{"id":"97c77b9c-ece7-4f99-8be8-3ac3de3ea493","arxiv_id":"2505.07039","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For all N≥3 and γ0≤γ<(N−2)^2/4, the infimum defining the Bianchi-Egnell constant C_BE(γ) for the Hardy-Sobolev inequality is achieved by some function.","lead":"This paper proves that the best Bianchi-Egnell stability constant for the Hardy-Sobolev inequality is attained for every Hardy parameter above an explicit threshold, extending König's result for the classical Sobolev inequality. The proof introduces a third critical level based on the ratio of the Hardy-Sobolev and Sobolev constants to handle the loss of translation invariance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition of γ0 for N≥4 assumes, without proof, that the equation Λ(γ0)=C^rad_BE(γ0) has a root; the paper's strict inequalities C_BE<Λ and C_BE<1−Sγ/S all start from this root.","rationale":"The reader's verdict is CONDITIONAL with the same weakest assumption. After full-text review, I agree that the root existence for γ0 in (1.10), N≥4, is the most load-bearing unproved step: every strict comparison in the proof of Theorem 1.1, including C_BE≤C^rad_BE<Λ and then C_BE<1−Sγ/S, depends on the existence of this threshold. The endpoint facts needed are available but are not stated. A secondary, smaller gap appears in Lemma 5.1, Step 1: the contradiction assumes the radial mode P0^1 also vanishes (it uses ||P0^1 φ_n||_{L^{2*}}→0), while the stated conclusion claims a mode with 1≤k≤k0 survives; if only the radial mode carries mass, the written argument does not apply. This is also fixable, but it is a separate proof detail. The main construction — the hidden critical level, the use of constancy of C^rad_BE, and the two-peak comparison — is coherent, and the cited external results appear to transfer to the Hardy-Sobolev setting. I therefore see no reason to move the verdict; CONDITIONAL remains appropriate until the root existence and the Lemma 5.1 modal case are explicitly justified.","tokens_in":23583,"tokens_out":36879,"duration_ms":359381,"concrete_test":"For N≥4, evaluate (3.2) at γ=0 to confirm Λ(0)=0; use [WW24, Proposition 4.1] at γ>γ*_c together with Proposition 3.1 to confirm C^rad_BE(γ)<Λ(γ*_c)=4/(N+4); then, by continuity and strict monotonicity of Λ on (0,γ*_c), invoke the intermediate value theorem to obtain a unique γ0∈(0,γ*_c) with Λ(γ0)=C^rad_BE. If either endpoint check fails, the definition (1.10) is invalid and the range in Theorem 1.1 is in question.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 is that C_BE(γ) is attained for every γ0≤γ<(N−2)^2/4. For N≥4, γ0 is introduced in (1.10) as 'the point Λ(γ0)=C^rad_BE(γ0)'. The text asserts that Λ is continuous and strictly increasing on (0,γ*_c) and that C^rad_BE is constant, but it never proves that the constant value C^rad_BE lies between Λ(0+) and Λ(γ*_c)=4/(N+4). The required strict inequalities C^rad_BE(γ)<Λ(γ) for γ>γ0, which are used to obtain C_BE≤C^rad_BE<Λ and then, via Lemma 3.3, C_BE<1−Sγ/S, only hold once such a root is known to exist. Existence is in fact recoverable from cited results: formula (3.2) gives Λ(0)=0, and [WW24, Proposition 4.1] together with the constancy of C^rad_BE gives C^rad_BE<4/(N+4). But the paper does not supply this intermediate-value argument, so the statement of Theorem 1.1 is not fully justified as written. This is a rigor gap rather than an indication that the theorem is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23860,"tokens_out":21600,"duration_ms":206440,"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":[{"comment":"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.","section":"Section 1, Eq. (1.10)"},{"comment":"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.","section":"Section 5, Lemma 5.2"}],"minor_comments":[{"comment":"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.","section":"Section 5, Lemma 5.1, Step 1"},{"comment":"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.","section":"Section 4, Proposition 4.2(b)"},{"comment":"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.","section":"Section 5, proof of Theorem 1.1 (N = 3)"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious and mostly careful contribution, and the central strategy is plausible. The two load-bearing points that need work are the existence of gamma_0 for N >= 4 and the omitted proof of Lemma 5.2; both appear fixable without changing the scope of the paper. The reliance on [WW24] for the spectral gap and the radial strict inequality is transparent and acceptable. I see no circularity or hidden parameter fitting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what to know: this paper resolves a real open question in the quantitative stability program—attainment of the best Bianchi-Egnell constant for the Hardy-Sobolev inequality—and it does so for a genuinely larger parameter range than the existing radial-only result of Wei-Wu and the later threshold of Deng-Tian-Wei. The central mechanism, the hidden critical level 1−Sγ/S, is new and well chosen: it is exactly what lets the authors extract a nonzero weak limit without translation invariance. The two-bubble expansion and the nonzero weak limit lemma are worked out carefully; the reliance on Wei-Wu for the spectral gap and for the strict radial inequality is explicit and legitimate, not a disguised assumption.