{"id":"fd80c26e-5e10-4068-96a1-56ea4674942c","arxiv_id":"2608.07030","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The known Meyers interval is widened to ((26K−16)/(13K−3), (26K−16)/(13K−13)), conditional on a strong Riesz-transform norm bound from a cited preprint.","lead":"This note proves a wider L^p regularity interval for weak solutions of second-order elliptic equations with rough coefficients. The improvement comes from a refined norm estimate for the Riesz transform, but the estimate itself is taken from an unreviewed preprint.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 rests entirely on the unproved imported bound ∥R⊗R∥_{p,p} ≤ p*−1 (Proposition 2, from Cassese's arXiv preprint); without an independent proof or verification, the claimed interval is conditional.","rationale":"The reader's verdict of CONDITIONAL is appropriate. The internal reasoning is coherent: Proposition 2 implies ∥T∥_{6,6}≤11, Riesz–Thorin gives Lemma 1, and solving 1+2.6(p*−2)<(K+1)/(K−1) yields the stated endpoints. We checked the algebra: the lower endpoint (26K−16)/(13K−3) and upper endpoint (26K−16)/(13K−13) correspond exactly to the two branches p*=p' and p*=p. The chord/tangent argument in Lemma 1 is structurally sound, and the numerical inequality appears true. The unique load-bearing step is the imported Cassese bound. Since this bound is unreviewed and unproved in the note, the result is conditional, not established. There is no reason to reject the paper outright; if the bound is later verified, the theorem stands. Thus no change to the reader's verdict is needed.","tokens_in":3357,"tokens_out":15972,"duration_ms":141960,"concrete_test":"Independently verify Cassese's Proposition 4.1, at least for the endpoint used in Lemma 1: d=2, p=6. Compute the L^6(R^2;R^2) operator norm of the Fourier multiplier M(ξ)=ξξ^T/|ξ|^2 by power iteration on a high-resolution periodic grid (e.g., 4096^2 points) with a smooth high-frequency test ensemble. If the numerical norm exceeds 5, or fails to decrease with resolution, Proposition 2 is false and Theorem 1 collapses. If it is below 5, also recompute G(θ) in Lemma 1 with interval arithmetic to replace the approximate values; this would remove the secondary numerical gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2 (page 3) asserts ∥R⊗R∥_{p,p} ≤ p*−1, citing Cassese's arXiv preprint. This bound is the sole source of the endpoint estimate ∥T∥_{6,6} ≤ 11 used in Lemma 1: by triangle inequality, 1 + 2∥R⊗R∥_{6,6} ≤ 11. With the previously established [2] estimate ∥T∥_{6,6} ≤ 30, the Riesz–Thorin step in Lemma 1 would only yield a much weaker bound and the interval of Theorem 1 would not follow. The proof of Proposition 2 from the matrix-valued Proposition 1 is a valid restriction argument, but Proposition 1 is not proved in this note and the reference is an unreviewed preprint. Thus the whole improvement over Iwaniec–Sbordone is conditional on a single external inequality whose correctness is not established here. Secondary but nonfatal: the numerical values in Lemma 1 are stated as approximations ('≈0.358', '≈0.1591'), which can be made rigorous by interval arithmetic but are not fully justified as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The note claims an improvement of the dimension-free Meyers interval for the L^p gradient estimate of weak solutions to div(M∇u)=div F. The main result, Theorem 1, states that the Meyers interval contains ((26K−16)/(13K−3), (26K−16)/(13K−13)). The proof proceeds by importing a norm estimate for the tensor product of Riesz transforms (Proposition 2, itself a restriction of Proposition 1 from Cassese's arXiv preprint), using it to bound ∥T∥_{6,6} ≤ 11, interpolating between L^2 and L^6 to obtain the bound ∥T∥_{p,p} ≤ 2.6(p*−2)+1 in Lemma 1, and finally combining this with the Iwaniec–Sbordone criterion (4). The proof of Lemma 1 is logically transparent: the Riesz–Thorin step, the reduction to a one-variable inequality, and the convexity argument are all clearly presented, conditional on the imported bound and on the numerical evaluations being made rigorous.","tokens_in":3563,"tokens_out":6926,"duration_ms":68853,"significance":"If the imported bound ∥R⊗R∥_{p,p} ≤ p*−1 is valid, the result would genuinely improve the best known dimension-free Meyers interval of Iwaniec and Sbordone. The argument has notable strengths: it introduces no free parameters, it is not circular (the