REVIEW 2 major objections 5 minor 21 references
Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights
T0 review · 2 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read The two-weight commutator of the Bessel Riesz transform is bounded exactly when the symbol lies in the Bloom space BMO_{ν,α}, and compact exactly when it lies in the corresponding VMO space.
desk verdict Genuine two-weight Bloom theorem for Bessel Riesz commutators in the Andersen–Kerman normalization, with a clearly stated but real dependence on a companion preprint for the quotient-kernel estimates; worth a serious referee. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The exact conjugation U_w(x)=$x^{{p−2α−1}}$w(x), which maps Andersen–Kerman weights w to A_p(ρ_α) weights while leaving the Bloom weight ν=(μ/λ)^{1/p} unchanged, and the quotient kernel K^ρ_α(x,y)=K^m_α(x,y)/x, which is proved to be a Calderón–Zygmund kernel and to be non-degenerate in the following sense: every interval I has a companion interval Î to its right, with comparable radius and ρ_α-measure, on which the kernel has fixed sign and magnitude at least c/ρ_α(I). This companion-interval non-degeneracy carries both the lower bound of the boundedness theorem and the necessity part of the compactness theorem.
What would settle it
Take α=1/2 and a pair (x,y) with y=κ_∞x as in (4.8), and evaluate the conjugated kernel K^ρ_α(x,y) numerically: if |K^ρ_α(x,y)| is not bounded below by a positive constant times $y^{{−(2α+2)}}$, or if the sign is not fixed, then the companion intervals of Proposition 4.3 cannot exist and both the lower-bound and compactness-necessity arguments fail.
Extended reading notes
Core claim
Theorems 1.1 and 1.2 characterize commutators of the Bessel Riesz transform in the Andersen–Kerman two-weight setting. For α>−1/2, α≠0, 1<p<∞, μ,λ∈A_{p,α}, and ν=(μ/λ)^{1/p}, the paper claims that [b,R_α]:L^p(μ dx)→L^p(λ dx) has a bounded extension if and only if b∈BMO_{ν,α}, with norm equivalence, and that it extends to a compact operator if and only if b∈VMO_{ν,α}. The BMO and VMO oscillations are computed with respect to dρ_α=$x^{{2α+1}}$dx, and the proof transfers the problem through the conjugation U_w(x)=$x^{{p−2α−1}}$w(x) to the quotient Bessel kernel on (R_+, |x−y|, ρ_α).
Load-bearing premise
The load-bearing premise is that the companion paper's quotient-kernel estimates — the size, smoothness, and lower-bound inequalities (4.1)–(4.8) and the asserted weak type (1,1) of the maximal truncation — are correct; if any of them fails, the sparse upper bound, the median lower bound, and the compactness necessity argument all collapse.
Editorial extensions
If this is right
- In the one-weight case μ=λ the theorem reduces to a single-weight BMO characterization, and since BMO(m_α)=BMO(ρ_α) for α>−1/2, it agrees with the standard Bessel BMO space.
- The operator norm of [b,R_α] is equivalent to ∥b∥_{BMO_{ν,α}} with constants depending only on p, α, and the A_{p,α} characteristics of μ and λ, so the characterization is quantitatively sharp.
- Compactness fails exactly when the symbol's oscillation fails to vanish in one of the three interval regimes (small intervals, large intervals, or intervals escaping to infinity); VMO_{ν,α} is the precise necessary and sufficient class.
- The admissible Andersen–Kerman weights allow stronger singularities at the origin than the usual Bessel weight class, so the theorem applies to pairs for which the corresponding Bloom BMO built on dm_α is not even locally finite; the choice of ρ_α is essential.
Reading between the lines
- The same conjugation scheme should extend to α=0, the half-line Hilbert-type case, once quotient-kernel estimates in that regime are established; the algebraic identities remain formally valid there.
- The companion-interval non-degeneracy replaces Fourier analysis and may apply to other transforms with explicit kernels, such as Dunkl or Jacobi settings, to yield analogous two-weight BMO/VMO characterizations.
- A general principle suggested here is that in two-weight commutator theory the Bloom base measure is selected by the weight conjugation, and using the 'natural' measure of the underlying operator can produce a strictly different function space.
- A testable extension is that vector-valued or fractional commutators of the same transform should be controlled by the same BMO_{ν,α} class whenever the kernel size and non-degeneracy estimates hold with the corresponding exponents.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper characterizes boundedness and compactness of commutators of the Bessel Riesz transform R_α on the half-line in a two-weight Andersen–Kerman setting. The key device is the exact change of measure U_w(x)=x^{p-2α-1}w(x), which converts the problem to a Bloom-type weighted problem on the homogeneous space (R_+, |x-y|, ρ_α), where ρ_α(x)=x^{2α+1}dx. The main results, Theorems 1.1 and 1.2, assert that [b,R_α] maps L^p(μdx) boundedly to L^p(λdx) if and only if b∈BMO_{ν,α}, with norm equivalence, and that it is compact if and only if b∈VMO_{ν,α}. The proofs rely on exact conjugation identities, quotient-kernel estimates, Calderón–Zygmund theory, non-degeneracy, sparse domination, and a weighted VMO density theorem.
