{"id":"4c9cfa09-8296-4847-b149-b2bf39406d3b","arxiv_id":"1908.05095","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a fractional Hardy-Sobolev minimization problem on a bounded domain with an interior singularity, the optimal constant is attained for parameter values below a threshold and not attained above it.","lead":"This paper studies when the best constant in a fractional Hardy-Sobolev inequality on a bounded domain is actually achieved by a function. It shows there is a critical value of the parameter such that minimizers exist below it and fail to exist above it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.2(i) does not prove strong convergence: the displayed chain using μ_{α,λ} permits a nonzero splitting, so attainment for λ<λ* is not established as written.","rationale":"The reader's conditional verdict is the right one. The central construction is promising and most estimates check out; the use of the Marano-Mosconi decay is legitimate because the cited theorem states the needed bound, and n>4s makes the relevant integrals converge. The real weakness is internal: Proposition 3.2(i) stops one line short of the standard contradiction. The chain with μ_{α,λ} only yields equality in the limit and cannot distinguish concentration from strong convergence; the later conclusion that u_k→u strongly is therefore unsupported. This is not a fundamental flaw, since the global constant μ is available and the repair is routine, but as written the proof of attainment for λ<λ* is incomplete. I therefore keep the conditional verdict and recommend adding the missing step. I partially agree with the reader: the reader's stated 'weakest_assumption' points to the external asymptotics, but the reader's rationale also names the Proposition 3.2(i) gap; I regard the latter as the load-bearing concern.","tokens_in":11559,"tokens_out":13711,"duration_ms":134532,"concrete_test":"Re-derive Proposition 3.2(i) with the following modification: assume for contradiction that limsup_{k→∞} ∫_Ω |u_k-u|^{2_{s,α}}/|x|^α dx > 0. Apply the global inequality (1.3) separately to u and u_k-u to get μ[(∫|u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}} + (∫|u_k-u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}}] ≤ [u_k]^2_{s,Ω}+o(1) ≤ μ_{α,λ}+o(1). Using (a+b)^r≤a^r+b^r with r=2/2_{s,α}<1 and Brézis-Lieb, the left side is at least μ+o(1); if this forces μ≤μ_{α,λ}, the contradiction is established and the missing strong-convergence step is repairable. If the inequality fails at any displayed step, the claimed attainment for λ<λ* would need another argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in Proposition 3.2(i). For a minimizing sequence u_k ⇀ u, after Brézis-Lieb the proof derives 1 ≤ (∫|u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}} + (∫|u_k-u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}} + o(1) ≤ [E(u)+E(u_k-u)]/μ_{α,λ} + o(1) ≤ E(u_k)/μ_{α,λ}+o(1) = 1+o(1). This chain is compatible with a nontrivial split: both weighted masses can tend to positive values whose exponents sum to 1. The sentence 'Since u ≠ 0, we conclude u_k → u strongly' is therefore a non sequitur. The strict inequality μ_{α,λ}<μ is not used at this point. The standard repair is to apply the global Hardy-Sobolev inequality (1.3) with constant μ to u and u_k-u rather than the local lower bound with μ_{α,λ}. Since [u]^2_{s,Ω}+[u_k-u]^2_{s,Ω} ≤ [u_k]^2_{s,Ω}+o(1) and (a+b)^r ≤ a^r+b^r for r=2/2_{s,α}<1, one obtains μ ≤ μ_{α,λ}, contradicting strictness. This is a short fix, but as written the proof of Proposition 3.2(i), and hence Theorem 1.1, is incomplete. The external decay assumption flagged in Lemma 2.2(1) is not the main issue: [18, Theorem 1.1] supplies exactly |u_0(x)|≤C/|x|^{n-2s}, so the upper-bound construction is supported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the minimization problem for the fractional Hardy-Sobolev type constant μ_{α,λ}(Ω), defined as the infimum over u∈H^s(Ω) of the fractional seminorm plus λ∫|u|², normalized by the weighted Hardy-Sobolev mass ∫|u|^{2_{s,α}}|x|^{-α}. The main result, Theorem 1.1, states the existence of a threshold λ*∈(0,∞] such that μ_{α,λ}(Ω) is attained for