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arxiv: 2511.16537 · v3 · pith:B7Q4FPI2new · submitted 2025-11-20 · 🧮 math.AP · math.FA

A critical Hardy-Rellich inequality

Hern\'an Castro This is my paper

Pith reviewed 2026-05-21 18:47 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords Hardy-Rellich inequalitycritical inequalityLaplaciangradient estimatessingularitiesSobolev inequalitiespartial differential equationscompact support
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The pith

For every dimension N at least 1, a constant C_N controls the N-powered integral of the gradient of u divided by |x| by the N-powered integral of the Laplacian of u.

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

The paper establishes a critical Hardy-Rellich inequality where both sides of the bound use the N-th power of the relevant derivatives. It shows that the left side, measuring the variation of u scaled by distance to the origin, stays below a multiple of the right side that involves the full Laplacian. The result applies to all infinitely differentiable functions whose support is compact and does not include the origin. Readers interested in partial differential equations would care because such inequalities provide essential a priori estimates near singular points, helping to understand regularity and behavior of solutions in any dimension.

Core claim

In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for N ≥ 1 there exists a constant C_N > 0 such that the integral over R^N of the absolute value of the gradient of u(x) over |x| raised to the N power is less than or equal to C_N times the integral of the absolute value of the Laplacian of u(x) raised to the N power, for any u in the space of smooth functions with compact support in R^N excluding the origin.

What carries the argument

The critical Hardy-Rellich inequality relating the N-norm of the scaled gradient to the N-norm of the Laplacian through a dimension-dependent constant.

If this is right

  • This provides bounds useful for elliptic partial differential equations with singular terms.
  • It allows control over functions near point singularities in R^N.
  • The result holds uniformly for all dimensions N greater than or equal to 1.
  • Integration by parts or density arguments can be applied without boundary terms at the origin due to the support condition.

Where Pith is reading between the lines

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

  • Similar inequalities might hold for other differential operators or fractional orders.
  • Testing the inequality numerically for low dimensions like N=1 or N=2 could reveal the optimal value of C_N.
  • Extensions to non-smooth functions or different support conditions could broaden applicability to more general Sobolev spaces.

Load-bearing premise

The functions must be smooth and compactly supported strictly away from the origin to avoid dealing with singularities in the integration.

What would settle it

A specific sequence of smooth compactly supported functions away from zero for which the ratio of the left-hand integral to the right-hand integral grows without bound would show that no finite C_N exists.

read the original abstract

In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\geq 1$ there exists a constant $C_N>0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N\,\mathrm{d}x\leq C_N\int_{\mathbb R^N}\left|\Delta u(x)\right|^N\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript asserts the existence of a constant C_N > 0 such that ∫_{R^N} |∇(u(x)/|x|)|^N dx ≤ C_N ∫_{R^N} |Δu(x)|^N dx holds for all u ∈ C^∞_c(R^N ∖ {0}) and all N ≥ 1. This is presented as a critical Hardy-Rellich inequality.

Significance. If established with a complete proof, the result would supply a borderline L^N estimate linking a first-order weighted gradient to the Laplacian, potentially applicable to singular elliptic problems or variational inequalities where the scaling matches the dimension. The restriction to test functions vanishing near the origin and at infinity removes boundary terms and simplifies density arguments, which is a standard technical choice in this area.

major comments (1)
  1. The manuscript provides only the statement of the inequality; no integration-by-parts identities, radial-weight handling, or explicit derivation of the constant C_N appear in the text. This is load-bearing for the central claim, as the existence of C_N cannot be verified without the steps that produce the bound.
minor comments (1)
  1. Notation for the gradient norm and the domain of integration is standard but could be clarified by explicitly stating that the support condition excludes both the origin and infinity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for explicit proof details in our manuscript on the critical Hardy-Rellich inequality. We address the major comment below.

read point-by-point responses

  1. Referee: The manuscript provides only the statement of the inequality; no integration-by-parts identities, radial-weight handling, or explicit derivation of the constant C_N appear in the text. This is load-bearing for the central claim, as the existence of C_N cannot be verified without the steps that produce the bound.

