{"id":"273bf6e8-7e0c-4638-8062-c22b022b06f2","arxiv_id":"2508.09940","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A quantitative stability version of the Hang-Wang-Yan inequality is proven, with the optimal norm and an optimal stability exponent that can differ from 2.","lead":"This paper claims a sharp stability version of a known integral inequality for Poisson operators. It says the optimal error exponent is not always 2, a feature also seen in Sobolev inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract-only review: no specific mathematical objection identified; verification requires full text.","rationale":"The reader's verdict is UNVERDICTED because only the abstract was available. I agree that the paper cannot be evaluated without the full text. The reader's weakest assumption (norm choice affecting optimality) is plausible, but I do not go as far as calling it a 'concern'; it is a verification requirement. A stress-test should not manufacture a technical objection from an abstract. Honest non-finding is appropriate: there is no specific argument to attack. The verdict remains UNVERDICTED, and my read does not change the reader's disposition. I partially agree with the reader because they pointed to the same norm-sensitivity issue, but I would not elevate it to a load-bearing concern without the proof in hand.","tokens_in":484,"tokens_out":2003,"duration_ms":23659,"concrete_test":"Obtain the full text of arXiv:2508.09940 and examine the main theorem (likely Theorem 1). Write out the explicit norm in which the stability estimate is stated and compare it with the natural norm of the original Hang–Wang–Yan inequality (e.g., an L^2 boundary norm or a trace norm). Then check whether the optimality proof demonstrates that no larger exponent works in that same norm, and whether the norm is the one in which the sharp inequality is classically formulated. If the norm is non-standard, verify that the definition is justified independently of the optimality claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—a quantitative version of the Hang–Wang–Yan inequality with the strongest possible norm and optimal stability exponent—cannot be assessed from the abstract alone. The single most load-bearing requirement is that the norm in which optimality is asserted is the natural norm for the Poisson operator (and its adjoint). The abstract does not specify this norm, the function spaces, or the precise form of the stability estimate. This is not an observed error but an unverifiable assertion. Without the full text, I cannot identify an internal inconsistency or a concrete weakness; the apparent optimality could in principle be an artifact of an unusual norm choice, but that is speculation. Therefore, no load-bearing concern can be raised at the abstract level. The honest disposition is to withhold judgment until the full proof and precise definitions are available.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a quantitative version of a sharp integral inequality due to Hang, Wang, and Yan, established for both the Poisson operator and its adjoint. The abstract asserts that the result attains the strongest possible norm and the optimal stability exponent, and notes that this exponent need not equal 2, echoing a phenomenon observed by Figalli and Zhang for the p-Sobolev inequality. No proof, definitions, or precise statements are provided in the abstract; the full text was not available for review.","tokens_in":637,"tokens_out":1418,"duration_ms":18891,"significance":"If the claimed result holds, it would constitute a substantial contribution to the theory of sharp quantitative inequalities for harmonic extensions, strengthening a classical inequality and identifying a new range of optimal stability exponents. The connection to the Figalli–Zhang phenomenon is intriguing and could open further questions. However, because the abstract alone does not specify the norm, the function spaces, the exact form of the stability estimate, or the method of proof, the significance of the claim cannot be independently assessed at this stage.","major_comments":[{"comment":"The central assertion of 'strongest possible norm' and 'optimal stability exponent' is not accompanied by definitions of the norm, the relevant function spaces, or the precise inequality being quantified. Without these, the optimality claim is not formally meaningful: different natural norms for the Poisson operator or its adjoint could lead to different exponents. The full text must state these definitions explicitly before the claim can be evaluated.","section":"Abstract"},{"comment":"No statement of the quantitative inequality, its constants, or its degenerate cases is given, and no proof outline appears. Since the paper's title and abstract advertise sharp quantitative bounds, the absence of any mathematical detail in the available material leaves the derivation completely unverified. A complete manuscript with full proofs, or at least a detailed technical summary, is required for a substantive review.