\n\nThe soft spots are minor. The definition of γ0 for N≥4 says 'the point Λ(γ0)=C^rad_BE(γ0)' without proving such a point exists. It does exist: Λ is continuous and strictly increasing on (0,γ*_c), Λ(0+)=0, Λ(γ*_c)=4/(N+4), and C^rad_BE is a positive constant strictly below 4/(N+4), so the intermediate value theorem gives a unique root. The ingredients are all in the cited material, but the paper should say this explicitly; as written, the statement of Theorem 1.1 jumps over a step. Lemma 5.2 is dismissed as following verbatim from König; that's probably true, but a few lines would make the paper self-contained. The proof of Lemma 5.1's Step 2 is telegraphic, though the localization argument via one-dimensional GNS is sound.\n\nNone of these come close to invalidating the theorem. The proof is coherent, the new content is identifiable, and the citation pattern is honest. Who is this for: people working on quantitative stability of Sobolev-type inequalities. It will not reshape the field, but it is a solid extension, and the hidden-level idea may be useful elsewhere. I would send it to peer review; ask the authors to add the IVT argument for γ0 and to expand Lemma 5.2. After that I expect it to be accepted.","headline":"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.","tokens_in":24418,"tokens_out":5465,"would_cite":true,"duration_ms":50098,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","46E35","49K40","47J20","49J20","49J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem: the Hardy-Sobolev Bianchi-Egnell constant is attained for every $\\gamma$ in $[\\gamma_0,(N-2)^2/4)$, for all $N\\ge3$.","keywords":["Hardy-Sobolev inequality","Bianchi-Egnell stability","stability extremizer","quantitative stability","spectral gap","concentration-compactness","Caffarelli-Kohn-Nirenberg inequality","extremal functions"],"falsifier":"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.","tokens_in":23381,"feed_emoji":"📐","tokens_out":10863,"duration_ms":95661,"temperature":0.7,"pith_summary":"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.","feed_headline":"Stability extremizer exists for the Hardy-Sobolev inequality","feed_subtitle":"For every gamma between a threshold and the Hardy limit, the optimal stability ratio is achieved by a real function.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the explicit spectral gap $\\Lambda(\\gamma)$ and the radial Bianchi-Egnell attainment for $\\gamma>\\gamma_c^\\star$ on which the strict inequality $C^{\\mathrm{rad}}_{\\mathrm{BE}}<\\Lambda$ rests.","marker":"[WW24]"},{"why":"provides the existence proof for the Sobolev case $\\gamma=0$ and the two-critical-level strategy that this paper adapts.","marker":"[K\\\"on25]"},{"why":"introduced the Bianchi-Egnell stability inequality for the Sobolev inequality, the baseline this paper extends to Hardy-Sobolev.","marker":"[BE91]"},{"why":"proved $C_{\\mathrm{BE}}(\\gamma)>0$ for the Hardy-Sobolev inequality, establishing that the stability quotient is well posed.","marker":"[WW22]"},{"why":"classifies the optimizers of the Hardy-Sobolev inequality, defining the manifold $\\mathcal{M}_{\\mathrm{HS}}$ from which the distance is measured.","marker":"[CC93]"},{"why":"concentration-compactness principle used to obtain weak limits; invoked for the non-zero-weak-limit lemma.","marker":"[Lio84a]"},{"why":"second part of the concentration-compactness principle used in the same extraction step.","marker":"[Lio84b]"}],"fun_headline_variants":["Hardy-Sobolev stability extremizer exists for high gamma","Bianchi-Egnell constant attained for Hardy-Sobolev inequality","Optimal stability ratio achieved for Hardy-Sobolev inequality","Existence of Hardy-Sobolev stability extremizer proven","Attainment of best Bianchi-Egnell constant for HS inequality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hardy-Sobolev stability extremizer exists for high gamma","Bianchi-Egnell constant attained for Hardy-Sobolev inequality","Optimal stability ratio achieved for Hardy-Sobolev inequality","Existence of Hardy-Sobolev stability extremizer proven","Attainment of best Bianchi-Egnell constant for HS inequality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2829,"prompt_tokens":1220,"completion_tokens":1609,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":836,"completion_tokens_details":{"reasoning_tokens":1517}},"tokens_in":836,"tokens_out":1609,"duration_ms":10930,"temperature":1.0,"reasoning_tokens":1517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:28:15.689290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}