{"id":"e6872ff6-6bdb-4a4f-8372-d8994483812d","arxiv_id":"1908.08235","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New explicit improved interpolation inequalities on the sphere give sharp stability estimates and lower bounds for optimal constants in the symmetry-breaking range.","lead":"The paper derives sharper versions of known interpolation inequalities on the sphere, with explicit remainder terms, and transfers them to Euclidean space via stereographic projection. A specialist reader learns new quantitative stability estimates and lower bounds for optimal constants in the symmetry-breaking range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exceptional-case inequality (10) in Theorem 1 has the wrong sign: since γ=2−p forces p<2, the stated constant 2d/(p−2) is negative, making the inequality trivial and the sharpness claim false; the correct constant is d/(2−p).","rationale":"The reader identified the imported tensor computation in Lemma 5 as the weakest assumption. That is a legitimate verification gap: the identity is stated as 'found in [19]' and the admissible range is imported from [18], so a wrong or misapplied computation there would undermine all of Theorem 1. However, I found no evidence of an error in that identity, and the paper's internal algebra from Lemma 5 to Theorem 1 is consistent. The load-bearing concern I would press is different and demonstrable: the exceptional-case formula (10) is trivially true with a nonpositive right-hand side, so the sharpness assertion in Theorem 1 is false as stated. This is a sign/constant error rather than a conceptual one, and the proof itself supplies the correct constant. The verdict should therefore be conditional rather than outright acceptance: the paper is acceptable once (10) and the corresponding formula in Proposition 8 are corrected, and once the γ=0 endpoint in Theorem 2(i) is handled or excluded. This does not question the authors' integrity and does not impugn the main Bakry-Emery argument; it is a concrete mathematical correction to a central stated theorem.","tokens_in":17409,"tokens_out":20886,"duration_ms":189753,"concrete_test":"Re-derive (10) as the limit γ→2−p of (9) for fixed u with ‖u‖_{Lp}=1 and A=‖u‖_{L2}^2≥1. Putting θ=γ/(2−p), the bracket in (9) equals A−A^θ, so the right-hand side of (9) is d/(2−p) · (A−A^θ)/(1−θ). Letting θ→1 via l'Hôpital gives (d/(2−p))A log A, i.e., the corrected inequality, not the printed 2d/(p−2)A log A. The same conclusion is reached by substituting φ from (7) into (5) and simplifying. This single algebraic check settles whether the printed exceptional-case constant is a typo: it is.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most concrete defect is in Theorem 1, exceptional case γ=2−p. This case forces p∈(1,2), namely p=p*(d) for d≥2 or d=1 with p=7/4. With the uniform probability measure on S^d, Hölder's inequality gives ‖u‖_{Lp} ≤ ‖u‖_{L2}, so log(‖u‖_{L2}^2/‖u‖_{Lp}^2) ≥ 0. Since p−2<0, the right-hand side of (10), namely 2d/(p−2)‖u‖_{L2}^2 log(‖u‖_{L2}^2/‖u‖_{Lp}^2), is nonpositive for every u. Thus (10) holds trivially and the asserted sharpness of the constant 2d/(p−2) is impossible. The intended inequality follows from the paper's own argument: substituting φ from (7) into (5), or taking the limit γ→2−p in (9), gives ‖∇u‖_{L2}^2 ≥ d/(2−p)‖u‖_{L2}^2 log(‖u‖_{L2}^2/‖u‖_{Lp}^2). This has the correct linearization: for u=1+εv with −Δv=dv, both sides equal d ε^2∫v^2 + o(ε^2). So Theorem 1 is false as printed in the exceptional case, although the underlying method is sound and the correction is straightforward. The same wrong sign appears in the exceptional-case formula of Proposition 8, which has 8d/(p−2) with p−2<0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies subcritical Gagliardo-Nirenberg-Sobolev interpolation inequalities on the sphere, using carré du champ and heat-flow methods. It proposes improved inequalities with explicit remainders (Theorem 1), derives lower bounds for the optimal constants in (1)-(2) (Theorem 2), and uses stereographic projection to obtain weighted Euclidean stability estimates (Theorems 3-4 and Proposition 8). A nonlinear-flow extension covers p in (2, 2*) (Theorem 14). The proof of Theorem 1 is based on a differential inequality for the entropy/Fisher-information pair along the heat