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Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities

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Pith's one-line read This paper proves that every positive finite-energy cylindrically symmetric solution of the critical p-Laplace Hardy-Sobolev-Maz'ya equation is an explicit rescaling and translation of one profile, which yields the best constant and…

desk verdict A genuinely new classification and identity, but the k≤2p branch of the proof has a μ=0 cutoff error that must be fixed before the main theorem is established. read the letter →

arxiv 2412.09033 v1 pith:Z73WIBBK submitted 2024-12-12 math.AP

Daowen Lin , Xi-Nan Ma This is my paper · ORCID
classification math.AP MSC 35J9235B3335B06
keywords Hardy-Sobolev-Maz'yainequalityp-LaplaceequationbestconstantextremalfunctionscylindricalsymmetrydifferentialidentityclassificationofsolutionscriticalweightedSobolev
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 aims to prove that the positive extremals of a class of Hardy-Sobolev-Maz'ya inequalities have a completely explicit shape. For $s=1$, $n\ge 4$, and $3\le k\le n-1$, the authors show that every positive finite-energy cylindrically symmetric solution of the associated $p$-Laplace equation must be a dilation and a $z$-translation of the single profile $u_0$ in (1.3). Combined with the cylindrically symmetric minimization result of Secchi-Smets-Willem, the classification gives the best constant and the extremal functions for inequality (1.1). If the proof is correct, the paper settles the Alvino-Ferone-Trombetti conjecture for $k\ge 3$ and leaves the two-dimensional singular set case $k=2$ open.

What carries the argument

The carrying mechanism is the differential identity in Proposition 3.1, a $p$-Laplace generalization of the Garofalo-Vassilev identity. After changing variables $u=\kappa\,\varphi^{-(n-p)/p}$, the identity writes $\mathrm{div}\,Y$ as the sum of three nonnegative squared terms: a traceless second-derivative term, a deviation term for $\Delta_p\varphi-\frac{2(p-1)}{p}|\nabla\varphi|^p/\varphi$, and a mixed radial-tangential term. Integration over cylindrical shells with cutoffs, together with the $C^0$ decay estimate and the gradient estimates in Lemmas 2.4 and 2.5, shows the total integral is smaller than any positive constant. Hence $\mathrm{div}\,Y=0$, and the nonnegativity of the three terms forces each to vanish, yielding the two relations (3.38) and (3.39) that integrate to the explicit profile.

What would settle it

Evaluate $\mu$ in (3.31) for $k\le 2p$ with $q=2(p-1)k/(k-2)$: substituting gives $\mu=0$, so the error term (3.32) is of order one as $\varepsilon\to 0$; that direct computation would falsify the claim that this term vanishes in that parameter range.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for $s=1$, $n\ge 4$, and $3\le k\le n-1$, any cylindrically symmetric solution $u(y,z)$ of (1.2) has the form $u(y,z)=\lambda^{(n-p)/p}u_0(\lambda y,\lambda z+z_0)$, where $u_0(x)=C_{p,n,k}[(1+|y|)^2+|z|^2]^{-(n-p)/(2(p-1))}$. The proof establishes an integral identity with a nonnegative right-hand side, integrates it against carefully chosen cutoffs, and uses the decay and gradient estimates to force the right-hand side to vanish. That forces the solution to satisfy two differential equations that integrate directly to the explicit profile. As a consequence, the best constant in (1.1) for $s=1$ and $k\ge 3$ is attained by these explicit functions, resolving the conjecture in [1] for that range.

Load-bearing premise

The whole argument rests on one limiting step: the inner-cutoff error term must shrink to zero as $\varepsilon\to 0$, which requires the exponent $\mu$ in (3.31) to be positive in every parameter range used.

Editorial extensions

If this is right

  • The best constant $S_{p,1}$ in (1.1) for $s=1$ and $k\ge 3$ is attained by the explicit profile $u_0$ under dilations and $z$-translations, making the extremal value effectively explicit.
  • The Alvino-Ferone-Trombetti conjecture is true for $k\ge 3$: the candidate $u_0$ is the only extremal profile up to the stated transformations.
  • Every positive finite-energy cylindrically symmetric solution of (1.2) is the explicit one, so the classification within this symmetry class is complete.
  • The case $k=2$ remains open because the a priori estimates used in the proof are not available for that singular set.

