{"id":"57711bd3-fa43-455d-924b-cb18ec92b216","arxiv_id":"2506.05993","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New necessary and sufficient side conditions characterize exactly when Fujii-Wilson, reverse Hölder, B∞, Bp, and reverse Jensen conditions are equivalent for arbitrary Békollé-Bonami weights.","lead":"This paper pins down exactly which extra conditions make the standard Békollé-Bonami weight classes equivalent for arbitrary weights in the unit disk. It introduces simple, testable side conditions and proves they are both necessary and sufficient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5's proof of B∞⇒FW uses the bound MQw ≤ w_Q λ^k on F_j^k, but the correct bound is ≤ w_Q λ^{k+1}; since λ>4, the displayed estimate is false and the proof chain for Theorems 1.1–1.2 is incomplete as written.","rationale":"The reader's weakest assumption was the quantitative range in Lemma 3.6. I checked Lemma 3.6 and found the layer-cake and Calderón–Zygmund argument correct: M_Qw ≤ M(w·X_Q) gives ∫_Q M_Qw ≤ Lw(Q), the local identity M_Qw = M_{Q_j^λ}w on CZ boxes is valid, and the absorption condition p < 4L/(4L−1) is exactly 4L(p−1)/p < 1. So I do not share that concern. The central claim is otherwise supported by consistent CZ arguments, with the side conditions used essentially. The soft spot I found is the level-set estimate in Lemma 3.5, which is load-bearing because B∞⇒FW is needed for Theorems 1.1 and 1.2. Since the B∞⇒FW implication is already known from the literature, the correct resolution is to repair the displayed bound or cite the known theorem; this does not overturn the central claim. The reader's conditional verdict remains appropriate, now with the additional Lemma 3.5 item to fix alongside the Theorem 5.1 sketch.","tokens_in":18068,"tokens_out":39663,"duration_ms":380501,"concrete_test":"Recompute the estimate in Lemma 3.5 with MQw ≤ w_Q λ^{k+1} on F_j^k. If the resulting sum is bounded by C(α,β)w(Q), using w(F_j^k) ≥ (1−β)w(Q_j^k) and |F_j^k|/|Q_j^k| ≥ 1−4/λ, then the lemma is valid and the flaw is a typographical error; otherwise the proof of B∞⇒FW must be replaced by a citation or a new argument. A secondary check: confirm that no later lemma uses the specific constant 1/(1−β), only its finiteness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 3.5 (proof that B∞ implies FW), the sets F_j^k = Q_j^k \\ ∪_l Q_l^{k+1} decompose {MQw > w_Q}. The proof asserts ∫_{F_j^k} MQw ≤ w_Q λ^k |F_j^k|. This is not justified: on F_j^k, x lies in a level-k Calderón–Zygmund cube but in no level-(k+1) cube, so the dyadic maximal function can be as large as w_Q λ^{k+1}, and may exceed w_Q λ^k. Since λ is chosen with 4/λ ≤ α < 1, we have λ > 4, so λ^{k+1} is genuinely larger. The subsequent absorption into w(F_j^k), with constant 1/(1−β), relies on the too-small bound. Replacing λ^k by λ^{k+1} introduces an extra factor λ and changes the constant to about λ/(1−β), which is still finite. Thus the lemma's statement is very likely correct (it is also known, cf. [7, Theorem 6.1]), but the proof supplied is incomplete as written, and it is the route used for B∞⇒FW in Theorems 1.1 and 1.2. This is a proof gap, not a counterexample.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript characterizes the precise relationships between several B∞-type conditions for Békollé-Bonami weights on the unit disc: the Fujii–Wilson condition, the reverse Hölder inequality, the B_q classes, the B∞ class, and the reverse Jensen (Blog) class. The authors introduce four new side conditions, denoted (M dw), (MLp), (mLp), and (m log), and prove that each of these conditions, when paired with a known condition, restores an equivalence that fails in general for Carleson-square bases. The main theorems are Theorems 1.1–1.4: (FW) + (M dw) characterizes B∞, (FW) + (MLp) characterizes the reverse Hölder inequality, B∞ + (mLp) characterizes membership in some B_q, and B∞ + (m log) characterizes Blog. Section 5 additionally discusses logarithms of B_p weights in