flow (Lemma 5), which is quoted from the authors' earlier work.","tokens_in":17756,"tokens_out":12651,"duration_ms":106066,"significance":"The paper contains genuinely useful explicit improved interpolation inequalities: the remainder terms are given in closed form in terms of L^2 and L^p norms, the constants in the main inequality are claimed sharp, and the stability estimates with explicit constants are of interest. The derivation is coherent and no numerical fitting is involved. However, the exceptional-case formulas in Theorem 1 and Proposition 8 contain a sign error, so the central theorem as printed is false in that case. The correction is local and the underlying method remains sound, which makes the paper suitable for publication after a major revision.","major_comments":[{"comment":"The exceptional case gamma = 2 - p is misstated. In that case p lies in (1,2) (namely p = p*(d), with p = 7/4 if d = 1), so p - 2 < 0. Since d mu is a probability measure, Hoelder's inequality gives ||u||_{L^p} <= ||u||_{L^2}, hence log(||u||_{L^2}^2 / ||u||_{L^p}^2) >= 0, and the right-hand side of (10), namely (2d/(p-2)) ||u||_{L^2}^2 log(...), is nonpositive for every u. Thus (10) holds trivially, and the asserted sharpness of the constant 2d/(p-2) is impossible: testing u = 1 + epsilon v with -Delta v = d v gives a left-hand side of order d epsilon^2 integral v^2 and a right-hand side of negative order epsilon^2. Substituting phi from (7) into (5), or taking the limit gamma -> 2 - p in (9), gives the correct inequality ||grad u||_{L^2}^2 >= (d/(2-p)) ||u||_{L^2}^2 log(||u||_{L^2}^2 / ||u||_{L^p}^2), which has the correct second-order expansion: both sides equal d epsilon^2 integral v^2 + o(epsilon^2). Theorem 1 needs this correction; the same error propagates to Remark 9, where the printed (10) is claimed to be stronger than (3).","section":"Section 2, Theorem 1, Eq. (10)"},{"comment":"The exceptional-case formula in Proposition 8 has the same sign error. In the gamma = 2 - p case one has p < 2, so the printed constant 8d/(p-2) is negative, the right-hand side is nonpositive for all v, and the inequality is trivial. The correct constant is 4d/(2-p), obtained as the limit gamma -> 2 - p of the first formula in Proposition 8 or by stereographic projection of the corrected inequality (10). As printed, the proposition is false in this case.","section":"Section 4, Proposition 8, exceptional case"},{"comment":"The central differential inequality (12), namely e'' + 2d e' - gamma |e'|^2 / (1 - (p-2)e) >= 0, is not proved in the manuscript. The text says that the tensor computation 'can be found in [19]' and that the admissible range (8) is 'as shown in [3, 18]', but the reader is not told exactly which statement in those references implies (12), nor are the hypotheses fully restated. Since Theorem 1 and all of its consequences rest on this lemma, the authors should either include the full computation or quote the precise lemma with hypotheses and proof location, so that the range (8) and the formula for gamma are independently verifiable without consulting the earlier papers.","section":"Section 3, Lemma 5"}],"minor_comments":[{"comment":"The displayed formula for lambda(mu) is hard to parse: '2 - p - gamma mu^{1 - (2-p)/gamma} / (2 - p - gamma)' should be written with parentheses, e.g., (2 - p - gamma mu^{1-(2-p)/gamma})/(2-p-gamma), to avoid ambiguity.","section":"Section 2, Theorem 2(i)"},{"comment":"The function phi defined by (19) is not explicit; the authors acknowledge this drawback, but it would be helpful to add a short comment on how the supremum over beta in B(p,d) can be evaluated or bounded in practice for p in (2#,2*).","section":"Section 4, Theorem 14"},{"comment":"The step 'e' = -2i' is stated without explanation; since e is defined with a denominator p-2, the relation depends on the normalization ||u||_{L^p}=1 and on the heat flow (11); a one-line derivation would improve readability.","section":"Section 3, proof of Lemma 5"},{"comment":"The caption says the curves p -> m_{\\pm}(p) enclose the admissible range, but the text defines m_{\\pm}(p,d); please make the dependence on d explicit in the figure or caption.","section":"Figure 2 and Lemma 13"}],"recommendation":"major_revision","confidential_remarks":"The sign error in the exceptional cases of