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Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities

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Pith's one-line read The paper proves that for n≥4, 1<p<n and 3≤k≤n−1, the deficit in the Hardy-Sobolev-Maz'ya inequality is bounded below by a constant times the Sobolev-gradient distance to the extremal manifold, raised to the sharp power max{2,p}.

desk verdict Substantial and probably true, but the spectral characterization in Theorem 1.1 has a real gap in the nodal-domain gluing step that needs fixing before the stability theorem rests on it. read the letter →

arxiv 2509.00814 v2 pith:2OPJ34JC submitted 2025-08-31 math.AP math.CA

Wei Dai , Jingze Fu , An Zhang This is my paper · ORCID
classification math.APmath.CA MSC 35A2335B3535J92
keywords Hardy-Sobolev-Maz'yainequalityquantitativestabilitysharpgradientnon-radialextremalscylindricalsymmetryp-Laplacespectralgapsingularweightconcentration-compactness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a sharp quantitative stability theorem for a class of Hardy-Sobolev-Maz'ya inequalities with the partial singular weight |y|^{-1}, where y lives in a k-dimensional subspace. It proves that if a function nearly attains the sharp constant in the inequality, then it must be close, in Sobolev-gradient norm, to one of the explicitly known cylindrically symmetric extremal functions. The closeness is measured with the optimal exponent max{2,p}, the same sharp exponent as in the classical Sobolev case. This appears to be the first stability result for non-radial extremals in this setting, and it matters because the partial singular weight breaks radial symmetry and rules out the usual ODE and Sturm-Liouville methods.

What carries the argument

The workhorse is the linearized p-Laplace operator L_v[φ]=−div(|Dv|^{p−2}Dφ+(p−2)|Dv|^{p−4}(Dv·Dφ)Dv), acting on L^2(R^n;|y|^{-1}v^{p^*_1−2}). Theorem 1.1 identifies its first two eigenvalues and eigenspaces: the first is (p−1)S^p∥v∥^{p−p^*_1}_{L^{p^*_1}(|y|^{-1})} with eigenspace span{v}, and the second is (p^*_1−1)S^p∥v∥^{p−p^*_1}_{L^{p^*_1}(|y|^{-1})} with eigenspace exactly T_vM. Because the second eigenspace is the tangent space, functions orthogonal to T_vM sit in a spectral gap, producing the refined spectral inequality of Proposition 3.4. A second ingredient is a weighted compact embedding D^{1,2}(R^n;|Dv|^{p−2})↪L^2(R^n;|y|^{-1}v^{p^*_1−2}), complemented for small p by a delicate Or

What would settle it

Produce a positive finite-energy solution of the Euler-Lagrange equation (1.6), for some k in [3,n−1], that is not of the form (1.5); that would invalidate the classification that defines M. A negative check would similarly be to classify k=2 extremals and test whether inequality (1.10) still holds there.

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Extended reading notes

Core claim

On the paper's own terms, Theorem 1.2 states that for n≥4, 1<p<n and 3≤k≤n−1, with γ=max{2,p}, every u∈D^{1,p}(R^n) satisfies δ(u) ≥ c inf_{v∈M} (∥D(u−v)∥_{L^p}/∥Du∥_{L^p})^γ, and the exponent γ is sharp. The extremal manifold M consists of the explicit family v_{a,λ,z'}(x)=aλ^{(n-p)/p} ((1+λ|y|)^2+|λz−z'|^2)^{-(n-p)/(2(p-1))}, with a≠0, λ>0, z'∈R^{n−k}. The proof establishes non-degeneracy of the Euler-Lagrange equation: the linearized p-Laplace operator L_v has discrete spectrum, its first eigenspace is span{v}, and its second eigenspace is exactly the tangent space T_vM=span{∂_λ v, ∂_{z'_1}v, …, ∂_{z'_{n−k}}v}. This spectral gap, combined with a new compact embedding adapted to the strong

Load-bearing premise

The theorem inherits the classification from [62] of all positive finite-energy extremals for 3≤k≤n−1; if that classification is incomplete, the manifold M would not be the true extremal set and the stated distance would point at the wrong target.