Reading between the lines

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

  • A reader checking (3.31) for $k\le 2p$ with the stated $q=2(p-1)k/(k-2)$ finds $\mu=0$, not $\mu>0$; the printed computation of $\mu$ in the $k\le 2p$ case does not close the limit $\varepsilon\to 0$ unless another choice of $q$ or an additional estimate is supplied.
  • If the classification can be repaired for $p\ge k/2$, the same identity method would yield the sharp constant by evaluating the Rayleigh quotient at $u_0$, giving a closed-form expression for $S_{p,1}$.
  • The differential-identity strategy is tied to the cylindrical symmetry and to the weight $|y|^{-1}$; extending it to other weights $s$ or to the case $k=2$ would require new pointwise estimates rather than only integral ones.
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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

2 major / 3 minor

Summary. The paper studies the Hardy-Sobolev-Maz'ya inequality (1.1) and its Euler-Lagrange equation (1.2) for the p-Laplace operator. The authors derive a differential identity for a transformed function φ, combine it with a priori C0 and gradient estimates, and aim to classify all positive finite-energy cylindrically symmetric solutions of (1.2) for 3 ≤ k ≤ n−1. From this classification and the symmetrization result of Secchi-Smets-Willem, they claim the best constant and all extremal functions for the inequality. The main theorem is stated as Theorem 1.2, and the proof occupies Section 3.

Significance. If the classification result were fully established, it would confirm the Alvino-Ferone-Trombetti conjecture for k ≥ 3 and extend the p = 2 result of Mancini-Fabbri-Sandeep to the p-Laplace setting. The differential identity in Proposition 3.1 is a nontrivial and interesting contribution, and the final ODE analysis is clean. However, the proof of Theorem 1.2 currently fails as written in the case p ≥ k/2 because a key error term is claimed to decay as ε→0 when, with the chosen parameter q, it does not. Since the gap is local and apparently repairable by a small modification of q, the paper is not without merit.

major comments (2)
  1. [Section 3, Eqs. (3.30)-(3.32)] For the case k ≤ 2p, the authors choose q = 2(p−1)k/(k−2) and compute μ in (3.31), concluding μ > 0. Direct substitution gives q − 2p + 2 = 4(p−1)/(k−2), so k(q−2p+2)/q = 2 and therefore μ = 0, not a positive number. The displayed computation in the manuscript, containing expressions such as '2.1 − 2 = 0.1', is algebraically incorrect. Consequently the estimate (3.32) is C(R)ε^0, which does not tend to zero as ε→0. The passage from (3.34) to (3.35), where the ε→0 limit is taken to obtain the ε-independent bound, is therefore unjustified. Since the conclusion divY = 0 at (3.37) rests on this passage, Theorem 1.2 is not proved for p ≥ k/2 as written. This gap appears repairable: Lemma 2.5 is stated for every q > 1, and choosing q_δ = 2(p−1)k/(k−2) + δ gives μ_δ = k(q_δ − 2p + 2)/q_δ − 2 > 0, while (3.27) and (3.29) remain unchanged. The differential identity (3.11), the boundary vanishing (3.17), and the final ODE step are not implicated in this objection.
  2. [Theorem 1.2 statement vs. proof] The theorem statement includes a free translation parameter z0 ∈ R^{n−k} in the conclusion u(y,z) = λ^{(n−p)/p} u0(λy, λz + z0), but the proof at the end concludes u(y,z) = λ^{(n−p)/p} u0(λy, λz) with no z0. If 'cylindrically symmetric' is understood with the symmetry axis fixed at the origin, the statement should say so explicitly; otherwise the proof should begin by translating the symmetry axis to the origin and then state the final conclusion with z0 restored. This is not load-bearing for the main argument but should be clarified.
minor comments (3)
  1. [Section 3, Eq. (3.31)] The computation of μ contains corrupted decimal-point notation ('2.1', '4.2', '0.1') that makes the displayed algebra impossible to follow; this should be corrected to the clean fractions and the true value μ = 0 should be acknowledged.
  2. [Lemma 2.5 proof] In the proof of Lemma 2.5, the word 'Anagin' should read 'Again'. Also, the iteration step leading to (2.23) is terse; a few words explaining how α_{i+1} is chosen from the two integrability conclusions would improve readability.
  3. [References] Reference [22] is cited in the introduction for the symmetrization result, but the statement of Proposition 2.1 as '[22]' could benefit from a page or theorem number to help the reader verify the exact claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the p-Laplace classification is derived from first principles; external citations are non-overlapping and not load-bearing.