relation to BMO-type conditions and gives two examples illustrating sharpness. The proofs use Calderón–Zygmund decompositions, layer-cake formulas, and dyadic maximal/minimal operators adapted to Carleson squares.","tokens_in":18357,"tokens_out":13659,"duration_ms":118900,"significance":"If the stated results hold, the paper gives a clean and testable set of side conditions that repair the known failures of the classical A∞ equivalences in the Békollé–Bonami setting. The introduction of the dyadic minimal operator and the explicit side conditions is natural and likely to be useful for applications to Bergman projection weighted estimates. The paper includes detailed proofs of the main equivalences, with the exception of Theorem 5.1, which is only sketched. The two examples in Section 5.2 are valuable because they show the necessity of the side conditions and the optimality of the exponent ranges. The manuscript also provides an alternative proof of the known implication B∞ ⇒ FW via Calderón–Zygmund decompositions, which is of independent interest once the gap discussed below is repaired.","major_comments":[{"comment":"The proof of B∞ ⇒ FW contains a false estimate. On the sets F_j^k = Q_j^k \\setminus \\bigcup_l Q_l^{k+1}, the manuscript asserts ∫_{F_j^k} M_Qw ≤ w_Q λ^k |F_j^k|. This is not justified: since x ∈ Q_j^k, the dyadic maximal function satisfies M_Qw(x) ≥ w_{Q_j^k} > w_Q λ^k, and it can be as large as w_Q λ^{k+1} (or at least 4 w_Q λ^k). Because λ > 4, the displayed inequality is strictly false. The subsequent absorption into w(F_j^k) with constant 1/(1−β) relies on this bound. The lemma is repairable by replacing λ^k with λ^{k+1} (or 4λ^k), which introduces an extra factor λ into the final Fujii–Wilson constant; the statement then still follows, consistent with [7, Theorem 6.1]. However, as written the proof is incomplete, and since Lemma 3.5 is used in the proofs of Theorems 1.1 and 1.2, this gap must be fixed.","section":"§3.3, Lemma 3.5"},{"comment":"Theorem 5.1 is stated as a characterization of logarithms of B_p weights, but its proof is only a sketch. In particular, the implication (a) ⇔ (b) is attributed to an 'identical proof' to the classical cube case without details, and the crucial estimate (5.1) is said to follow by 'imitating the arguments of [20, pp. 64–66]' with no concrete verification of how the Carleson-square basis and Lemma 3.2 are used. For a stated theorem in a research paper, this is insufficient. The authors should either provide a complete proof, state the result as a conjecture or remark with a clear indication of the missing details, or remove the theorem from the main text. Since the abstract does not advertise this result, it is not load-bearing for the central claims, but it is a gap in the paper as presented.","section":"§5.1, Theorem 5.1"}],"minor_comments":[{"comment":"In the statement of Lemma 4.9, equation (4.6), the term λ^{1−β} should read λ/(1−β), as is evident from the proof. Please correct this typo.","section":"§4.3, Lemma 4.9"},{"comment":"The proof of Lemma 4.9 cites Lemma 2.9 for the doubling property of the measure w dx, but Lemma 2.9 only proves that RHI implies B∞. The doubling property under B∞ is proved in Lemma 3.5. The citation should be to Lemma 3.5.","section":"§4.3, Lemma 4.9"},{"comment":"The statement 'If w ∈ Blog(D) for 1 ≤ p < ∞' contains a grammatical error; the phrase 'for 1 ≤ p < ∞' is dangling. The statement should read 'If w ∈ Blog(D), then w ∈ B∞(D).'","section":"§2, Lemma 2.7"},{"comment":"There are several typographical and formatting issues, including 'W say that' in Definition 2.6, 'lemmatas' in Section 4.3, and inconsistent use of the slashed and unslashed integral notation. A careful proofreading pass is recommended.","section":"Throughout"},{"comment":"The remark that condition (mLp) implies (m log) 'by a standard limiting argument' is stated without proof. Since this implication is used only as a supporting observation and not in the main theorems, it is acceptable, but