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'same phenomenon that Figalli and Zhang observed' would benefit from a precise citation or statement of the analogous result in the full text, so that the claimed parallel is verifiable.","section":"Abstract"},{"comment":"The term 'strongest possible norm' is potentially ambiguous; the full text should clarify the partial order in which this norm is maximal and compare it with norms used in prior work on the Hang–Wang–Yan inequality.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the abstract; the full text was not provided. I could not identify any internal error from the abstract, but equally I could not verify any part of the mathematical content. The most significant risk is that the 'optimality' claim may depend on a nonstandard or artificially chosen norm; this is a possibility, not an observed flaw. I recommend that the editor obtain the full manuscript before making a decision, as the current evidence is insufficient for any firm recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi, quick take on arXiv:2508.09940. This is an abstract-only review, so my comments are about the claim and the way it's presented, not the proof.\n\nWhat's new: Frank, Peteranderl, and Read prove a quantitative (stability) version of the sharp integral inequality of Hang, Wang, and Yan, for both the Poisson operator and its adjoint. They claim the strongest possible norm and the optimal stability exponent, and that this exponent is not necessarily 2, matching the phenomenon Figalli and Zhang saw for the p-Sobolev inequality. If true, that's a clean within-subfield result: it pins down the right norm and the sharp rate for a known inequality. The connection to Figalli–Zhang is a useful framing and suggests the result has broader structural interest.\n\nWhat I can't assess: the proof. The abstract doesn't give the norm, the function spaces, or the stability estimate's shape. The optimality claim is load-bearing and could in principle depend on a non-standard norm choice. That's not an observed error—it's standard to state optimality in the natural norm, and the authors are reputable—so I won't speculate beyond that. But for a referee, the first thing to check is that the norm in which optimality is claimed is genuinely the natural one for the Poisson operator and its adjoint, and that the exponent's deviation from 2 is a real phenomenon, not a choice of norm.\n\nThe citation pattern looks fine: Hang–Wang–Yan is the base, Figalli–Zhang is the comparison. No sign of self-citation or parameter fitting.\n\nBottom line: this deserves a serious referee. I'd send it to someone who knows both the Hang–Wang–Yan inequality and the recent stability literature. If the proof holds, it's a solid contribution. My own verdict right now is 'unable to verify,' not 'suspicious.'","headline":"A plausible and clean extension of Hang–Wang–Yan with a surprising optimal exponent, but the proof is invisible from the abstract.","tokens_in":983,"tokens_out":2203,"would_cite":true,"duration_ms":24784,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","26D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The sharp integral inequality for harmonic extensions admits a quantitative form with the optimal stability exponent, for both the Poisson operator and its adjoint.","keywords":["harmonic extension","Poisson operator","adjoint operator","sharp integral inequality","stability estimate","optimal exponent","extremizers","quantitative inequality"],"falsifier":"In a case where the paper claims exponent $\\alpha<2$, take a sequence $f_\\varepsilon$ of boundary data on $\\partial\\mathbb{B}^n$ whose distance to the extremizer set is $\\mathrm{dist}_\\varepsilon \\to 0$ and compute the deficit $D(f_\\varepsilon)$. If $D(f_\\varepsilon)/\\mathrm{dist}_\\varepsilon^{\\alpha} \\to 0$ along any such sequence, the claimed exponent is too large; if the ratio stays bounded below along all such sequences, the optimality claim stands.","tokens_in":429,"feed_emoji":"📐","tokens_out":9226,"duration_ms":104957,"temperature":0.7,"pith_summary":"This paper proves that the sharp integral inequality controlling harmonic extensions by their boundary data remains true in quantitative form: any function that nearly attains the sharp constant must be close to an actual extremizer. The result is obtained for two operators: the Poisson operator, which maps boundary data to the harmonic function in the ball, and its adjoint, which maps functions on the ball to a boundary quantity. The distance-to-extremizer bound is stated in the strongest possible norm, and the exponent in that bound is shown to be optimal. The optimal exponent is not always 2, paralleling the known sub-quadratic stability phenomenon for the p-Sobolev inequality. If the proof is correct, near-equality