every λ<λ*, and, if λ*<∞, is not attained for every λ>λ*. The proof strategy is to compare μ_{α,λ}(Ω) with the global constant μ_α on R^n: Lemma 2.2 establishes μ_{α,λ}(Ω)≤μ_α, monotonicity, continuity, and the limit as λ→0; Lemma 3.1 proves a local fractional Hardy-Sobolev inequality with a lower-order L² term; Proposition 3.2 then claims attainment when μ_{α,λ}(Ω)<μ_α and non-attainment when equality holds; Theorem 1.1 follows from a corollary of Lemma 2.2. The central claim of the paper is thus a threshold phenomenon for the attainability of this optimal constant.","tokens_in":11957,"tokens_out":3605,"duration_ms":38143,"significance":"If the proof is repaired, the result is a natural and clean fractional analogue of the local Hardy-Sobolev minimization problem studied by Hashizume and Ghoussoub-Kang, and it gives a precise threshold in λ for the loss of compactness caused by the interior singularity. The paper is concise and its comparison lemmas are mostly sound. A notable strength is that the upper-bound construction in Lemma 2.2 is explicitly benchmarked against the external result of Marano-Mosconi, including the polynomial decay of the global extremal, rather than relying on an ad-hoc ansatz. The main weakness is concentrated in Proposition 3.2(i): as written, the proof of strong convergence of the minimizing sequence is incomplete, and since that step is the core of the attainment claim, the paper requires a substantive but local revision.","major_comments":[{"comment":"The proof that the minimizing sequence converges strongly is not complete. After Brezis-Lieb, the displayed chain establishes only that the two weighted masses, one for u and one for u_k-u, have exponents summing to 1, namely lim_k [(∫|u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}} + (∫|u_k-u|^{2_{s,α}}/|x|^α)^{2/2_{s,α}}] = 1. This is compatible with a nontrivial split in which both terms have positive limits. The sentence 'Since u ≠ 0, we conclude u_k → u strongly' is therefore a non sequitur. The strict inequality μ_{α,λ}(Ω)<μ_α is not used at this point, and the displayed estimate with μ_{α,λ}(Ω) in the denominator cannot rule out a split. The standard repair is to apply the global Hardy-Sobolev inequality (1.3) with constant μ_α to u and to u_k-u separately, using [u]^2+[u_k-u]^2 ≤ [u_k]^2+o(1) and (a+b)^r ≤ a^r+b^r with r=2/2_{s,α}<1. This yields μ_α ≤ μ_{α,λ}(Ω), contradicting the strict inequality assumed in Part (1). Because this step is exactly what proves attainment for λ<λ*, the proof of Theorem 1.1 is incomplete as written, although the gap is local and repairable.","section":"Section 3, Proposition 3.2(i)"}],"minor_comments":[{"comment":"In the estimate of I2, the exponent in the factor d_1^{-2α/2_{s,α}} is misprinted once as '2*_{s,α}' instead of '2_{s,α}'; the surrounding lines show the intended exponent.","section":"Section 3, Lemma 3.1"},{"comment":"The statement 'It is clear that φu∈L^2(Ω)' appears twice in the proof; the second occurrence seems intended for the seminorm finiteness and should be rephrased to avoid repetition.","section":"Section 2, Lemma 2.1"},{"comment":"The abstract speaks of 'existence of nontrivial solutions' to a minimization problem, but the paper actually proves existence and non-existence of minimizers; the wording could be aligned with the theorem statement.","section":"Abstract and Introduction"},{"comment":"Several reference entries appear to contain typographical artifacts (for example, entries [7] and [8]); the final version should ensure the bibliographic data is clean.","section":"References"},{"comment":"The constant test function c is used without explicitly noting that for a bounded domain the constant function belongs to H^s(Ω); a one-line justification would improve clarity.","section":"Section 2, proof of Lemma 2.2(3)"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem is defensible and the identified gap in Proposition 3.2(i) is fixable by a short argument using the global constant μ_α. I would support publication once the author supplies the missing contradiction argument. The reliance on Marano-Mosconi for the external extremal is legitimate and not circular. No concerns about novelty or citation practice."