a brief justification or reference would improve readability.","section":"§1, Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong and the main ideas are sound, but the proof of Lemma 3.5 contains a false inequality that must be corrected before the main theorems can be considered fully proved. The fix is straightforward and does not change the qualitative results. The other issue is the insufficiently proved Theorem 5.1, which should be upgraded to a full proof or explicitly downgraded to a sketch/remark. The examples in Section 5.2 are a nice addition and support the sharpness of the main results. In my view, after these revisions the paper would be suitable for publication in a good analysis journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper settles the right version of the question \"which side conditions restore the classical A∞ equivalences for Békollé–Bonami weights?\". The four new side conditions (MLp), (M dw), (mLp), and (m log) are genuinely new, and the main equivalence theorems (1.1–1.4) are proved in detail via Calderón–Zygmund machinery. This is a real advance over the counterexamples in [11] and the restricted class in [1].\n\nThe proofs are mostly careful, and the discussion of the side conditions is illuminating. Lemma 3.6, which gives the quantitative range p < 4L/(4L−1) for the Fujii–Wilson bound on ∫(M_Q w)^p, is load-bearing for Theorem 1.2, and it appears correct. The examples in Section 5.2 are useful and support the sharpness claims.\n\nThere are two soft spots worth naming. First, the stress-test note about Lemma 3.5 is right: the proof writes ∫_{F_j^k} M_Q w ≤ w_Q λ^k |F_j^k|, but on F_j^k the dyadic maximal function can be as large as w_Q λ^{k+1}, because those points are excluded from the level-(k+1) cubes. The fix is to replace λ^k by λ^{k+1} in that line, which adds a factor λ to the constant. The lemma is still true (it is known from [7, Theorem 6.1]), so this is a correction, not a counterexample, but it should be fixed before publication. Second, Theorem 5.1 about logarithms of B_p weights and BMO is only given a proof sketch. It is honestly labeled as a sketch, and it is a secondary result, but a referee will want either a complete proof or an explicit statement that it is a sketch.\n\nOverall, the main theorems hold up. The paper deserves a serious referee. I would send it out, asking for the Lemma 3.5 inequality to be corrected and the BMO result to be expanded or explicitly deferred. If those are addressed, it is a strong paper for math.CA.","headline":"The paper supplies the right side conditions to restore the classical A∞ equivalences for Békollé–Bonami weights; the main theorems are solid, with one fixable proof slip in Lemma 3.5.","tokens_in":18873,"tokens_out":4132,"would_cite":true,"duration_ms":35403,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E30","42B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Four side conditions restore the classical equivalences for B∞ weights","keywords":["Békollé-Bonami weights","Fujii-Wilson condition","reverse Hölder inequality","B∞ weights","Blog weights","Carleson squares","dyadic maximal function","dyadic minimal function"],"falsifier":"Construct a weight $w$ on the unit disk satisfying the Fujii-Wilson condition with constant $L$ such that, for some exponent $p$ with $1<p<4L/(4L-1)$ and some Carleson square $Q$, the inequality $\\int_Q (M_Q w)^p \\le C (\\int_Q w)^p$ fails for every $C$; this would refute Lemma 3.6 and the forward direction of Theorem 1.2.","tokens_in":17878,"feed_emoji":"⚖️","tokens_out":14224,"duration_ms":108824,"temperature":0.7,"pith_summary":"For Békollé-Bonami weights on the unit disk, the classical equivalences among reverse Hölder, Fujii-Wilson, $B_\\infty$, $B_p$, and reverse Jensen fail in general. This paper identifies four simple side conditions, each an integral version of bounded hyperbolic oscillation, and proves that adding the appropriate one restores each equivalence. Concretely, Fujii-Wilson plus one