Editorial extensions

If this is right

  • Nearly extremal functions in the Hardy-Sobolev-Maz'ya inequality are quantitatively close, in Sobolev-gradient norm, to the explicit extremal family, with the optimal rate ε^{max{2,p}}.
  • The sharp exponent is the same as for the unweighted Sobolev inequality and does not depend on the weight dimension k, even though the extremal manifold itself depends on k.
  • The linearized problem is spectrally non-degenerate: the only directions that do not generate a positive quadratic cost are precisely the symmetry directions of the extremal manifold.
  • The theorem is global, holding for every u∈D^{1,p}(R^n), not just in a neighborhood of the extremal set.
  • The range currently covered is 3≤k≤n−1; the case k=2 is explicitly left open because the classification of extremals is itself unknown.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a classification of extremals for k=2 is ever completed, the same spectral-gap and compactness architecture would plausibly yield the same sharp exponent max{2,p} for the full range 2≤k≤n−1; this is an extrapolation, not a theorem in the paper.
  • The independence of the exponent from k hints that the partial weight changes the geometry of the extremal family but not the scaling rate of the deficit, suggesting the same exponent for other partially weighted inequalities with cylindrical symmetry.
  • The nodal-domain proof of the second eigenspace may transfer to linearized operators for other non-radial extremal problems, offering a template where ODE methods are unavailable.
  • The behavior of the optimal stability constant c(n,p,k) as a function of k and of the sharp constant S(n,p,k) is not addressed here and is a natural next question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper claims a sharp gradient stability estimate for a class of Hardy-Sobolev-Maz'ya inequalities with partial singular weight |y|^{-1}. For n≥4, 1<p<n, 3≤k≤n−1 and γ=max{2,p}, it asserts that the deficit δ(u) is bounded below by c(n,p,k) times the γ-th power of the Sobolev-gradient distance to the extremal manifold M given by the cylindrically symmetric functions (1.5), and that γ is optimal. The proof strategy follows Bianchi–Egnell and Figalli–Zhang: expand around a nearest extremal v∈M, prove non-degeneracy and a spectral gap for the linearized operator L_v (Theorem 1.1, Propositions 3.2 and 3.4), then use concentration-compactness and a Taylor expansion to obtain the global estimate (Theorem 1.2). The classification of all extremals is imported from Lin–Ma [62]; the case k=2 is left open.

Significance. If the proof is correct, this is an important result: it would be the first sharp gradient stability theorem for a Sobolev-type inequality with non-radial extremal functions, and the sharp exponent γ=max{2,p} being independent of the weight dimension k is a surprising and valuable feature. The paper contains substantial technical contributions, including a compact embedding adapted to the strong partial singularity, a spectral analysis of the linearized operator without the ODE reduction available in radial problems, and refined Taylor-type inequalities. The dependence on the external classification [62] is explicitly acknowledged and is a natural conditional limitation rather than an internal inconsistency. However, the central spectral characterization contains a gap that currently prevents the main theorem from being fully established.

major comments (3)
  1. [Section 3.1, Step 2 (proof of Theorem 1.1)] The argument that the local constants on the subdomains A_j coalesce is not valid. The text asserts that because φ̃≠0 in Ω1, continuity of g=φ/φ̃ forces adjacent constants to agree. But the common boundary of two A_j inside Ω1 is precisely a nodal set of φ̃, so φ̃=0 there; g may blow up and no unique continuation or matching argument is supplied. Moreover, the orthogonality computation gives c1 I1 + c2 I2 = 0 together with I1 + I2 = 0, which does not imply c1=c2 unless I1≠0 is shown. Since Propositions 3.2 and 3.4 require exactly E2=span{∂λv,∂_{z'}v}, this gap is load-bearing for Theorem 1.2.
  2. [Section 3.1, Step 1, case p=2n/(n+1)] The proof of (3.12) under g>0 is omitted in the critical case p=2n/(n+1). The sentence 'We omit the quite similar calculations here' is not acceptable for a parameter range that is included in Theorem 1.2 and in Proposition 3.4. The claimed contradiction from (p−2)(n^2+3n−4)X1 + (p−2)(k−1)X2 = 0 needs to be shown in detail, since the coefficient (p−2) changes sign in this range and the cancellation structure is delicate. Provide the full computation or a precise reference.
  3. [Section 3.1, Step 2 (nodal domain argument)] The assertion that every second eigenfunction of L_v has exactly two nodal domains is imported from Courant–Hilbert [23, VI.6] without addressing the fact that L_v is a degenerate/singular operator on R^n: the coefficients behave like powers of W=(1+|y|)^2+|z|^2 and contain 1/|y| terms. The standard nodal-domain count is for uniformly elliptic operators on bounded domains. A justification, or a reference valid in this degenerate setting, is needed before the two-domain decomposition and the subsequent matching argument can be used.
minor comments (6)
  1. [Section 4 heading] The heading reads 'Poof of Theorem 1.2'; it should be 'Proof of Theorem 1.2'.
  2. [Proposition 3.4 proof] The proof references 'Definition 3.7' for orthogonality, but no Definition 3.7 appears in the manuscript. The intended reference is presumably Definition 3.3.
  3. [Section 2 / Appendix A] Y0 is said to be defined at the beginning of Section 2, but the definition is not explicit in the text. Since Y0 plays a key role in Theorem A.1 and Lemma A.2, it should be stated clearly.
  4. [Proposition 3.4(1)] The weighted quotient in the right-hand side is typeset ambiguously. It should be written with explicit parentheses, e.g. |y|^{-1} ((v+C1|φ|)^{p*_1}/(v^2+|φ|^2))|φ|^2.
  5. [Remark 1.4] The sharpness constructions are described informally ('one can check', 'one can discover'). Since sharpness of the exponent is part of the theorem, the asymptotic expansions for δ(ui) and the right-hand side of (1.10) should be written out or placed in an appendix.
  6. [Lemma 2.3, Step 1] The symbol φ is used both for the weak limit in D^{1,p}(R^n) and for the local strong limit obtained by Rellich–Kondrachov; different notation would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the stability inequality is derived from the external Lin–Ma classification, a spectral gap proven independently, and standard inequalities; the named limitations are correctness risks, not circular inputs.