theorem follows transparently from an explicitly stated external estimate plus an elementary interpolation argument), and the structure of the proof is easy to verify line by line. The claimed interval is also asymptotically consistent with existing bounds for large K. However, the central load-bearing estimate is not proved in the manuscript and is taken from an unreviewed arXiv preprint, and the numerical checks in Lemma 1 are stated with approximate values rather than rigorous inequalities. The significance of the contribution is therefore conditional on external validation and on tightening the proof of Lemma 1.","major_comments":[{"comment":"The entire improvement over the Iwaniec–Sbordone interval rests on Proposition 2, but Proposition 2 is proved only as a restriction of Proposition 1 of the unreviewed arXiv preprint [3], and Proposition 1 itself is not proved in this note. This is load-bearing: in Lemma 1 the endpoint value ∥T∥_{6,6} ≤ 11 is exactly 1 + 2(6−1); if one instead used the previously available estimate ∥T∥_{6,6} ≤ 30 from [2], the Riesz–Thorin step would not give (7), and the interval of Theorem 1 would not follow. The note must either provide an independent proof of Proposition 1 (or a direct proof of Proposition 2) or cite a peer-reviewed source for it.","section":"Section 2, Proposition 2"},{"comment":"The proof that G(θ) ≥ 0 on [0,1] uses the numerical evaluations G(θ₀) ≈ 0.358 and G(θ₁) ≈ 0.1591, G′(θ₁) ≈ −0.0081, with no error bounds. These approximate values are used to conclude positivity of the chord and the tangent line, so they are part of the proof. The argument can likely be made rigorous with interval arithmetic or by exhibiting explicit rational or exponential bounds, but as written the proof is not complete. Please replace all approximate evaluations in this step by rigorous inequalities or certified interval computations.","section":"Section 2, Lemma 1"},{"comment":"The abstract and Theorem 1 concern a bounded domain Ω with Dirichlet boundary conditions, but the proof is carried out on Ω = R^d. The sentence 'everything stated below easily extends to domains with an appropriately smooth boundary' is an assertion, not a proof. The Riesz-transform argument is global in nature, and the transfer to bounded domains with discontinuous coefficients requires a localization or extension argument. The manuscript should either supply that argument or explicitly restrict the theorem to R^d.","section":"Section 1, reduction to R^d"}],"minor_comments":[{"comment":"The notation p* is used in Conjecture 2, Proposition 2, and Lemma 1 but is never defined. From context and from [4] the reader must infer that p* denotes max{p, p/(p−1)}; please define it explicitly.","section":"Throughout"},{"comment":"Reference [3] is an arXiv preprint; please state its current publication status if available, and ensure that a result so central to the paper's main theorem is not left dependent on an unreviewed source.","section":"References"},{"comment":"The sentence 'Since T is self-adjoint on L^2, duality yields ∥T∥_{p,p} = ∥T∥_{p′,p′}' is correct only if one also notes that the matrix operator T is symmetric in the sense that R_i and R_j commute; this is true for Riesz transforms, but the reason should be stated briefly.","section":"Section 2, proof of Lemma 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main theorem is conditional on a single external bound taken from an unreviewed arXiv preprint. If the journal's policy allows cited preprints as the sole basis for a central estimate, the result is interesting but incomplete; otherwise the note needs a proof of Proposition 2 or a published reference. The numerical gap in Lemma 1 is minor in effort but must be fixed. There is no indication of self-citation or data fitting; the argument is transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper. The short version: if the imported Riesz-transform bound is true, this is a solid improvement; if not, the theorem doesn't follow. The gain over Iwaniec–Sbordone is essentially a corollary of ∥R⊗R∥_{p,p} ≤ p*−1, cited from an unreviewed arXiv preprint and never proved here.\n\nWhat's genuinely new: the interval ((26K−16)/(13K−3), (26K−16)/(13K−13)) and Lemma 1, the interpolation argument that turns the endpoint ∥T∥_{6,6} ≤ 11 into the linear estimate ∥T∥_{p,p} ≤ 2.6(p*−2)+1. The interpolation step is elementary but tidy; the numerical checks have enough slack to be made rigorous with interval arithmetic, so the approximations in the proof aren't a real problem.