condition is equivalent to $B_\\infty$; Fujii-Wilson plus another is equivalent to reverse Hölder; $B_\\infty$ plus a third is equivalent to being a $B_p$ weight; $B_\\infty$ plus a fourth is equivalent to being a $B_{\\log}$ weight. The conditions are testable from the weight's dyadic maximal and minimal functions over Carleson squares.","feed_headline":"Four side conditions restore the classical equivalences for B∞ weights","feed_subtitle":"One simple integral condition per case restores the classical equivalence for arbitrary disk weights.","key_machinery":"The machinery is the pair of dyadic maximal and minimal operators $M_Q w$ and $m_Q w$ over Carleson squares, together with the Calderón-Zygmund decomposition adapted to Carleson squares for doubling measures. The new side conditions $(M dw)$, $(MLp)$, $(mLp)$ and $(m\\log)$ compare $w$ to $M_Q w$ or $m_Q w$ in measure, $L^p$, or logarithmic integral form; they are exactly the integral substitutes for the pointwise domination $w\\lesssim M_Q w$ that holds for bounded-hyperbolic-oscillation weights but fails for general $B_p$ weights. The load-bearing estimate is Lemma 3.6: under the Fujii-Wilson condition with constant $L$, for every $1<p<4L/(4L-1)$ one has $\\int_Q (M_Q w)^p \\lesssim (\\int_Q w)^p$, proved by a Calderón-Zygmund decomposition over Carleson squares.","core_discovery":"The central discovery is that, for arbitrary weights in the unit disk, the failure of the classical equivalences between the various $B_\\infty$-type conditions is exactly captured by four side conditions built from the dyadic maximal operator $M_Q$ and the dyadic minimal operator $m_Q$ over Carleson squares. Theorem 1.1 states that the Fujii-Wilson condition plus the measure-tail condition $(M dw)$ characterizes $B_\\infty(D)$. Theorem 1.2 states that the Fujii-Wilson condition plus the integral condition $(MLp)$ characterizes the reverse Hölder inequality. Theorem 1.3 states that $B_\\infty(D)$ plus $(mLp)$ characterizes membership in some $B_q(D)$ for $1\\le q<\\infty$, and Theorem 1.4 states that $B_\\infty(D)$ plus $(m\\log)$ characterizes membership in $B_{\\log}(D)$. The side conditions are necessary as well as sufficient, and the paper's examples show that without them the corresponding implications genuinely fail.","pith_inferences":["The paper does not pursue it, but the same four side-condition scheme should transfer to other bases of sets without pointwise domination $w\\lesssim Mw$, yielding necessary and sufficient side conditions for the analogous $A_\\infty$ equivalences.","A natural test is whether the reverse-Hölder exponent in Theorem 1.2 is governed exactly by the range $1<p<4L/(4L-1)$; if so, the Fujii-Wilson constant would directly bound the sharp integrability gain of a $B_\\infty$ weight.","The radial examples in Section 5 suggest that the side conditions are genuinely independent of the classical conditions; one could quantify, for the family of Example 5.2, how the admissible RHI exponent range and $B_q$ range depend on the oscillation parameter $x$."],"forward_implications":["A weight satisfying the Fujii-Wilson condition and the measure-tail condition $(M dw)$ is exactly a $B_\\infty$ weight, and every $B_\\infty$ weight satisfies both.","The Fujii-Wilson condition together with the integral domination condition $(MLp)$ characterizes the reverse Hölder inequality, and it is enough to verify $(MLp)$ for a single exponent in the range $1<p<4L/(4L-1)$, where $L$ is the Fujii-Wilson constant.","Every $B_q$ weight for some $1\\le q<\\infty$ satisfies the dual condition $(mLp)$, and any $B_\\infty$ weight satisfying $(mLp)$ belongs to some $B_q$.","Every $B_{\\log}$ weight satisfies the logarithmic condition $(m\\log)$, and any $B_\\infty$ weight satisfying $(m\\log)$ belongs to $B_{\\log}$.","Combining Theorems 1.2 and 1.3, a weight satisfies a reverse Hölder inequality plus $(mLp)$ if and only if it belongs to some $B_q$ and satisfies $(MLp)$."],"supporting_citations":[{"why":"introduced the