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Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality

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Pith's one-line read Theorem: the Hardy-Sobolev Bianchi-Egnell constant is attained for every $\gamma$ in $[\gamma_0,(N-2)^2/4)$, for all $N\ge3$.

desk verdict A solid extension of the Bianchi-Egnell attainment program to the Hardy-Sobolev case; the new hidden critical level works, and the main flaw—unproved existence of γ0 for N≥4—is a trivial fix, not a real gap. read the letter →

arxiv 2505.07039 v2 pith:N3JOTKV3 submitted 2025-05-11 math.AP math.FA

Souptik Chakraborty , Monideep Ghosh , Debabrata Karmakar This is my paper · ORCID
classification math.APmath.FA MSC 26D1046E3549K4047J2049J2049J40
keywords Hardy-SobolevinequalityBianchi-Egnellstabilityextremizerquantitativespectralgapconcentration-compactnessCaffarelli-Kohn-Nirenbergextremalfunctions
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 proves that the best Bianchi-Egnell constant for the Hardy-Sobolev inequality, the infimum of the deficit ratio $\delta(u)/\mathrm{dist}(u,\mathcal{M}_{\mathrm{HS}})^2$ over functions that are not optimizers, is attained for every Hardy parameter $\gamma$ in the range $\gamma_0\le\gamma<(N-2)^2/4$, for every $N\ge3$. This extends to the singular Hardy-Sobolev case the existence of a stability extremizer that was known for the classical Sobolev inequality at $\gamma=0$. The interest is that the standard route fails here: the usual non-vanishing third-eigenfunction criterion does not hold, and the inequality is not translation invariant, so extracting a non-zero weak limit from a minimizing sequence is a real obstacle. The paper identifies a third critical level, $C_{\mathrm{BE}}(\gamma)<1-S_\gamma/S$, that supplies the missing weak limit, and then rules out loss of compactness by comparing with the local and two-bubble critical levels.

What carries the argument

The core machinery is a chain of three comparison levels for the best constant. The local Bianchi-Egnell level $\Lambda(\gamma)=1-\mu_2^\gamma/\mu_3^\gamma$ is the spectral gap of the linearized Hardy-Sobolev operator; the two-peak level is computed by testing with a sum of two weakly interacting optimizers $U_\gamma+U_\gamma[\lambda]$ as $\lambda\to0$; and the hidden level $1-S_\gamma/S$ compares the Hardy-Sobolev best constant $S_\gamma$ with the classical Sobolev constant $S$. The proof uses the isometric lift to the cylinder $\mathbb{R}\times S^{N-1}$, where dilations become translations and the optimizer becomes a sech profile; there, spherical-harmonic projections and an improved Hardy inequality produce a non-zero low-mode projection for any minimizing sequence, which is what extracts a non-zero weak limit without translation invariance.

What would settle it

Compute $\Lambda(\gamma)$ and $C^{\mathrm{rad}}_{\mathrm{BE}}(\gamma)$ numerically at $\gamma=\gamma_0$ for $N=4$; if the defining equality $\Lambda(\gamma_0)=C^{\mathrm{rad}}_{\mathrm{BE}}(\gamma_0)$ has no solution or is violated, the strict comparison $C_{\mathrm{BE}}(\gamma)<\Lambda(\gamma)$ fails and the proof of attainment collapses.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $N\ge3$ and every $\gamma_0\le\gamma<(N-2)^2/4$, the infimum $C_{\mathrm{BE}}(\gamma)$ is attained. A minimizing sequence for the quotient is bounded, and after suitable dilations it has a non-zero weak limit; the proof shows the remainder vanishes strongly and that the limit is not a Hardy-Sobolev optimizer, so the limit is a genuine extremizer. In particular, for $N\ge4$ the threshold $\gamma_0$ lies below $\gamma_c^\star$, so the result covers the region where the third eigenspace of the linearized operator contains only spherical harmonics of degree $1$ and the earlier non-vanishing criterion is unavailable. The paper also shows that the radial Bianchi-Egnell constant is independent of $\gamma$, and that the comparisons $C_{\mathrm{BE}}(\gamma)\le C^{\mathrm{rad}}_{\mathrm{BE}}(\gamma)\le\Lambda(\gamma)$ together with $C_{\mathrm{BE}}(\gamma)<1-S_\gamma/S$ force the attainment.

