On the Constant and Extremal Function for Weighted Hardy Inequality in $L_p$

Ivan Gadjev

arxiv: 2606.09067 · v1 · pith:AAM6CJGUnew · submitted 2026-06-08 · 🧮 math.CA

On the Constant and Extremal Function for Weighted Hardy Inequality in L_p

Ivan Gadjev This is my paper

Pith reviewed 2026-06-27 14:46 UTC · model grok-4.3

classification 🧮 math.CA

keywords Hardy inequalityweighted inequalitybest constantextremal functionconvergence rateintegral inequality

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The pith

The smallest constant d(a,b,p,ε) in the weighted Hardy inequality converges at an exact rate, with an almost extremal function identified.

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

This paper studies the infimum constant d(a,b,p,ε) that makes the weighted Hardy inequality hold on a finite interval [a,b] for 2 ≤ p < ∞. It determines the precise asymptotic behavior of this constant as the parameters vary. The work also produces a concrete function that nearly attains equality in the inequality. A reader would care because the result gives a sharp quantitative version of a classical integral inequality used throughout analysis.

Core claim

The paper establishes the exact rate of convergence of the smallest constant d(a,b,p,ε) in the weighted Hardy inequality ∫_a^b (1/x ∫_a^x f(t) dt)^p x^ε dx ≤ d(a,b,p,ε) ∫_a^b [f(x)]^p x^ε dx, and identifies the almost extremal function that nearly saturates the bound.

What carries the argument

The best constant d(a,b,p,ε), defined as the infimum of multipliers making the weighted Hardy inequality true, together with the almost extremal function constructed to approach equality.

If this is right

The weighted Hardy inequality holds with a constant whose dependence on a, b, p, and ε is asymptotically sharp.
The almost extremal function supplies a concrete test case for near-equality in the inequality.
The established convergence rate permits precise estimates when the interval length or the parameter ε approaches limiting values.

Where Pith is reading between the lines

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

The rate of convergence could be used to obtain error bounds when the Hardy inequality is applied to approximate problems on shrinking intervals.
The construction of the almost extremal function may suggest candidate optimizers for related inequalities with different weights.

Load-bearing premise

The integrals in the weighted Hardy inequality are assumed to be well-defined and finite for the functions f under consideration.

What would settle it

A function f on [a,b] for which the ratio of the left-hand side to the right-hand side exceeds d(a,b,p,ε) by more than the stated convergence rate would disprove the result.

read the original abstract

We study the behaviour of the smallest possible constant $d(a,b, p,\epsilon)$ in Hardy inequality $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^px^{\epsilon}\,dx\leq d(a,b,p,\epsilon)\,\int_a^b [f(x)]^px^{\epsilon}\, dx, \quad 2\le p<\infty. $$ The exact rate of convergence of $d(a,b,p,\epsilon)$ is established and the ``almost extremal'' function 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 an exact asymptotic rate for the best constant in this finite-interval weighted Hardy inequality plus an almost extremal function, but the abstract alone gives no evidence the derivation holds.

read the letter

The new piece is the claimed exact rate of convergence for d(a,b,p,ε) as the weight parameter ε varies, together with an explicit almost extremal function for the inequality on a bounded interval [a,b]. That is narrower than the usual Hardy setting but could matter for people who need sharp constants on finite domains.

The work is honest in scope: it stays inside the stated range 2 ≤ p < ∞ and focuses on making the constant precise rather than reproving the inequality itself. If the proofs are complete and the rate is indeed sharp, that is a modest but usable refinement.

The main soft spot is that nothing in the abstract or the supplied description shows how the rate is derived. Without a proof sketch or even a hint at the method, it is impossible to judge whether the argument is correct or whether hidden assumptions on a, b, and ε are being used. The finite-interval restriction also means the result does not speak to the more common infinite-interval or power-weight cases that dominate the literature.

A reader who already works on sharp constants in one-dimensional inequalities might find the explicit almost extremal function worth checking. For everyone else the paper is too narrow and too lightly documented to be worth time.

I would send it to a referee only if the full manuscript contains a self-contained, verifiable derivation of the rate; otherwise it should be desk-rejected for lack of evidence.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the smallest constant d(a,b,p,ε) in the weighted Hardy inequality ∫_a^b [(1/x)∫_a^x f(t) dt]^p x^ε dx ≤ d(a,b,p,ε) ∫_a^b [f(x)]^p x^ε dx for 2 ≤ p < ∞. It claims to establish the exact rate of convergence of d(a,b,p,ε) and to construct an almost extremal function.