full rationale

The main theorem is derived from the Hardy–Sobolev–Maz'ya inequality (1.4), the external classification of extremals by Lin–Ma [62], and a spectral analysis of the linearized p-Laplacian L_v. The extremal family (1.5) and sharp constant S are imported from [62] as an external theorem; this is load-bearing, but it is not a self-citation and not an input fitted to the target stability inequality. The proof of Theorem 1.1 characterizes E_1 and E_2 via the Rayleigh quotient, second-variation computations, and PDE identities; the subsequent spectral gap (Proposition 3.2) and the expansion in Section 4 do not presuppose the target inequality (1.10). The sharp exponent gamma = max{2,p} is tested independently by explicit sequences in Remark 1.4. The paper's own Remark 1.3 explicitly limits the result to 3 <= k <= n-1, and the skeptical concern about the nodal-domain gluing step in Section 3.1 Step 2 is a possible correctness gap, not a circularity: no quantity in that step is defined in terms of the target inequality, and the step does not reduce to a fit or to a self-citation. Self-citations by the current authors appear only as background literature (e.g., [10,11,24-29,36,37,67]) and are not load-bearing. Thus the derivation chain is not circular; the central result has independent mathematical content relative to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new fitted parameters and no new physical or mathematical entities. Its central claim rests on the external classification of extremals [62] and on standard weighted inequalities and spectral theory. The proof itself is a mathematical derivation from these inputs.

assumptions (5)
  • domain assumption The Hardy-Sobolev-Maz'ya inequality (1.4) with sharp constant S and the complete classification of its extremal functions as the manifold M in (1.5), taken from Lin-Ma [62].
    The stability theorem is stated with respect to the extremal manifold M, and the proof of the spectral gap uses the explicit form of v in (1.5). This classification is not re-proved here, and k=2 is left open in Remark 1.3.
  • standard math Weighted functional inequalities used in Section 2: the weighted Hardy-Sobolev inequality (B.1), the Caffarelli-Kohn-Nirenberg inequality (B.2), and the weighted Hardy-type lemma B.4 from [31].
    These are cited as known results and are used to prove the compact embedding Proposition 2.1 and the estimates in Lemma 2.3.
  • standard math Second-order Taylor estimates for p-convex functionals and the pointwise expansion Lemma B.2 from Figalli-Zhang [56].
    These estimates are used throughout Section 4 to expand the deficit and isolate the leading spectral term.
  • standard math Spectral theory for self-adjoint compact inverses, the Rayleigh quotient characterization, Courant nodal domain theory, and the Liouville-type theorem [38, Theorem 9.11].
    Used in Proposition 3.1 and Theorem 1.1 to prove discreteness of the spectrum and to characterize the first two eigenspaces.
  • standard math The concentration-compactness principles of Lions, as adapted in Appendix A to the weighted setting.
    The proof of Theorem A.1 is a sketch that explicitly states it follows by the methods of [65, Theorem I.1] and [66, Theorem 2.4]; it underpins Lemma 4.1 and is load-bearing for the main theorem.

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Cite this review

Pith. "Pith review of Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities." pith.science (2026). https://pith.science/paper/2OPJ34JC

@misc{pith2026250900814,
  author       = {Pith},
  title        = {Pith review of: Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OPJ34JC}},
  note         = {Machine review of arXiv:2509.00814}
}
abstract

In this paper, we proved the sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities with partial (stronger) singular weight and non-radial extremal functions. Our result seems to be the first stability result for non-radial extremal functions. The presence of partial (stronger) singular weight brings substantial new challenges, requiring us to significantly refine the techniques from Deng-Tian 2025, Figalli-Neumayer 2019 and Figalli-Zhang 2022, and introduce some new ideas to handle both the cylindrical symmetry of non-radial extremal functions and the partial (stronger) singular weight structure. Key technical innovations include new compact embedding with strong singularity, non-degeneracy and spectral property of the linearized operator $\mathcal{L}_{v}$ generated by non-radial extremal function $v$ and new refined spectral inequalities, which are crucial for our analysis. Since the extremal function $v$ is non-radial, ODE approach fails, we use binary PDE to prove the spectral property of $\mathcal{L}_{v}$. Surprisingly, the sharp exponent $\gamma=\max\{2,p\}$ in our sharp gradient stability inequality (1.12) is independent of the partial weight dimension $k$, while the extremal manifold depends on $k$.

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