Load-bearing premise

The argument relies on imported spectral facts about the linearized Hardy-Sobolev operator, namely that the gap between its second and third eigenvalues is exactly the stated $\Lambda(\gamma)$ and that the radial stability constant stays strictly below that gap for large enough $\gamma$; if either fact failed, the chain of strict inequalities that keeps minimizers away from the optimizer manifold would break.

Editorial extensions

If this is right

  • For every $N\ge3$ and $\gamma_0\le\gamma<(N-2)^2/4$, the stability inequality is sharp: some function outside the optimizer family attains $C_{\mathrm{BE}}(\gamma)$.
  • Because $C_{\mathrm{BE}}(\gamma)<1-S_\gamma/S$ holds on the stated range, a minimizing sequence cannot lose its mass to infinity by pure rescaling; the weak limit is non-zero.
  • The constant $\gamma_0$ for $N\ge4$ lies below $\gamma_c^\star$, so existence of an extremizer is established precisely in the parameter region where the degree-$2$ spherical-harmonic non-vanishing criterion is absent.
  • The radial Bianchi-Egnell constant is the same for every admissible $\gamma$, so radial stability is scale-rigid, while the full non-radial constant is achieved for the newly covered range.
  • The strict bounds below both the local and two-bubble critical levels imply the minimizer is isolated from both degeneracies: it neither converges to an optimizer nor splits into two bubbles.

Reading between the lines

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

  • If the hidden-critical-level idea is transferable, the same kind of bound $C<\text{one constant}-\text{another}$ should locate the obstruction to non-zero weak limits in other non-translation-invariant sharp inequalities, such as weighted Caffarelli-Kohn-Nirenberg inequalities.
  • For $N=3$, the threshold $\gamma_0$ exceeds $\gamma_c^\star$; a numerical computation of $C_{\mathrm{BE}}(\gamma)$ just above $\gamma_c^\star$ would indicate whether that restriction is artificial or a genuine dimensional transition.
  • The spherical-harmonic projection step suggests a general recipe: when translations are absent, project a minimizing sequence onto low spherical-harmonic modes and use an improved Hardy inequality; the number of modes needed may quantify the failure of translation invariance.
  • A direct corollary the paper does not spell out: the attained extremizer gives an explicit function at which the Hardy-Sobolev deficit is quadratically controlled by distance to the optimizer family, so the stability inequality is optimal with a realized equality case.
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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 / 4 minor

Summary. The paper studies the sharp quantitative stability (Bianchi-Egnell) inequality for the Hardy-Sobolev inequality. It defines the best Bianchi-Egnell constant C_BE(gamma) as the infimum of the deficit functional divided by the squared distance to the manifold of all Hardy-Sobolev optimizers, and proves (Theorem 1.1) that for N >= 3 and gamma >= gamma_0, with gamma_0 defined in (1.10), this infimum is attained. The strategy follows the now-standard framework of König: the authors identify a local critical level, a two-bubble critical level, and an additional 'hidden' critical level 1 - S_gamma/S. Working in cylindrical coordinates, they prove that the radial best constant C_BE^rad(gamma) is independent of gamma, compute explicit two-bubble expansions, and use the hidden level to extract a nonzero weak limit from a minimizing sequence despite the lack of translation invariance. A case analysis then rule out loss of compactness and shows the weak limit is a genuine extremizer.

Significance. If the result holds, it is a substantial extension of König's existence theorem for the Sobolev inequality to the Hardy-Sobolev inequality, where translation invariance is lost and the third eigenspace of the linearized operator can contain only spherical harmonics of degree one. The paper also improves the radial-only attainability results of Wei-Wu and Deng-Shengbing-Wei for the Hardy-Sobolev case. The manuscript has clear strengths: Proposition 4.2 gives a self-contained two-bubble expansion in cylindrical coordinates that does not rely on the Kelvin transform, Proposition 3.1 proves the constancy of the radial Bianchi-Egnell constant by a clean scaling argument, and Lemma 5.1 exploits the hidden critical level to obtain a nonzero weak limit in a non-translation-invariant setting. The external inputs from Wei-Wu are explicitly identified, and the proof is not circular: no parameter is fitted to the targeted conclusion.

