{"id":"905c8ea2-9f61-45a4-9b16-6ac9fd66af26","arxiv_id":"2504.12407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove off-diagonal and general-basis Rubio de Francia extrapolation for matrix weights, and show multiparameter bases satisfy the required maximal operator bound.","lead":"This paper proves new 'extrapolation' theorems for matrix-valued weights, showing that a single weighted inequality for an operator implies a whole family of inequalities. It also extends the theory to general collections of sets called bases, including multiparameter rectangles.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.1 rests on an unpublished matrix self-improvement result ([12]); if that result is unavailable, the multiparameter basis verification and hence the main extrapolation theorem's main example are unsupported.","rationale":"Reader's verdict is CONDITIONAL, and my read agrees. The main off-diagonal extrapolation theorem (Theorem 1.10/4.1) is proven in detail and the algebra of exponents checks out; I did not find an internal inconsistency there. The weakest spot is the verification that multiparameter bases are matrix Muckenhoupt bases, because Lemma 6.1 is the only place where the slice weights are shown to be one-parameter A_p weights. That lemma needs F and G in L^s_loc for JMZ differentiation, and as written this rests on [12, Theorem 1.5], an unpublished preprint by the first author and Penrod. This is genuinely load-bearing: without it, the uniform control of the reducing operators W^p_{R_l} in (6.2) is not established, and therefore Theorem 1.12—the only nontrivial example family for Theorem 1.10 beyond cubes—is unsupported. The concern is not that the theorem is false; it is that a key ingredient is not independently verifiable from the published literature. In fact, a published route may exist: W∈A_{p,Q} should imply |W|_{op}∈A_{p,Q} and |W^{-1}|_{op}∈A_{p',Q}, and scalar reverse Hölder would give the required L^s_loc. The paper does not take this route, so the concrete test is to see whether it works; if yes, replace the citation and the concern disappears. I therefore keep the reader's CONDITIONAL verdict.","tokens_in":30997,"tokens_out":29008,"duration_ms":276076,"concrete_test":"Independently re-derive the L^s_loc step of Lemma 6.1 without invoking [12, Theorem 1.5]. Specifically, check whether W∈A_{p,Q} implies |W|_{op}∈A_{p,Q} and |W^{-1}|_{op}∈A_{p',Q} (via [2, Cor. 6.7] or a direct estimate), and then use scalar reverse Hölder to get |W|_{op}∈L^{p(1+ε)} and |W^{-1}|_{op}∈L^{p'(1+ε)} for some ε>0. If this places F_{Q_k,v} and G_{Q_k,v} in L^s_loc with s=1+ε' >1, rewrite the paragraph replacing the [12] citation; the multiparameter proof is then self-contained modulo [2]. If the scalar reduction does not give L^s_loc, supply a full proof or published reference for [12, Theorem 1.5] before Theorem 1.12 can be considered verified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point is Lemma 6.1. To prove each slice W_{\\hat{x}_k}(·) is in A_p with uniform constant, the authors must apply the Jessen–Marcinkiewicz–Zygmund theorem to the functions F_{Q_k,v}(\\hat{x}_k)=∫_{Q_k}|W(x_k,\\hat{x}_k)v|^p dx_k and G_{Q_k,v} defined analogously with W^{-1}. JMZ requires F,G ∈ L^s_loc for some s>1. The paper obtains this solely from [12, Theorem 1.5], an unpublished preprint by the first author and M. Penrod, asserting that W∈A_{p,Q} gives W∈A_{sp,Q} and W^{-1}∈A_{sp',Q} for some s>1 (Section 6, p. 26). If that theorem is false, or merely unavailable to a referee, there is no demonstrated s>1, strong differentiation with respect to Rα rectangles may fail (Saks), and the uniform bound on the reducing operators W^p_{R_l} does not follow. Since Theorem 6.3 and Theorem 1.12 are the only nontrivial examples of matrix Muckenhoupt bases given, Theorem 1.10 loses its multiparameter application. The concern is verification rather than an identified contradiction: the needed integrability might follow from the standard scalar fact |W|_{op}∈A_{p,Q}, |W^{-1}|_{op}∈A_{p',Q} plus scalar reverse Hölder, but the paper does not make that argument and instead cites the unpublished matrix self-improvement result.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-16T12:35:49.018730+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}