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper does something clean: it extends Hashizume's local Hardy-Sobolev minimization threshold to the fractional setting with an interior singularity, adding an L^2 term and proving that the optimal constant μ_{α,λ}(Ω) is attained exactly below a critical λ*. That is a natural and useful result, and the paper is honest about where the inputs come from: the global constant μ_α and its extremal u_0 are taken from Marano-Mosconi, with the decay estimate |u_0(x)| ≤ C/|x|^{n-2s} that makes the upper-bound construction work. The comparison with an external result is exactly what keeps the circularity burden low.\n\nWhat the paper does well: the local-global comparison Lemma 2.2, the fractional Hardy-Sobolev inequality Lemma 3.1 with the ε-splitting trick, and the nonexistence part of Proposition 3.2 are all standard but correctly executed. The scaling in the upper-bound argument checks out, and n>4s is used where needed. The writing is clear.\n\nThe soft spot is real, and it is the same one I think you will find when you read Proposition 3.2(i). After showing the weak limit u is nonzero, the proof uses the local lower bound with μ_{α,λ} in the Brezis-Lieb chain. That gives 1 ≤ A^{2/q} + B^{2/q} + o(1) ≤ [E(u)+E(u_k-u)]/μ_{α,λ} + o(1) ≤ 1+o(1). This is compatible with a nontrivial split where both A and B are positive and their exponents sum to 1. The sentence \"Since u ≠ 0, we conclude u_k → u strongly\" simply does not follow. The standard repair is to apply the global Hardy-Sobolev inequality (1.3) with constant μ_α to u and u_k-u separately, then use the split-norm inequality and the elementary (a+b)^r ≤ a^r + b^r for r=2/2_{s,α}<1 to get μ_α ≤ μ_{α,λ}, contradicting the strict inequality. That is a short fix, but as written the attainment part of Theorem 1.1 is not proven.\n\nI do not see this as a fatal flaw. The strategy is sound, the missing step is well-understood in the local literature, and the paper is otherwise careful. The external decay assumption flagged by the reader is not an issue: [18, Theorem 1.1] supplies exactly the decay needed.\n\nWho is this for? Anyone working on fractional Hardy-Sobolev inequalities or nonlocal concentration-compactness arguments. It deserves a serious referee, and I would recommend conditional acceptance: ask the author to complete Proposition 3.2(i) with the concentration-exclusion argument, and perhaps to add a remark clarifying the role of the strict inequality. I would not desk-reject this.\n\nRecommendation: send to peer review; likely minor revision.","headline":"A solid, squarely-posed fractional analogue of Hashizume's threshold result, but the attainment proof in Proposition 3.2(i) has a repairable gap that as written leaves Theorem 1.1 incomplete.","tokens_in":12486,"tokens_out":1954,"would_cite":true,"duration_ms":16794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A threshold parameter controls when minimizers exist for the fractional Hardy-Sobolev inequality.","keywords":["fractional Hardy-Sobolev inequality","minimization problem","attainability of optimal constants","inner singularity","fractional Sobolev space","threshold parameter","extremal functions"],"falsifier":"Check the decay of the whole-space extremal used in Lemma 2.2: if the true bound is weaker than $C/|x|^{n-2s}$ at infinity, or if $n\\le 4s$, the integral $\\int_{|y|\\ge 1}|y|^{-2(n-2s)}\\,dy$ diverges and the test function $v_\\varepsilon$ is not in $H^s(\\Omega)$, so the upper-bound comparison collapses. Alternatively, compute $\\mu_{\\alpha,\\lambda}(\\Omega)$ for an explicit domain such as a ball and test numerically whether it ever equals $\\mu_\\alpha$; equality at some $\\lambda$ confirms the nonexistence branch, while strict inequality for all $\\lambda$ would mean $\\lambda_*=\\infty$.","tokens_in":11346,"feed_emoji":"🧮","tokens_out":8484,"duration_ms":70753,"temperature":0.7,"pith_summary":"This paper studies a constrained minimization problem on a bounded domain containing the origin: minimize the fractional Gagliardo seminorm plus $\\lambda$ times the $L^2$ norm, subject to a fixed