major comments (2)
  1. [Section 1, Eq. (1.10)] For N >= 4, the threshold gamma_0 is defined as 'the point Lambda(gamma_0) = C_BE^rad(gamma_0)', which presupposes that this equation has a solution. The text states that Lambda is continuous and strictly increasing on (0, gamma_c^*) and that C_BE^rad is constant, but it does not prove that the constant value of C_BE^rad lies strictly between Lambda(0+) and Lambda(gamma_c^*) = 4/(N+4). Without this intermediate-value argument, the inequalities C_BE^rad(gamma) < Lambda(gamma) for gamma > gamma_0, and consequently the comparisons C_BE(gamma) < Lambda(gamma) and C_BE(gamma) < 1 - S_gamma/S, are not justified. The missing argument is short and appears recoverable from the explicit formula (3.2) together with [WW24, Proposition 4.1], but it must be included in the proof.
  2. [Section 5, Lemma 5.2] The proof of Lemma 5.2 is omitted with the statement that it follows from [Kön25] verbatim. This lemma is load-bearing: it is used in every case of the strong-convergence argument to decompose m(u_k) and hence dist(u_k, M_HS)^2 as max{m(v), m(f_k)} plus a vanishing error. Because the Hardy-Sobolev problem is not translation invariant and m(u) is a supremum over dilations only, the adaptation of König's proof to this setting is not self-evident. Please supply a proof, or give the exact statement in [Kön25] that covers this setting and explain the modifications needed for the Hardy-Sobolev metric and the reduced parameter family.
minor comments (4)
  1. [Section 5, Lemma 5.1, Step 1] The assumption stated as 'for every i in {1, ..., k_0}' should also include the radial projection i = 0: the subsequent estimate of m(P_0^1 phi_n) and the conclusion that phi_n -> 0 in L^{2*} require the i = 0 component to vanish in L^{2*}. Please correct the index range.
  2. [Section 4, Proposition 4.2(b)] The intermediate display ||v_s||_{L^{2*}}^{2*} = 2 + 2Q(s) + o(Q(s)) appears to be missing the factor 2* in the linear term. The final exponentiated formula is consistent with the correct expansion 2 + 2*2* Q(s) + o(Q(s)); please correct the intermediate display.
  3. [Section 5, proof of Theorem 1.1 (N = 3)] For N = 3, the proof states that Lemma 3.3 gives Lambda(gamma) < 1 - S_gamma/S for gamma in [gamma_0, ...), but Lemma 3.3(b) gives the strict inequality only for gamma > gamma_0; at gamma = gamma_0 equality holds in Lambda = 1 - S_gamma/S. The endpoint is still harmless because C_BE^rad(gamma_0) < Lambda(gamma_0), but this should be stated explicitly.
  4. [References] The bibliography entry [Kön25] has irregular formatting ('Tobias k¨onig, stability for the Sobolev inequality: Existence of a minimizer') and should be cleaned up; please also ensure that all citations to [K¨23] and [K¨25] match the reference list consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all load-bearing comparisons use externally sourced spectral gap and two-bubble levels, not the target constant.

full rationale

The derivation chain is self-contained against external benchmarks and does not reduce to its own inputs. The target quantity C_BE(γ) is bounded above by C^rad_BE(γ), which is shown in Proposition 3.1 to be constant by an explicit scaling equivalence (Lemma 3.2), and below the local critical level via the spectral gap Λ(γ) imported from Wei-Wu [WW24, Proposition 3.1] and the two-bubble level derived directly in Proposition 4.2. The threshold γ0 in (1.10) is defined by the equation Λ(γ0)=C^rad_BE(γ0); neither side of this equation is the target constant C_BE(γ), and C_BE enters afterwards only through the elementary inequality C_BE≤C^rad_BE. The hidden critical level C_BE<1−Sγ/S is obtained from Λ(γ)<1−Sγ/S proved elementarily in Lemma 3.3 together with C^rad_BE<Λ(γ), again with no parameter fitted to C_BE. The nonzero weak limit then follows from this strict inequality, and the exclusion of optimizers follows from C_BE<Λ(γ). All cited inputs are external works by Wei-Wu and Konig, not by the present authors, and they provide explicit formulas or machine-independent estimates rather than a self-citation chain. The only caveat is a rigor gap: for N≥4 the existence of a root of Λ(γ0)=C^rad_BE(γ0) is asserted without displaying the intermediate-value argument, although the root is recoverable from the cited monotonicity of Λ and constancy of C^rad_BE. That is an omitted justification, not a circular reduction, and it does not affect the circularity score.

