inequality, Math. Inequal. Appl. 1(1998), no. 4, 491-506

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Stampach, Asymptotic spectral properties of the Hilbert L-matrix, SIAM J

F. Stampach, Asymptotic spectral properties of the Hilbert L-matrix, SIAM J. Matrix Anal. Appl. 43 (2022), 1658-1679

2022
[12]

Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen Uluchev,Hardy’s inequali- ties in finite dimensional Hilbert spaces, Proc

Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen Uluchev,Hardy’s inequali- ties in finite dimensional Hilbert spaces, Proc. Amer. Math. Soc. 149 (2021), 2515-2529, DOI:https://doi.org/10.1090/proc/15467

work page doi:10.1090/proc/15467 2021
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Dai, Y-T

W-Gao Long, D. Dai, Y-T. Li, X-S. Wang, Asymptotics of orthogonal polynomials with asymptotic Freud-like weights, Stud. Appl. Math. 144 (2020), 133–163

2020
[14]

Dimitrov, Ivan Gadjev, and Mourad E

Dimitar K. Dimitrov, Ivan Gadjev, and Mourad E. H. Ismail, Sharp Hardy’s inequalities in Hilbert spaces, J. Spectr. Theory 14 (2024), no. 3, pp. 1243-1256

2024
[15]

Ivan Gadjev, Vasil Gochev, On the constant in the Hardy inequality for finite sequences, Mathematical Inequalities & Applications, vol:26, issue: 2, 2023, pages: 493-498 WEIGHTED HARDY INEQUALITY INl 2 9

2023
[16]

Opic and A

B. Opic and A. Kufner,Hardy-type inequalities, Longman Scientific & Technical, 1990

1990
[17]

Kufner, L.-E

A. Kufner, L.-E. Person,Weighted Inequalities of Hardy Type, 2nd ed., World Scientific, Singapore, 2017

2017
[18]

Kufner, L

A. Kufner, L. Maligranda, and L.-E. Person,The Hardy Inequality: About its History and Some Related Results, Vydavatelsk´ y servis, 2007. Department of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Email address:gadjev@fmi.uni-sofia.bg

2007

[1] [1]

G. H. Hardy,Notes on some points in the integral calculus, LI. On Hilbert’s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals, Messenger Math. 48 (1919), 107–112

1919

[2] [2]

G. H. Hardy,Notes on a theorem of Hilbert, Math. Z. 6 (1920), 314–317

1920

[3] [3]

G. H. Hardy,Notes on some points in the integral calculus, LX. An inequality between inte- gral, Messenger Math. 54 (1925), 150–156

1925

[4] [4]

Landau,Letter to G

E. Landau,Letter to G. H. Hardy, June 21, 1921

1921

[5] [5]

Landau,A note on a theorem concerning series of positive terms: Extract from a letter of Prof

E. Landau,A note on a theorem concerning series of positive terms: Extract from a letter of Prof. E. Landau to Prof. I. Schur, J. London Math. Soc. 1 (1926), 38–39

1926

[6] [6]

Widom, H., On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc., 88(1), 491-522 (1958)

1958

[7] [7]

Widom, H., Extreme eigenvalues of translation kernels, Trans. Amer. Math. Soc., 100(2), 252-262 (1961)

1961

[8] [8]

Wilf, H., On finite sections of the classical inequalities, Nederl

H.S. Wilf, H., On finite sections of the classical inequalities, Nederl. Akad. Wet. Amsterdam Proc. Ser. A, 1962 - core.ac.uk

1962

[9] [9]

Bruijn, N. G. DE and H. S. Wilf, On Hilbert’s Inequality in n dimensions, Bull. Amer. Math. Soc

[10] [10]

ˇCiˇ zmeˇ sija, J

A. ˇCiˇ zmeˇ sija, J. Peˇ cari´ c, Mixed means and Hardy’s inequality, Math. Inequal. Appl. 1(1998), no. 4, 491-506

1998

[11] [11]

Stampach, Asymptotic spectral properties of the Hilbert L-matrix, SIAM J

F. Stampach, Asymptotic spectral properties of the Hilbert L-matrix, SIAM J. Matrix Anal. Appl. 43 (2022), 1658-1679

2022

[12] [12]

Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen Uluchev,Hardy’s inequali- ties in finite dimensional Hilbert spaces, Proc

Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen Uluchev,Hardy’s inequali- ties in finite dimensional Hilbert spaces, Proc. Amer. Math. Soc. 149 (2021), 2515-2529, DOI:https://doi.org/10.1090/proc/15467

work page doi:10.1090/proc/15467 2021

[13] [13]

Dai, Y-T

W-Gao Long, D. Dai, Y-T. Li, X-S. Wang, Asymptotics of orthogonal polynomials with asymptotic Freud-like weights, Stud. Appl. Math. 144 (2020), 133–163

2020

[14] [14]

Dimitrov, Ivan Gadjev, and Mourad E

Dimitar K. Dimitrov, Ivan Gadjev, and Mourad E. H. Ismail, Sharp Hardy’s inequalities in Hilbert spaces, J. Spectr. Theory 14 (2024), no. 3, pp. 1243-1256

2024

[15] [15]

Ivan Gadjev, Vasil Gochev, On the constant in the Hardy inequality for finite sequences, Mathematical Inequalities & Applications, vol:26, issue: 2, 2023, pages: 493-498 WEIGHTED HARDY INEQUALITY INl 2 9

2023

[16] [16]

Opic and A

B. Opic and A. Kufner,Hardy-type inequalities, Longman Scientific & Technical, 1990

1990

[17] [17]

Kufner, L.-E

A. Kufner, L.-E. Person,Weighted Inequalities of Hardy Type, 2nd ed., World Scientific, Singapore, 2017

2017

[18] [18]

Kufner, L

A. Kufner, L. Maligranda, and L.-E. Person,The Hardy Inequality: About its History and Some Related Results, Vydavatelsk´ y servis, 2007. Department of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Email address:gadjev@fmi.uni-sofia.bg

2007