weighted Hardy–Sobolev norm with weight $|x|^{-\\alpha}$. The main result is a threshold phenomenon: there is a $\\lambda_* \\in (0,\\infty]$ such that the optimal constant $\\mu_{\\alpha,\\lambda}(\\Omega)$ is attained for every $0<\\lambda<\\lambda_*$, and, if $\\lambda_*$ is finite, it is not attained for every $\\lambda>\\lambda_*$. The proof splits on whether the local constant is strictly below the whole-space constant $\\mu_\\alpha$ or equal to it. This gives a complete qualitative description of when minimizers exist for the fractional Hardy–Sobolev inequality with an interior singularity, extending the local picture to the nonlocal setting.","feed_headline":"Critical parameter decides existence of Hardy-Sobolev minimizers","feed_subtitle":"Below a critical λ the optimal constant is attained; above it, only concentrating sequences remain.","key_machinery":"The central object is the comparison between the local constant $\\mu_{\\alpha,\\lambda}(\\Omega)$ and the whole-space constant $\\mu_\\alpha$. Three pieces carry the argument. First, Lemma 2.2 shows $\\mu_{\\alpha,\\lambda}(\\Omega)$ is monotone, continuous, and bounded above by $\\mu_\\alpha$, using a scaled and cut-off whole-space extremal whose decay makes the test functions admissible. Second, Lemma 3.1 establishes a fractional Hardy–Sobolev inequality with an $\\varepsilon$-loss, $\\frac{\\mu_\\alpha}{1+\\varepsilon} \\|u\\|_{s,\\alpha,\\Omega}^2 \\le [u]_{s,\\Omega}^2 + C(\\varepsilon)\\|u\\|_{L^2(\\Omega)}^2$, which prevents a minimizing sequence from vanishing at the singularity. Third, a decomposition identity for weakly convergent sequences upgrades weak convergence to strong convergence once the strict inequality below $\\mu_\\alpha$ holds. The sign of $\\mu_{\\alpha,\\lambda}(\\Omega)-\\mu_\\alpha$ is the switch: negative means attainment, zero means non-attainment.","core_discovery":"The paper's central claim is Theorem 1.1: for $0<s<1$, $n>4s$, $0<\\alpha<2s$ and a bounded domain $\\Omega$ with $0\\in\\Omega$, there exists $\\lambda_*\\in(0,\\infty]$ such that $\\mu_{\\alpha,\\lambda}(\\Omega)$ is attained for every $0<\\lambda<\\lambda_*$, and if $\\lambda_*<\\infty$ it is not attained for every $\\lambda>\\lambda_*$. The mechanism is Proposition 3.2: strict inequality $\\mu_{\\alpha,\\lambda}(\\Omega)<\\mu_\\alpha$ forces the minimizing sequence to converge strongly in the weighted space, so a minimizer exists; equality $\\mu_{\\alpha,\\lambda}(\\Omega)=\\mu_\\alpha$ makes attainment impossible, because any would-be minimizer would have to beat the whole-space constant while living in a bounded domain. The threshold $\\lambda_*$ is defined as the first value of $\\lambda$ where the local constant meets the whole-space constant $\\mu_\\alpha$.","pith_inferences":["If the whole-space extremal's decay were weaker than $|x|^{-(n-2s)}$, the cut-off construction in Lemma 2.2(1) would break down; testing the theorem under a slower decay would show whether the threshold phenomenon depends on that specific estimate.","For the local (non-fractional) analogue, the geometry of the boundary at $0$ can decide attainability; by analogy, whether $\\lambda_*$ is finite for a given $\\Omega$ may depend on the shape of $\\Omega$ near $0$, which the paper does not address.","One direct extension is to compute $\\mu_{\\alpha,\\lambda}(\\Omega)$ for an explicit domain such as a ball; if equality with $\\mu_\\alpha$ never occurs, then $\\lambda_*=\\infty$ and minimizers exist for every $\\lambda$, whereas a finite $\\lambda_*$ would exhibit the predicted nonexistence branch."],"forward_implications":["For every bounded domain containing $0$, the set of parameters for which $\\mu_{\\alpha,\\lambda}(\\Omega)$ is attained is an interval $(0,\\lambda_*)$, possibly the whole half-line.","Whenever $\\mu_{\\alpha,\\lambda}(\\Omega)<\\mu_\\alpha$, every minimizing sequence converges strongly in the weighted Hardy–Sobolev space, so the minimizer is a genuine function in $H^s(\\Omega)$.","If equality $\\mu_{\\alpha,\\lambda}(\\Omega)=\\mu_\\alpha$ holds at some $\\lambda$, no minimizer exists for any larger $\\lambda$; the infimum can only be approached by sequences