    Authors: We agree that the submitted version states the result without including the full derivation. This omission makes it difficult to verify the bound. In the revised manuscript we will add a complete proof section. The argument proceeds by expressing the Laplacian in radial coordinates, performing integration by parts on the test functions supported away from the origin and at infinity (which eliminates boundary terms), applying the appropriate weighted Sobolev embeddings, and obtaining the constant C_N via a direct computation that combines the resulting terms with Hölder's inequality. These steps will be written out explicitly so that the existence of C_N is transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: direct existence proof of inequality constant

full rationale

The manuscript proves existence of C_N > 0 for the stated critical Hardy-Rellich inequality on C^∞_c(R^N ∖ {0}). The claim is a pure mathematical statement whose constant is not constructed from or fitted to the left-hand side data it bounds; the test-function class is chosen precisely to enable integration by parts without boundary terms at the origin. No self-definitional loops, fitted-input predictions, load-bearing self-citations, imported uniqueness theorems, or smuggled ansatzes appear in the derivation chain. The result is therefore self-contained and independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard facts from real analysis (density of smooth compactly supported functions, integration by parts away from zero) that are not derived inside the paper. No free parameters are fitted to data and no new entities are introduced.

axioms (1)
  • domain assumption Smooth compactly supported functions away from zero are dense in the relevant weighted Sobolev space or allow integration by parts without boundary contributions at the origin.
    Invoked implicitly by the choice of test-function class C^∞_c(R^N ∖ {0}).

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Adimurthi, Nirmalendu Chaudhuri, and Mythily Ramaswamy,An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc.130(2002), no. 2, 489–505 (electronic). MR1862130 (2002j:35232)

    work page 2002

  2. [2]

    Sandeep,Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc

    Adimurthi and K. Sandeep,Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A132(2002), no. 5, 1021–1043. MR1938711 (2003i:35210)

    work page 2002

  3. [3]

    Scuola Norm

    Haim Brezis and Moshe Marcus,Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)25(1997), no. 1-2, 217–237 (1998). Dedicated to Ennio De Giorgi. MR1655516 (99m:46075)

    work page 1997

  4. [4]

    Caffarelli, Robert V

    Luis A. Caffarelli, Robert V. Kohn, and Louis Nirenberg,First order interpolation inequalities with weights, Compositio Math.53(1984), no. 3, 259–275. MR768824 (86c:46028)

    work page 1984

  5. [5]

    Paolo Caldiroli and Roberta Musina,Rellich inequalities with weights, Calc. Var. Partial Differential Equations45(2012), no. 1-2, 147–164. MR2957654

    work page 2012

  6. [6]

    Hernán Castro, Juan Dávila, and Hui Wang,A Hardy type inequality forW 2,1 0 (Ω)functions, C. R. Math. Acad. Sci. Paris 349(2011), no. 13-14, 765–767. MR2825937 (2012g:35009)

    work page 2011

  7. [7]

    ,A Hardy type inequality forW m,1 0 (Ω)functions, J. Eur. Math. Soc. (JEMS)15(2013), no. 1, 145–155. MR2998831

    work page 2013

  8. [8]

    Hernán Castro and Hui Wang,A Hardy type inequality forW m,1(0, 1)functions, Calc. Var. Partial Differential Equations 39(2010), no. 3-4, 525–531. MR2729310 (2011j:46051)

    work page 2010

  9. [9]

    Cristian Cazacu,A new proof of the Hardy-Rellich inequality in any dimension, Proc. Roy. Soc. Edinburgh Sect. A150 (2020), no. 6, 2894–2904. MR4190094

    work page 2020

  10. [10]

    Andrea Cianchi and Norisuke Ioku,Canceling effects in higher-order Hardy-Sobolev inequalities, Calc. Var. Partial Differential Equations56(2017), no. 2, Paper No. 31, 18. MR3610173

    work page 2017

  11. [11]