full rationale

The paper's central claim is a classification of cylindrically symmetric solutions to the p-Laplace Hardy-Sobolev-Maz'ya equation. The key differential identity, Proposition 3.1, is proved directly in the paper: it is obtained from the equation (1.2), the change of variable u = kappa * phi^{-(n-p)/p}, and a sequence of explicit algebraic manipulations, including the definition of the trace-free tensor E, the vector field Y in (3.10), and the identity (3.11). There is no fitted parameter renamed as a prediction, and no self-citation by the authors (Lin and Ma). The cited results used as ingredients — Badiale-Tarantello for the inequality, Secchi-Smets-Willem for cylindrical symmetry of extremals, Dutta for the C0 estimate, Xiang and Antonini-Ciraolo-Farina for gradient estimates, and standard p-Laplace regularity — are external to the present paper and are not used to assume the classification being proved. In particular, the Secchi-Smets-Willem symmetrization result is invoked only to convert the classification of cylindrically symmetric solutions into a statement about best constants and extremals; it does not itself assert the form of the extremal and is not an assumption embedded in Proposition 3.1. The proof of the identity and the subsequent ODE reduction are self-contained from (1.2). The reviewer-noted issue that the exponent mu in (3.31)–(3.32) appears to vanish for k <= 2p with the chosen q, making the epsilon -> 0 passage at (3.34)–(3.35) questionable, is a potential technical or algebraic gap in the limiting argument; it is not circularity, because the missing step would be an estimate, not an assumption of the conclusion. Accordingly, no circular step can be exhibited, and the appropriate score is 0.

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

The paper introduces no free parameters fitted to data; constants kappa, a, b, lambda are determined by normalization and the ODE. It relies on external theorems for regularity, symmetrization, and a priori estimates.

assumptions (5)
  • domain assumption Cylindrically symmetric minimizers exist, i.e. S_{p,s}=S^{**}_{p,s}
    Quoted from Secchi-Smets-Willem [22, Prop 2.1]; it reduces the extremal problem to cylindrically symmetric functions.
  • domain assumption A priori C0 estimate (2.3) of Dutta [15]
    Used throughout to bound u and phi and to justify the integral estimates.
  • domain assumption Gradient estimates of Xiang [27] and Antonini-Ciraolo-Farina [2]
    Used to derive the integral gradient estimates (2.6) and (2.10) that control the cutoff errors.
  • standard math Regularity results for p-Laplace equations from [3], [10], [13], [24]
    Used for C^{1,alpha} and W^{1,2} regularity of |nabla u|^{p-2} nabla u and for vanishing of the critical set.
  • domain assumption Finite energy and cylindrical symmetry of the solution class
    The theorem only treats positive finite energy cylindrically symmetric solutions of (1.2).

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Cite this review

Pith. "Pith review of Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities." pith.science (2026). https://pith.science/paper/Z73WIBBK

@misc{pith2026241209033,
  author       = {Pith},
  title        = {Pith review of: Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z73WIBBK}},
  note         = {Machine review of arXiv:2412.09033}
}
abstract

We derive an integral identity for a class $p$-Laplace equation, and then classify all positive finite energy cylindrically symmetric solutions of the equation (\ref{1.2}) for $3\leq k\leq n-1,$ with the help of some a prior estimates. Combining this with the result of Secchi-Smets-Willem{\cite{SSW03}}, as a consequence, we obtain the best constant and extremal functions for the related Hardy-Sobolev-Maz'ya inequalities.

Discussion (0). Continue with ORCID to comment.

Forward citations

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