concentrating near the singularity.","The fractional Hardy–Sobolev inequality of Lemma 3.1 supplies the quantitative control needed to rule out vanishing, so the threshold result holds for all bounded domains with $0\\in\\Omega$, not just for special geometries.","The theorem leaves open whether $\\lambda_*$ is finite or infinite; both alternatives occur through Corollary 2.3."],"supporting_citations":[{"why":"Supplies the whole-space extremal $u_0$ for $\\mu_\\alpha$ and the decay estimate $|u_0(x)|\\le C/|x|^{n-2s}$ used to build admissible cut-off test functions in Lemma 2.2(1).","marker":"[18]"},{"why":"Provides the local-domain minimization strategy that Lemma 3.1 adapts to the fractional setting.","marker":"[15]"},{"why":"Gives the decomposition identity used in Proposition 3.2 to turn weak convergence into strong convergence of minimizing sequences.","marker":"[2]"},{"why":"Supplies the compact embedding $H^s(\\Omega)\\hookrightarrow L^p(\\Omega)$ for $p<2_s^*$ used to extract convergent subsequences.","marker":"[5]"},{"why":"Provides the definition and basic properties of fractional Sobolev spaces used throughout the paper.","marker":"[6]"}],"fun_headline_variants":["Fractional Hardy-Sobolev minimizers exist below critical λ","Threshold λ decides attainment of Hardy-Sobolev constant","Inner singularity: minimizers appear below a critical parameter","Hardy-Sobolev minimization: existence hinges on λ threshold","Attainment of optimal constant in fractional Hardy-Sobolev problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the known whole-space extremal decays as $|u_0(x)|\\le C/|x|^{n-2s}$ at infinity and that $n>4s$; if that decay were slower, the scaled cut-off functions used to prove $\\mu_{\\alpha,\\lambda}(\\Omega)\\le\\mu_\\alpha$ would not be admissible, and the threshold comparison could fail.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Hardy-Sobolev minimizers exist below critical λ","Threshold λ decides attainment of Hardy-Sobolev constant","Inner singularity: minimizers appear below a critical parameter","Hardy-Sobolev minimization: existence hinges on λ threshold","Attainment of optimal constant in fractional Hardy-Sobolev problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000959,"raw_usage":{"total_tokens":4073,"prompt_tokens":919,"completion_tokens":3154,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":3067}},"tokens_in":535,"tokens_out":3154,"duration_ms":22284,"temperature":1.0,"reasoning_tokens":3067,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:25:09.587812+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the decay of the whole-space extremal used in Lemma 2.2: if the true bound is weaker than $C/|x|^{n-2s}$ at infinity, or if $n\\le 4s$, the integral $\\int_{|y|\\ge 1}|y|^{-2(n-2s)}\\,dy$ diverges and the test function $v_\\varepsilon$ is not in $H^s(\\Omega)$, so the upper-bound comparison collapses. Alternatively, compute $\\mu_{\\alpha,\\lambda}(\\Omega)$ for an explicit domain such as a ball and test numerically whether it ever equals $\\mu_\\alpha$; equality at some $\\lambda$ confirms the nonexistence branch, while strict inequality for all $\\lambda$ would mean $\\lambda_*=\\infty$.","supporting_citations":[{"cited_title":"Marano and Sunra J","cited_arxiv_id":null,"evidence_quote":"Supplies the whole-space extremal $u_0$ for $\\mu_\\alpha$ and the decay estimate $|u_0(x)|\\le C/|x|^{n-2s}$ used to build admissible cut-off test functions in Lemma 2.2(1)."},{"cited_title":"24 (2017), no","cited_arxiv_id":null,"evidence_quote":"Provides the local-domain minimization strategy that Lemma 3.1 adapts to the fractional setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the decomposition identity used in Proposition 3.2 to turn weak convergence into strong convergence of minimizing sequences."},{"cited_title":"MR 28951788","cited_arxiv_id":null,"evidence_quote":"Supplies the compact embedding $H^s(\\Omega)\\hookrightarrow L^p(\\Omega)$ for $p<2_s^*$ used to extract convergent subsequences."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the definition and basic properties of fractional Sobolev spaces used throughout the paper."}],"review_version":1}