Theorem 1 and Proposition 8 is straightforward to fix and does not undermine the method, but as printed the statements are false. The authors should also make the proof of Lemma 5 self-contained or precisely referenced. Aside from these points, the paper contains useful explicit inequalities and stability estimates and would be suitable for publication after the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, honest extension of the authors' own Bakry-Emery/carré du champ program, and the main non-exceptional results are new and useful. But the printed exceptional case in Theorem 1 is wrong, and the stress-test note has it right. For gamma = 2 - p, p is forced into (1,2), so p - 2 < 0; the constant in (10) is negative and the inequality holds trivially with a false sharpness claim. The correct inequality is ||grad u||_2^2 >= d/(2-p) ||u||_2^2 log(||u||_2^2 / ||u||_p^2), obtained by taking the limit gamma -> 2-p in (9) or directly from (5) with the log-form phi. The same sign error appears in the exceptional case of Proposition 8. This is a statement-level bug in a central theorem, but it is local and the underlying derivation is sound.\n\nWhat is genuinely new: explicit improved inequalities (9) with sharp constants and equality only at constants; lower bounds for mu(lambda) and lambda(mu) in the symmetry-breaking range, including p in [1,2) which previously had no explicit estimates; quantitative stability on R^d (Theorem 3); Section 4 extends to p in (2, 2*) via nonlinear flows, with phi non-explicit but honestly so. The algebra in Theorem 2 checks out, and the paper is careful to note where the method degenerates (p = p*(d), and non-optimality near critical).\n\nSoft spots, in proportion: the key differential inequality (12) in Lemma 5 is imported from [18,19]; the paper is not self-contained at its load-bearing point. That is acceptable if the cited computation is right, but a referee should verify or at least demand a sketch. The admissible-range analysis also comes from the authors' earlier work; heavy self-citation is here legitimate, not circular, because the new inequalities are derived from, not identical to, the prior results. The exceptional-case error, however, is not minor: as printed, Theorem 1 is false for gamma = 2-p and the sharpness statement is meaningless. It clearly needs correction to d/(2-p), and Proposition 8 needs the same fix.\n\nWho is it for: people working on functional inequalities, sharp constants, stability, and spectral estimates on the sphere and R^d. The paper deserves a serious referee; the sign bug does not sink the method, but it must be fixed before publication.\n\nRecommendation: send to peer review with a request for a corrected v2. I would not cite (10) as printed, but I would cite (9) and Theorem 2.","headline":"Solid and honest, but the exceptional-case inequality in Theorem 1 is printed with a wrong sign; fix that and it is a citable paper.","tokens_in":18272,"tokens_out":3927,"would_cite":false,"duration_ms":38304,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","46E35","58E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Improved interpolation inequalities on the sphere have sharp constants, equality only at constants, and yield stability estimates in Euclidean space.","keywords":["interpolation inequalities","Gagliardo-Nirenberg-Sobolev inequalities","sphere","carré du champ","heat flow","stability","optimal constants","stereographic projection"],"falsifier":"Take a concrete admissible case, say $d=3$ and $p=3$, and a nonconstant initial datum on $S^3$; evolve it by (11) and compute $e(t)$ and $i(t)$ directly. If inequality (12) is violated at any finite time, Theorem 1 is false. A cheaper independent check is to verify the tensor identity in the proof of Lemma 5 for this case by direct coordinate computation, since any sign error in $\\gamma$ would visibly break (12).","tokens_in":17205,"feed_emoji":"📐","tokens_out":13271,"duration_ms":115804,"temperature":0.7,"pith_summary":"This paper establishes a family of improved interpolation inequalities on the d-dimensional sphere. For every exponent p in the admissible subcritical range, the gradient energy is bounded below not just by the usual entropy term but by a strictly larger explicit nonlinear function of that entropy, so the slack is a