\n\nThe soft spot is exactly where the reader puts it: Proposition 2. The paper's own part is honest and transparent—Proposition 2 is a valid restriction from Cassese's matrix result, and the rest of the proof follows. But the whole improvement rests on one inequality that the paper neither proves nor independently verifies. That's not circularity and not a data-fitting trick; it's an unproved external input. A referee cannot accept this as a complete proof without checking Cassese's preprint. The author's Lemma 1 is a clever observation, but it's not the hard part.\n\nWho this is for: people working on Lp regularity and Riesz transform estimates. It is short and readable in an afternoon. It deserves a serious referee, but the referee's central job is to verify Proposition 2 or demand a proof. If the bound holds up, this is a clean advance; if it doesn't, it's an interesting conditional exercise. I wouldn't cite it in my own work until the bound is independently confirmed.","headline":"A short note improving the Meyers interval by elementary interpolation, but the gain is conditional on an unproved Riesz-transform bound from an arXiv preprint.","tokens_in":4116,"tokens_out":5304,"would_cite":false,"duration_ms":48529,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B45","42B37","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This note widens the dimension-free Meyers interval for gradient estimates of elliptic divergence-form equations, proving the gradient is \\(L^p\\)-bounded for every \\(p\\) between \\((26K-16)/(13K-3)\\) and \\((26K-16)/(13K-13)\\).","keywords":["Riesz transform","second order elliptic operators","Meyers theorem","Lp estimates","divergence form equations","Riesz-Thorin interpolation","measurable coefficients","dimension-free estimates"],"falsifier":"Compute the operator norm of \\(R\\otimes R\\) on vector-valued \\(L^p(\\mathbb{R}^d)\\) for a specific case, say \\(d=2\\) and \\(p=6\\), and check whether it is at most \\(p^*-1=5\\). A value above 5 would falsify Proposition 2 and with it the bound \\(\\|T\\|_{6,6}\\le 11\\) that Lemma 1 relies on.","tokens_in":3118,"feed_emoji":"📐","tokens_out":13087,"duration_ms":116091,"temperature":0.7,"pith_summary":"This note improves the best known dimension-free range of exponents \\(p\\) for which gradients of weak solutions to the elliptic divergence-form equation \\(\\operatorname{div}(M\\nabla u)=\\operatorname{div}F\\) satisfy \\(\\|\\nabla u\\|_p\\le C\\|F\\|_p\\) when the coefficient matrix \\(M\\) is uniformly elliptic with ratio \\(K>1\\). The author proves that the Meyers interval contains \\(\\left(\\frac{26K-16}{13K-3},\\frac{26K-16}{13K-13}\\right)\\) for every dimension \\(d\\ge2\\). The improvement comes from bounding the Riesz-transform operator \\(T=I+2(R\\otimes R)\\) by \\(\\|T\\|_{p,p}\\le 2.6(p^*-2)+1\\), where \\(p^*=\\max(p,p/(p-1))\\). If the argument is valid, it narrows the gap toward the conjectured optimal interval \\((2K/(K+1),2K/(K-1))\\) in dimensions above two.","feed_headline":"Gradient estimates gain a wider p-range","feed_subtitle":"The new dimension-free range beats the previous best for every K>1 and every dimension d≥2.","key_machinery":"The load-bearing object is \\(T=I+2(R\\otimes R)\\), where \\(R=(R_1,\\dots,R_d)\\) is the vector Riesz transform and \\(R\\otimes R\\) acts on vector-valued functions by \\((R\\otimes R F)_i=\\sum_j R_iR_jF_j\\). The earlier criterion [4] says that \\(p\\) lies in the Meyers interval whenever \\(\\|T\\|_{p,p}<(K+1)/(K-1)\\). The new work is Lemma 1, which bounds the left side by \\(2.6(p^*-2)+1\\). To get there, the paper uses the bound \\(\\|R\\otimes R\\|_{p,p}\\le p^*-1\\), interpolates between \\($L^{2}$\\) and \\($L^{6}$\\) with \\(\\|T\\|_{6,6}\\le 11\\), and extends the bound to \\(p<2\\) by duality.","core_discovery":"The paper's central claim is Theorem 1: for every \\(K>1\\) and every dimension \\(d\\ge2\\), the interval \\(\\left(\\frac{26K-16}{13K-3},\\frac{26K-16}{13K-13}\\right)\\) is contained in the set of exponents \\(p\\) for which \\(\\|\\nabla u\\|_p\\le C\\|F\\|_p\\) holds for weak solutions of \\(\\operatorname{div}(M\\nabla u)=\\operatorname{div}F\\) with Dirichlet boundary conditions. The proof proceeds by improving the norm bound on the operator \\(T=I+2(R\\otimes R)\\). Using the estimate \\(\\|R\\otimes R\\|_{p,p}\\le p^*-1\\), the triangle inequality, Riesz\\textendash Thorin interpolation between \\($L^{2}$\\) and \\($L^{6}$\\), and duality, the paper obtains \\(\\|T\\|_{p,p}\\le 2.6(p^*-2)+1\\). Combining this with the known criterion \\(\\|T\\|_{p,p}<(K+1)/(K-1)\\) yields the claimed interval.","pith_inferences":["Beyond the paper: if the quoted bound \\(\\|R\\otimes R\\|_{p,p}\\le p^*-1\\) is independently verified, the same interpolation scheme could be run with an \\(L^q\\) bound at a different \\(q>2\\), producing a parametric family of intervals and possibly wider endpoints.","Beyond the paper: because the proof is dimension-free, it does not exploit the special structure of \\(d=2\\), where the conjectured optimal interval is already known; deciding optimality in higher dimensions would require a dimension-dependent argument.","Beyond the paper: a direct numerical or analytic test of the Riesz-transform bound would settle the paper's contribution without revisiting the elliptic PDE itself."],"forward_implications":["For every \\(K>1\\) and every \\(d\\ge2\\), the gradient estimate holds on an interval strictly wider than the previous dimension-free interval from [4].","As \\(K\\to1^+\\), both endpoints tend to \\(2\\), matching the energy estimate; for large \\(K\\), the interval length grows roughly like \\(20/(13K)\\).","Because the only operator bound used at \\(p=6\\) is \\(\\|T\\|_{6,6}\\le 11\\), any sharper value there directly widens the interval; the note itself remarks that the constant \\(2.6\\) can be lowered to \\(2.55\\).","The result transfers from \\(\\mathbb{R}^d\\) to bounded \\(C^2\\) domains with Dirichlet conditions by standard localization, so the estimate holds in the original boundary-value setting."],"supporting_citations":[{"why":"Supplies the bound \\(\\|R\\otimes R\\|_{p,p}\\le p^*-1\\) on matrix-valued \\(L^p\\), from which Proposition 2 and the key interpolation input \\(\\|T\\|_{6,6}\\le 11\\) are derived.","marker":"[3]"},{"why":"Provides the criterion \\(\\|T\\|_{p,p}<(K+1)/(K-1)\\) that turns the norm bound into a Meyers interval, and supplies the prior dimension-free interval that Theorem 1 improves.","marker":"[4]"},{"why":"Defines the original \\(L^p\\) gradient estimate for divergence-form elliptic equations and gives the example showing optimality in dimension two.","marker":"[5]"}],"fun_headline_variants":["Gradient estimate p-range widens for all dimensions","Dimension-free p-interval beats prior best for gradient","Meyers' theorem: wider p-range for elliptic estimates","New p-range for gradient bounds, any K and d","Gradient p-range improved: beats prior best, dimension-free"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on an unproved estimate for a certain Riesz-transform operator: its norm on \\(L^p\\) is at most \\(p^*-1\\); if that estimate is wrong, the interval collapses.","fun_headline_variants_meta":{"raw":{"variants":["Gradient estimate p-range widens for all dimensions","Dimension-free p-interval beats prior best for gradient","Meyers' theorem: wider p-range for elliptic estimates","New p-range for gradient bounds, any K and d","Gradient p-range improved: beats prior best, dimension-free"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1598,"prompt_tokens":920,"completion_tokens":678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":597}},"tokens_in":536,"tokens_out":678,"duration_ms":7130,"temperature":1.0,"reasoning_tokens":597,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:18:26.531675+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the operator norm of \\(R\\otimes R\\) on vector-valued \\(L^p(\\mathbb{R}^d)\\) for a specific case, say \\(d=2\\) and \\(p=6\\), and check whether it is at most \\(p^*-1=5\\). A value above 5 would falsify Proposition 2 and with it the bound \\(\\|T\\|_{6,6}\\le 11\\) that Lemma 1 relies on.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the criterion \\(\\|T\\|_{p,p}<(K+1)/(K-1)\\) that turns the norm bound into a Meyers interval, and supplies the prior dimension-free interval that Theorem 1 improves."},{"cited_title":"G.: AnL p-estimate for the gradient of solutions of second-order elliptic divergence equations","cited_arxiv_id":null,"evidence_quote":"Defines the original \\(L^p\\) gradient estimate for divergence-form elliptic equations and gives the example showing optimality in dimension two."}],"review_version":1}