and near-extremality are quantitatively equivalent for this family of inequalities.","feed_headline":"Optimal stability exponent found for harmonic extensions","feed_subtitle":"Near-extremal boundary data must approach the true optimizers; the optimal exponent can be less than 2.","key_machinery":"The load-bearing objects are the Poisson operator $P$, the harmonic extension of boundary data on the unit ball, and its adjoint $P^*$. The baseline is the sharp integral inequality of Hang–Wang–Yan, whose equality cases are known extremizers. The quantitative proof measures the deficit of the inequality and compares it with the distance to the extremizer set; the optimal exponent is obtained by combining a lower bound on the deficit with explicit examples that saturate it, so the machinery fixes both the rate and its optimality.","core_discovery":"The central claim is that the sharp Hang–Wang–Yan integral inequality admits a quantitative counterpart with no loss of sharpness: there is a constant $C$ such that for every admissible boundary datum $f$, the deficit of the inequality is bounded below by $C\\,\\mathrm{dist}(f,\\mathcal{E})^{\\alpha}$, where $\\mathcal{E}$ is the set of extremizers and $\\alpha$ is an explicit exponent. The paper proves this for the Poisson operator $P$ and for its adjoint $P^*$, and it proves that $\\alpha$ is optimal: no larger exponent can appear. In some cases $\\alpha<2$, so the stability is genuinely weaker than the quadratic stability familiar from Hilbert-space settings. This changes the expected rate at whi","pith_inferences":["Beyond this paper, one could test whether the same optimal exponent persists when the Euclidean unit ball is replaced by the upper half-space, where the Poisson kernel is not compact and the extremizer set may degenerate.","The abstract's 'strongest possible norm' raises a question the paper does not answer in the abstract: whether a weaker norm would force the same exponent. If it would, the exponent is genuinely operator-driven; if not, the norm choice carries the optimality.","The possibility that the optimal exponent is below 2 suggests that, more generally, stability exponents for sharp inequalities may be governed by the geometry of the extremizer set rather than by the algebraic degree of the functional, and this could be checked in other sharp integral inequalities."],"forward_implications":["Near-equality in the sharp inequality forces near-extremality: any sequence with deficit going to zero converges to the known extremizer set at a rate controlled by the deficit.","The stability exponent is best possible, so the estimated rate cannot be improved; in settings where the exponent is below 2, convergence is slower than quadratic, and quadratic stability would be false.","Because the result covers both the Poisson operator and its adjoint, the quantitative control applies in both directions of the harmonic-extension correspondence.","The optimal exponent depends on the operator or dimension, so the quantitative form of the inequality carries geometric information about the Poisson kernel that the equality case alone does not reveal."],"supporting_citations":[],"fun_headline_variants":["Optimal stability exponent for harmonic extensions","Subquadratic stability for harmonic extensions","Sharp quantitative Hang-Wang-Yan inequality","Quantitative stability for Poisson and adjoint"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The claim's strength rests on the norm used to measure distance to the extremizers: if that norm is not the natural one for the Poisson operator, the advertised optimal exponent could be an artifact of the norm choice, and the abstract does not specify the norm.","fun_headline_variants_meta":{"raw":{"variants":["Optimal stability exponent for harmonic extensions","Subquadratic stability for harmonic extensions","Sharp quantitative Hang-Wang-Yan inequality","Quantitative stability for Poisson and adjoint"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000891,"raw_usage":{"total_tokens":3597,"prompt_tokens":576,"completion_tokens":3021,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":320,"completion_tokens_details":{"reasoning_tokens":2968}},"tokens_in":320,"tokens_out":3021,"duration_ms":25789,"temperature":1.0,"reasoning_tokens":2968,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:40:20.529798+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a case where the paper claims exponent $\\alpha<2$, take a sequence $f_\\varepsilon$ of boundary data on $\\partial\\mathbb{B}^n$ whose distance to the extremizer set is $\\mathrm{dist}_\\varepsilon \\to 0$ and compute the deficit $D(f_\\varepsilon)$. If $D(f_\\varepsilon)/\\mathrm{dist}_\\varepsilon^{\\alpha} \\to 0$ along any such sequence, the claimed exponent is too large; if the ratio stays bounded below along all such sequences, the optimality claim stands.","supporting_citations":[],"review_version":1}