quantitative measure of distance to constant functions. The constants in the improved inequalities are sharp, and constant functions are the only equality cases. A sympathetically minded reader would care because the result converts a qualitative inequality into a stability statement: it yields explicit lower bounds on the optimal constants in the standard interpolation inequalities and, through the stereographic projection, quantitative stability estimates for weighted inequalities in Euclidean space.","feed_headline":"Sharper sphere inequalities come with exact constants","feed_subtitle":"The gap to the optimal constants is measured explicitly; only constants attain equality.","key_machinery":"The load-bearing device is the carré du champ method, a way of extracting inequalities from the second derivative of an entropy along a Markov diffusion; here it is run along the heat flow $\\partial_t u = \\Delta u + (p-1)|\\nabla u|^2/u$, which moves $u^p$ by the heat equation. The entropy $e=\\frac{1}{p-2}(\\|u\\|^2_{L^p}-\\|u\\|^2_{L^2})$ and Fisher information $i=\\|\\nabla u\\|^2_{L^2}$ satisfy the differential inequality (12), imported from a tensor computation; the function $\\phi$ defined by $\\phi'(s)=1+\\frac{\\gamma\\phi(s)}{1-(p-2)s}$, $\\phi(0)=0$, and given explicitly in (7), is then shown by a monotonicity argument to satisfy $i\\ge d\\phi(e)$ for every admissible $p$. The coefficient $\\gamma$ in (6) is exactly what makes the trace-free Hessian computation nonnegative in the admissible range; convexity of $\\phi$ converts the entropy bound into the sharper inequalities (9) and (10).","core_discovery":"On the sphere $S^d$ with uniform probability measure, the paper's central claim is that for every $p\\neq 2$ in the admissible range (8), with $\\gamma$ defined by (6), the inequalities (9) and (10) hold for all $u\\in H^1(S^d)$: the gradient energy is at least $\\frac{d}{2-p-\\gamma}\\left(\\|u\\|^2_{L^2} - \\|u\\|^{2-\\frac{2\\gamma}{2-p}}_{L^p}\\|u\\|^{\\frac{2\\gamma}{2-p}}_{L^2}\\right)$ in the generic case, and at least $\\frac{2d}{p-2}\\|u\\|^2_{L^2}\\log\\left(\\frac{\\|u\\|^2_{L^2}}{\\|u\\|^2_{L^p}}\\right)$ in the exceptional case $\\gamma=2-p$. The constants $\\frac{d}{2-p-\\gamma}$ and $\\frac{2d}{p-2}$ are sharp, as shown by testing with $u=1+\\varepsilon v$ where $-\\Delta v=d v$ in the limit $\\varepsilon\\to 0$, and equality is attained only by constant functions. Because the function $\\phi$ in (7) is convex with $\\phi'(0)=1$, these right-hand sides dominate the classical entropy term, so the inequalities are genuine improvements and the gap between the two sides measures the distance to the optimal functions.","pith_inferences":["Editorial inference: because the method proves $i\\ge d\\phi(e)$, the same inequality controls the decay rate of the entropy along the flow; combining it with the spectral gap should give explicit two-phase convergence rates for the heat flow (11).","Editorial inference: the proof uses only the tensor identities on the sphere, so the same argument is likely to hold on any compact manifold with the corresponding curvature-dimension condition; the sphere is the case where the constants can be computed explicitly.","Editorial inference: a straightforward numerical experiment would minimize the quotient in (9) over spherical harmonics of increasing degree; this would check uniqueness of constants as extremals and reveal whether the next critical value has a simple expression."],"forward_implications":["For every admissible $p$, the classical interpolation inequality (3) is strict away from constant functions, with a quantitative gap that scales like the square of the entropy near $p=2$.","Theorem 2 provides explicit lower bounds on the optimal constants in (1) and (2) for all $\\lambda,\\mu\\ge 1$, including exponents $p\\in[1,2)$ for which no explicit estimate was previously available.","Theorem 3 gives Euclidean stability estimates with explicit constants: the deficit of any admissible $v$ is controlled by a positive expression involving the distance to the extremal $v_\\star = \\langle x\\rangle^{2-d}$.","Under the orthogonality conditions of Theorem 4, the stability term is proportional to the entropy itself rather than its square, matching the structure of the critical Sobolev stability result.","The