Characterizations for arbitrary B\'ekoll\'e-Bonami weights

Carlos Mudarra; Karl-Mikael Perfekt

REVIEW 2 major objections 5 minor 1 cited by

Characterizations for arbitrary B\'ekoll\'e-Bonami weights

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Four side conditions restore the classical equivalences for B∞ weights

desk verdict 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. read the letter →

arxiv 2506.05993 v2 pith:4LJ3W326 submitted 2025-06-06 math.CA math.CVmath.FA

Carlos Mudarra , Karl-Mikael Perfekt This is my paper

classification math.CAmath.CVmath.FA MSC 46E3042B25

keywords Békollé-BonamiweightsFujii-WilsonconditionreverseHölderinequalityB∞BlogCarlesonsquaresdyadicmaximalfunctionminimal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $(MLp)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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 $x$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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].

The 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.

There 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.

Overall, 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[§3.3, Lemma 3.5] 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.
[§5.1, Theorem 5.1] 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.

minor comments (5)

[§4.3, Lemma 4.9] 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.
[§4.3, Lemma 4.9] 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.
[§2, Lemma 2.7] 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).'
[Throughout] 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.
[§1, Introduction] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the side conditions are independently defined and both directions of each equivalence are proved from external or self-contained lemmas.

full rationale

The paper's central claims (Theorems 1.1–1.4) are equivalences between independently defined conditions. The side conditions (MLp), (M dw), (mLp), and (m log) are formulated directly in terms of the dyadic maximal and minimal operators MQw and mQw, not in terms of B∞, Bq, Blog, FW, or RHI. Each implication is proved in the text: Theorem 1.1 from Lemmas 3.5, 4.2, and 4.3; Theorem 1.2 from Lemmas 3.6, 4.5, 2.9, and 3.5; Theorem 1.3 from Lemmas 4.7, 4.9, and 4.10(i); and Theorem 1.4 from Lemmas 4.8, 4.9, and 4.10(ii). No step reduces a conclusion to its own hypothesis by construction. The only self-citation, reference [6], appears in survey sentences in the introduction and Section 5, and it is not used as evidence for any theorem. The known implication B∞ ⇒ FW is attributed to the external reference [7, Theorem 6.1], and an alternative proof is supplied. A referee-level concern about the constant in Lemma 3.5's proof would be a correctness or proof-completeness issue, not a circularity issue, and it does not make the paper's derivation equivalent to its inputs. Accordingly, no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants or invented entities. The central proofs rely on standard inequalities, the dyadic geometry of Carleson squares, and two cited characterizations of B∞/B_log from the literature. The new side conditions are definitions rather than axioms.

assumptions (5)

standard math Jensen's inequality and Hölder's inequality are used throughout.
Invoked in Lemmas 2.9, 4.5, 4.7, 4.8 and elsewhere to pass between L^p norms and averages.
domain assumption Known characterization of B∞ as a quantitative absolute continuity condition (Lemma 2.8).
Taken from [7, Theorem 3.1]; used to prove Lemma 4.2 and Lemma 4.3 and to establish doubling from B∞.
domain assumption Inclusion B_log ⊂ B_∞ (Lemma 2.7).
Taken from [13, Theorem 1] and [7, Theorem 4.1]; used to structure the implication chains in Theorems 1.3 and 1.4.
standard math Layer-cake representation for L^p norms.
Used in Lemma 3.6 and Lemma 4.10 to convert L^p norm estimates into distribution function estimates.
standard math The dyadic interval structure of Carleson squares supports a Calderón-Zygmund decomposition with the doubling property.
Proven as Lemma 3.2, but the underlying dyadic nesting property of the intervals is taken as a geometric fact.

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Cite this review

Pith. "Pith review of Characterizations for arbitrary B\'ekoll\'e-Bonami weights." pith.science (2026). https://pith.science/paper/4LJ3W326

@misc{pith2026250605993,
  author       = {Pith},
  title        = {Pith review of: Characterizations for arbitrary B\'ekoll\'e-Bonami weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LJ3W326}},
  note         = {Machine review of arXiv:2506.05993}
}

abstract

We precisely characterize the relationships between the reverse H\"older inequality, the Fujii-Wilson condition, the B\'ekoll\'e-Bonami $\mathrm{B}_p$ condition, the $\mathrm{B}_\infty$ condition, and the reverse Jensen inequality, for arbitrary weights in the unit disc. This is achieved by introducing new side conditions that turn out to be necessary and sufficient. The side conditions are simple and testable, and can be interpreted as integral versions of the much stronger condition of bounded hyperbolic oscillation, which has been considered earlier in the literature.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Weak-type estimates for the Bergman projection on planar domains
math.CV 2026-07 accept novelty 6.0 of 10

If |ψ'| is in B₁ then the Bergman projection Π_Ω is weak-type (1,1); a necessary Lorentz-type condition and sharpened p>1 sufficient conditions are also proved via mixed-weighted estimates on the disk.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [11]

Alptekin Can Goksan, Sharpness of the side condition in a characterization of Bék ollé–Bonami weights , https://arxiv.org/abs/2505.05303

work page arXiv

[2] [1]

Alexandru Aleman, Sandra Pott, and María Carmen Reguera, Characterizations of a limiting class B∞ of Békollé- Bonami weights , Rev. Mat. Iberoam. 35 (2019), no. 6, 1677–1692

work page 2019

[3] [2]

71 (1981/82), no

David Bekollé, Inégalité à poids pour le projecteur de Bergman dans la boule unité de Cn, Studia Math. 71 (1981/82), no. 3, 305–323

work page 1981

[4] [3]

David Bekollé and Aline Bonami, Inégalités à poids pour le noyau de Bergman , C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 18, A775–A778

work page 1978

[5] [4]

Alexander Borichev, On the Bekollé-Bonami condition , Math. Ann. 328 (2004), no. 3, 389–398

work page 2004

[6] [5]

David Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes , Trans. Amer. Math. Soc. 347 (1995), no. 8, 2941–2960

work page 1995

[7] [6]

Alberto Dayan, Adrián Llinares, and Karl-Mikael Perfekt , Restrictions of Békollé-Bonami weights and Bloch functions, https://arxiv.org/abs/2308.04859

work page Pith review arXiv

[8] [7]

Martín-Reyes, and S heldy Ombrosi, On the A∞ conditions for general bases , Math

Javier Duoandikoetxea, Francisco J. Martín-Reyes, and S heldy Ombrosi, On the A∞ conditions for general bases , Math. Z. 282 (2016), no. 3-4, 955–972

work page 2016