bounded-hyperbolic-oscillation class for which the B∞ equivalences already hold; the paper's new side conditions are integral versions of this stronger condition.","marker":"[1]"},{"why":"proved that B∞ implies the Fujii-Wilson condition for Carleson-square bases and supplied the equivalent form of B∞ used in Lemma 2.8; the paper adapts this general-base framework.","marker":"[7]"},{"why":"the recent preprint demonstrating by examples that the classical equivalences among B∞-type conditions fail for arbitrary Békollé-Bonami weights; it sets up the gap the paper closes.","marker":"[11]"},{"why":"its Lemma 2.2 is the cube-setting maximal estimate that Lemma 3.6 adapts to Carleson squares, carrying the quantitative range that makes Theorem 1.2 work.","marker":"[15]"},{"why":"introduced the minimal operator and the structural analysis of reverse-Hölder classes that motivate the dual side conditions (mLp) and (m log).","marker":"[5]"},{"why":"introduced the Békollé-Bonami classes and the weighted Bergman projection problem that these weights were designed to classify.","marker":"[2, 3]"},{"why":"introduced the Fujii-Wilson condition, the baseline property to which Theorems 1.1 and 1.2 add their side conditions.","marker":"[9, 22]"}],"fun_headline_variants":["Four side conditions restore B∞ equivalences","Simple side conditions fix B∞ weight equivalences","Four testable side conditions restore classical equivalences","Four side conditions close B∞ weight gaps","New dyadic side conditions revive B∞ equivalences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.6, which asserts that a weight satisfying the Fujii-Wilson condition with constant $L$ obeys $\\int_Q (M_Q w)^p \\le C (\\int_Q w)^p$ precisely for $1<p<4L/(4L-1)$, and Theorem 1.2 collapses if that range or its constant cannot be established.","fun_headline_variants_meta":{"raw":{"variants":["Four side conditions restore B∞ equivalences","Simple side conditions fix B∞ weight equivalences","Four testable side conditions restore classical equivalences","Four side conditions close B∞ weight gaps","New dyadic side conditions revive B∞ equivalences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":2976,"prompt_tokens":835,"completion_tokens":2141,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":2071}},"tokens_in":451,"tokens_out":2141,"duration_ms":47251,"temperature":1.0,"reasoning_tokens":2071,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T06:02:14.108965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a weight $w$ on the unit disk satisfying the Fujii-Wilson condition with constant $L$ such that, for some exponent $p$ with $1<p<4L/(4L-1)$ and some Carleson square $Q$, the inequality $\\int_Q (M_Q w)^p \\le C (\\int_Q w)^p$ fails for every $C$; this would refute Lemma 3.6 and the forward direction of Theorem 1.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduced the bounded-hyperbolic-oscillation class for which the B∞ equivalences already hold; the paper's new side conditions are integral versions of this stronger condition."},{"cited_title":"Martín-Reyes, and S heldy Ombrosi, On the A∞ conditions for general bases , Math","cited_arxiv_id":null,"evidence_quote":"proved that B∞ implies the Fujii-Wilson condition for Carleson-square bases and supplied the equivalent form of B∞ used in Lemma 2.8; the paper adapts this general-base framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the recent preprint demonstrating by examples that the classical equivalences among B∞-type conditions fail for arbitrary Békollé-Bonami weights; it sets up the gap the paper closes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"its Lemma 2.2 is the cube-setting maximal estimate that Lemma 3.6 adapts to Carleson squares, carrying the quantitative range that makes Theorem 1.2 work."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduced the minimal operator and the structural analysis of reverse-Hölder classes that motivate the dual side conditions (mLp) and (m log)."}],"review_version":1}