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

The central claim rests on standard Hardy-Sobolev theory, the cited spectral gap result of Wei-Wu, and the improved Hardy inequality proved in the appendix. No parameter is fitted to data. The only potentially understated premise is the existence of the root defining γ0 for N≥4.

assumptions (5)
  • domain assumption Spectral gap formula Λ(γ) and C^rad_BE(γ)<Λ(γ) for γ>γ*_c, imported from Wei-Wu [WW24, Prop 3.1 and 4.1].
    Used to place C_BE(γ) below the local Bianchi-Egnell level and below the hidden level, ensuring the weak limit is not an optimizer (Eq. 3.2, Section 5).
  • domain assumption Existence and uniqueness of γ0 for N≥4 as the root of Λ(γ0)=C^rad_BE(γ0) with Λ increasing on (0,γ*_c).
    Definition (1.10) relies on this root without proof in the text; if it did not exist, the threshold and the proof for N≥4 would need modification.
  • standard math Improved Hardy inequality on the cylinder for functions orthogonal to low spherical harmonics (Lemma 6.6).
    Proved in the appendix; used in Lemma 5.1 to force a low spherical harmonic component in a minimizing sequence.
  • standard math Concentration-compactness principle of Lions and the 1D Gagliardo-Nirenberg-Sobolev inequality.
    Used in Lemma 5.1 and in bounding the minimizing sequence; standard tools.
  • domain assumption Characterization of optimizers of the Hardy-Sobolev inequality (Chou-Chu, Terracini).
    Defines M_HS and underlies the distance formulation in Lemma 2.2.

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Pith. "Pith review of Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality." pith.science (2026). https://pith.science/paper/N3JOTKV3

@misc{pith2026250507039,
  author       = {Pith},
  title        = {Pith review of: Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N3JOTKV3}},
  note         = {Machine review of arXiv:2505.07039}
}
abstract

In this article, we prove the best Bianchi-Egnell constant for the Hardy-Sobolev (HS) inequality \begin{align*} C_{\tiny\mbox{{BE}}}(\gamma) := \inf_{{u \ \small \mbox{not an optimizer}}} \frac{\int_{\mathbb{R}^n} \left(|\nabla u|^2 - \frac{\gamma}{|x|^2}u^2\right) \ {\rm d}x - S_{\gamma}\|u\|_{L^{2^{\star}}}^2}{\mbox{dist} (u, \ \mbox{set of optimizers})^2}, \end{align*} is attained, extending the result of K\"onig [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to $\gamma = 0$). One of the main difficulties is that the third eigenspace of the linearized operator may contain only spherical harmonics of degree $1$, and hence, an essential non-vanishing criterion fails [arXiv:2210.08482]. This non-vanishing criterion is indispensable for proving the best Bianchi-Egnell constant $C_{\tiny\mbox{{BE}}}(\gamma) < C_{\tiny\mbox{{BE}}}^{\tiny\mbox{{loc}}}(\gamma)$ that prevents a minimizing sequence converging to one of the optimizers. In addition, not being translation invariant, extracting a non-zero weak limit from a minimizing sequence presents difficulties. We found another hidden critical level $C_{\tiny\mbox{{BE}}}(\gamma) <1 - \frac{S_{\gamma}}{S},$ where $S$ is the best Sobolev constant that plays a significant role in proving the existence of an extremizer. In particular, we show that there exists a $\gamma_0>0$ such that for $\gamma \geq \gamma_0,\ C_{\tiny\mbox{{BE}}}(\gamma)$ is attained. Moreover, we remark that there is a region $\gamma_0 \leq \gamma < \gamma_c^{\star},$ where the third eigenspace of the linearized operator contains only spherical harmonics of degree $1.$ Our result improves some of the results in Wei-Wu [arXiv:2308.04667] corresponding to the HS inequality.

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Reviewed August 15, 2026 · model on record in the stance chip above.