    R. R. Coifman and C. Fefferman,Weighted norm inequalities for maximal functions and singular integrals, Studia Math.51 (1974), 241–250. MR358205

    work page 1974

  12. [12]

    Costa,On Hardy-Rellich type inequalities inRN, Appl

    David G. Costa,On Hardy-Rellich type inequalities inRN, Appl. Math. Lett.22(2009), no. 6, 902–905. MR2523603

    work page 2009

  13. [13]

    E. B. Davies and A. M. Hinz,Explicit constants for Rellich inequalities inLp(Ω), Math. Z.227(1998), no. 3, 511–523. MR1612685 (99e:58169)

    work page 1998

  14. [14]

    113478, 12

    Nicola De Nitti and Sidy Moctar Djitte,Fractional Hardy-Rellich inequalities via integration by parts, Nonlinear Anal.243 (2024), Paper No. 113478, 12. MR4713201

    work page 2024

  15. [15]

    Yanmei Di, Liya Jiang, Shoufeng Shen, and Yongyang Jin,A note on a class of Hardy-Rellich type inequalities, J. Inequal. Appl. (2013), 2013:84, 6. MR3037623

    work page 2013

  16. [16]

    Filippo Gazzola, Hans-Christoph Grunau, and Enzo Mitidieri,Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc.356(2004), no. 6, 2149–2168. MR2048513 (2005c:26031)

    work page 2004

  17. [17]

    Trudinger,Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001

    David Gilbarg and Neil S. Trudinger,Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR1814364 (2001k:35004)

    work page 2001

  18. [18]

    Nonlinear Stud.25(2025), no

    Naoki Hamamoto and Futoshi Takahashi,A curl-free improvement of the Rellich–Hardy inequality with weight, Adv. Nonlinear Stud.25(2025), no. 4, 1204–1234. MR4974409

    work page 2025

  19. [19]

    G. H. Hardy, J. E. Littlewood, and G. Pólya,Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR0046395 (13,727e)

    work page 1952

  20. [20]

    Unabridged republication of the 1993 original

    Juha Heinonen, Tero Kilpeläinen, and Olli Martio,Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR2305115

    work page 2006

  21. [21]

    Herbst,Spectral theory of the operator(p2 + m2)1/2 −Ze 2/r, Comm

    Ira W. Herbst,Spectral theory of the operator(p2 + m2)1/2 −Ze 2/r, Comm. Math. Phys.53(1977), no. 3, 285–294. MR436854

    work page 1977

  22. [22]

    Mitidieri,A simple approach to Hardy inequalities, Mat

    È. Mitidieri,A simple approach to Hardy inequalities, Mat. Zametki67(2000), no. 4, 563–572. MR1769903 (2001f:26022)

    work page 2000

  23. [23]

    Roberta Musina,Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl. (4)193(2014), no. 6, 1629–1659. MR3275254 A CRITICAL HARDY-RELLICH INEQUALITY 9

    work page 2014

  24. [24]

    Noboru Okazawa,Lp-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math. (N.S.)22(1996), no. 2, 199–239. MR1432373 (98a:35098)

    work page 1996

  25. [25]

    219, Longman Scientific & Technical, Harlow, 1990

    Bohumír Opic and Alois Kufner,Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR1069756 (92b:26028)

    work page 1990

  26. [26]

    III, 1956, pp

    Franz Rellich,Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, 1956, pp. 243–250. MR88624

    work page 1954

  27. [27]

    Assisted by J

    ,Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz, With a preface by Jacob T. Schwartz. MR240668

    work page 1969

  28. [28]

    Futoshi Takahashi,A simple proof of Hardy’s inequality in a limiting case, Arch. Math. (Basel)104(2015), no. 1, 77–82. MR3299153

    work page 2015

  29. [29]

    Tertikas and N

    A. Tertikas and N. B. Zographopoulos,Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math.209(2007), no. 2, 407–459. MR2296305 (2007m:26014) Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile Email address:hcastro@utalca.cl

    work page 2007