nonlinear-flow extension in Section 4 carries the improvement to the whole subcritical range $p\\in(2,2^*)$, at the cost of an implicit function $\\phi$ built from a supremum of primitives."],"supporting_citations":[{"why":"It supplies the lengthy tensor computation behind Lemma 5, from which the differential inequality (12) is imported.","marker":"[19]"},{"why":"It fixes the admissible range, supplies the orthogonality estimates, and contains the nonlinear-flow tensor computations used in Section 4.","marker":"[18]"},{"why":"It introduces the carré du champ method and proves the base interpolation inequalities in the admissible range.","marker":"[3]"},{"why":"It provides the hypercontractive diffusion setting where the carré du champ method applies, including the endpoint $p=2^\\#$.","marker":"[2]"},{"why":"It contains earlier improved interpolation inequalities on the sphere and the nonlinear-flow estimates that Section 4 extends.","marker":"[14]"},{"why":"It gives earlier improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds of positive curvature that motivate the improvement term.","marker":"[9]"},{"why":"It supplies the sharp Sobolev inequality on the sphere that fixes the limit case $p=2^*$.","marker":"[5]"},{"why":"It provides the critical Sobolev stability result against which the linear stability term of Theorem 4 is compared.","marker":"[6]"}],"fun_headline_variants":["Sphere inequalities sharpened with exact optimal constants","Sharper constants and stability for sphere interpolation","New stability estimates for optimal sphere inequalities","Explicit gaps tighten sphere interpolation inequalities","Improved sphere bounds with measured distance to optimizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 5: along the heat flow (11), the entropy satisfies $e''+2d e'-\\frac{\\gamma |e'|^2}{1-(p-2)e}\\ge 0$. The paper does not prove this from scratch but imports it from the tensor computation in references [19] and [18], so if that computation is wrong for any admissible exponent, the main theorem loses its foundation.","fun_headline_variants_meta":{"raw":{"variants":["Sphere inequalities sharpened with exact optimal constants","Sharper constants and stability for sphere interpolation","New stability estimates for optimal sphere inequalities","Explicit gaps tighten sphere interpolation inequalities","Improved sphere bounds with measured distance to optimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1408,"prompt_tokens":881,"completion_tokens":527,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":462}},"tokens_in":497,"tokens_out":527,"duration_ms":5705,"temperature":1.0,"reasoning_tokens":462,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:46:21.559205+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete admissible case, say $d=3$ and $p=3$, and a nonconstant initial datum on $S^3$; evolve it by (11) and compute $e(t)$ and $i(t)$ directly. If inequality (12) is violated at any finite time, Theorem 1 is false. A cheaper independent check is to verify the tensor identity in the proof of Lemma 5 for this case by direct coordinate computation, since any sign error in $\\gamma$ would visibly break (12).","supporting_citations":[{"cited_title":"Dolbeault, M","cited_arxiv_id":null,"evidence_quote":"It supplies the lengthy tensor computation behind Lemma 5, from which the differential inequality (12) is imported."},{"cited_title":"Dolbeault, M","cited_arxiv_id":null,"evidence_quote":"It fixes the admissible range, supplies the orthogonality estimates, and contains the nonlinear-flow tensor computations used in Section 4."},{"cited_title":"Bakry and M","cited_arxiv_id":null,"evidence_quote":"It introduces the carré du champ method and proves the base interpolation inequalities in the admissible range."},{"cited_title":"Dolbeault, M","cited_arxiv_id":null,"evidence_quote":"It contains earlier improved interpolation inequalities on the sphere and the nonlinear-flow estimates that Section 4 extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives earlier improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds of positive curvature that motivate the improvement term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the sharp Sobolev inequality on the sphere that fixes the limit case $p=2^*$."}],"review_version":1}