Significance. If the derivations hold, the result would supply precise asymptotics for the best constant together with nearly optimal functions in a weighted Hardy inequality, which is of interest in real analysis for sharpening constants and understanding extremals.

minor comments (2)

[Abstract] Abstract: the statement that the 'exact rate of convergence of d(a,b,p,ε)' is established does not specify the limiting process (e.g., ε→0, b-a→0, or a→0 with b fixed). This parameter must be stated explicitly in the abstract and introduction.
The assumption that the integrals are finite for the admissible f and parameters 2≤p<∞ should be stated with the precise range of a,b,ε for which the expressions are well-defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript and recommending minor revision. No major comments appear in the report, indicating that the core claims on the asymptotic rate of d(a,b,p,ε) and the construction of the almost extremal function are viewed as sound.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the exact rate of convergence for the best constant d(a,b,p,ε) in the weighted Hardy inequality and constructs an almost extremal function. No equations, definitions, or steps in the provided abstract or reader's summary reduce any claimed prediction or result to a fitted input, self-definition, or self-citation chain. The central result is presented as a direct mathematical derivation from the inequality statement itself, with no load-bearing reliance on prior author work or renaming of known patterns. This is the expected outcome for a self-contained analysis in real analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard real-analysis background; no free parameters, invented entities, or ad-hoc axioms are indicated.

axioms (1)

standard math Standard properties of Lebesgue integration and L_p norms on finite intervals
The inequality is written in integral form and assumes the expressions are defined for f in L_p.

pith-pipeline@v0.9.1-grok · 5606 in / 1149 out tokens · 26892 ms · 2026-06-27T14:46:46.921785+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 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

1919
[2]

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 integral, Messenger Math. 54 (1925), 150–156

1925
[4]

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

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Tomaselli, G, A class of inequalities, Boll. Un. Mat. Ital. 2 (1969), 622-631. MR 41 # 411

1969
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Talenti, G, Observazioni sopra una classe di disuguaglianze (Italian), Rend. Sem. mat. Fis. Milano 39 (1969), 171-185. MR 43 # 6380

1969
[7]

44 (1972), 31-38

Muckenhoupt, B., Hardy’s inequality with weights, Studia math. 44 (1972), 31-38. MR 47 # 418

1972
[8]

Kufner, L

A. Kufner, L. Maligranda, and L.-E. Person,The Hardy Inequality: About its History and Some Related Results, Vydavatelsk´ y servis, 2007

2007
[9]

Kufner, L.-E

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

2017
[10]

Dimitrov, Ivan Gadjev, Mourad E

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

2024
[11]

St. Kliment Ohrid- ski

F. Gesztesy and M.M.H. Pang, Optimal power-weighted Birman-Hardy-Rellich-type inequal- ities on finite Intervals and annuli. 2025, https://doi.org/10.1112/blms.70063 Department of Mathematics and Informatics, Sofia University “St. Kliment Ohrid- ski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Email address:gadjev@fmi.uni-sofia.bg

work page doi:10.1112/blms.70063 2025

[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 integral, Messenger Math. 54 (1925), 150–156

1925

[4] [4]

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

1958

[5] [5]

Tomaselli, G, A class of inequalities, Boll. Un. Mat. Ital. 2 (1969), 622-631. MR 41 # 411

1969

[6] [6]

Talenti, G, Observazioni sopra una classe di disuguaglianze (Italian), Rend. Sem. mat. Fis. Milano 39 (1969), 171-185. MR 43 # 6380

1969

[7] [7]

44 (1972), 31-38

Muckenhoupt, B., Hardy’s inequality with weights, Studia math. 44 (1972), 31-38. MR 47 # 418

1972

[8] [8]

Kufner, L

A. Kufner, L. Maligranda, and L.-E. Person,The Hardy Inequality: About its History and Some Related Results, Vydavatelsk´ y servis, 2007

2007

[9] [9]

Kufner, L.-E

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

2017

[10] [10]

Dimitrov, Ivan Gadjev, Mourad E

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

2024

[11] [11]

St. Kliment Ohrid- ski

F. Gesztesy and M.M.H. Pang, Optimal power-weighted Birman-Hardy-Rellich-type inequal- ities on finite Intervals and annuli. 2025, https://doi.org/10.1112/blms.70063 Department of Mathematics and Informatics, Sofia University “St. Kliment Ohrid- ski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Email address:gadjev@fmi.uni-sofia.bg

work page doi:10.1112/blms.70063 2025