Significance. If the cited estimates hold, the paper gives a complete two-weight Bloom theory—both boundedness and compactness—for a nontrivial Riesz transform in the Bessel setting, with the correct base measure ρ_α(x)=x^{2α+1}dx. The paper carefully distinguishes ρ_α from the usual Bessel measure dm_α and proves the non-degeneracy of the conjugated kernel, the VMO density theorem, and several measure comparisons in detail. These are substantive contributions to the two-weight commutator literature.
major comments (2)
- [Sections 2 and 4, Propositions 2.4, 2.5, 4.1, 4.2] The proofs of Theorems 1.1 and 1.2 are conditional on the companion preprint [21], which supplies the exact conjugation identities (Propositions 2.4 and 2.5), the quotient-kernel estimates (4.1)–(4.6) from Lemma 3.1 of [21], and the standard Calderón–Zygmund estimate (4.9) together with L^2(ρ_α) boundedness from Proposition 3.2 of [21]. These results are load-bearing: Theorem 5.1 uses them through the sparse domination argument, and the lower-bound and compactness arguments in Sections 5 and 6 rely on the non-degeneracy proved here only for the lower bounds. As submitted, the main theorems are not self-contained, and the referee cannot verify the central claim without access to [21] or the omitted proofs. The authors should include proofs of these propositions in an appendix or explicitly state that the results depend on the companion paper's unproved machinery.
- [Section 4, after Proposition 4.2] The assertion that the maximal truncation of R_α is of weak type (1,1) with respect to ρ_α is made with the sentence “Based on the above argument, we note... via the standard results,” but no proof or precise reference is given. This weak-type estimate is explicitly an assumption of Theorem 5.1, so it is load-bearing for the upper bound. The authors should either supply a proof or cite a theorem that applies to R_α on the space (R_+, |x-y|, ρ_α), and should indicate why the abstract Calderón–Zygmund theory of [6] is applicable in this quotient-kernel normalization.
minor comments (5)
- [Section 4, proof of Proposition 4.1] The symbol ν is reused for α−1/2 in the heat-kernel computation, while ν denotes the Bloom weight throughout the rest of the paper. Please use a different letter to avoid ambiguity.
- [Section 4, after Proposition 4.2] The phrase “Based on the above argument” is vague; it would be clearer to say “By Proposition 4.2 and the standard Coifman–Weiss theory (see [6]).”
- [Proposition 3.15] The proof of VMO(m_α)=VMO(ρ_α) is only sketched with a reference to a dyadic-annulus layer-cake argument. Since the statement is not immediate, the authors should either expand the proof or give a more detailed reference.
- [Lemma 3.2] The proof of (3.3) contains a chain of inequalities whose direction is not immediately transparent; a short explanatory note would improve readability.
- [Introduction] The claim that the paper “isolates” the required Calderón–Zygmund estimates is overstated, because the estimates are cited from [21] rather than proved in the present text.
Circularity Check
The two-weight theorems are not definitionally circular, but their upper-bound and compactness-sufficiency proofs hinge on quotient-kernel estimates and L^2(rho_alpha) boundedness imported from an unpublished companion preprint by co-author Wen; this is load-bearing self-citation.
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self citation load bearing
[Section 4, proof of Proposition 4.1]
"The kernel K^rho_alpha is the kernel denoted by K_alpha in [21]. Hence (4.1)–(4.6) follow from Lemma 3.1 of [21]."
The size and derivative estimates (4.1)–(4.6), including the delicate far-region bounds for y >= 2x and x >= 2y, are the Calderón–Zygmund input for the sparse upper bound (Theorem 5.1) and for the compactness of smooth-symbol commutators (Lemma 6.2). They are not proved in this paper; they are imported from [21], an unpublished companion paper by co-author C. Wen. The paper itself stresses that these are quotient-kernel estimates rather than consequences of the usual Bessel estimates divided by x, so the central upper-bound chain reduces to the companion's Lemma 3.1. This is load-bearing self-citation rather than definitional circularity.
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self citation load bearing
[Section 4, Proposition 4.2 and the paragraph after it]
"Proposition 4.2 ([21], Proposition 3.2). The kernel K^rho_alpha is a standard Calderón–Zygmund kernel on (R_+, |x−y|, rho_alpha). ... The operator R_alpha is bounded on L^2(rho_alpha). Based on the above argument, we note that the maximal truncation of R_alpha is also of weak type (1,1) with respect to rho_alpha via the standard results (see for example [6])."
This proposition is a second load-bearing import from the companion preprint [21] by co-author C. Wen. It supplies the standard kernel estimate (4.9) and L^2(rho_alpha)-boundedness that make R_alpha admissible for the abstract Bloom Theorem 5.1 and for Lemma 6.2. The weak-type (1,1) assertion, also required by Theorem 5.1, is added by the present authors without proof. Thus the sufficiency halves of Theorems 1.1 and 1.2 are conditional on [21, Prop. 3.2] together with an unstated verification that the standard Coifman–Weiss argument applies to this maximal truncation.
full rationale
The main characterization is not circular in the strict sense: BMO_{nu,alpha} is defined by oscillation integrals and not from the commutator norm; the lower bound uses the median/non-degeneracy argument; no parameter is fitted and then renamed a prediction. The core boundedness upper bound, however, is routed through the abstract Bloom Theorem 5.1, whose hypotheses are exactly the standard Calderón–Zygmund kernel estimate (4.9), L^2(rho_alpha) boundedness, and weak type (1,1) of the maximal truncation. Proposition 4.2 obtains these from [21, Prop. 3.2], and the weak type is asserted 'via the standard results' rather than proved. Likewise, the size and derivative estimates (4.1)–(4.6) in Proposition 4.1 are quoted from [21, Lemma 3.1]; the paper's own text emphasizes that these are non-trivial quotient-kernel estimates, not consequences of the usual Bessel estimates divided by x. Since [21] is an unpublished companion paper by co-author C. Wen, this is load-bearing self-citation. If those estimates are accepted as independently proved in [21], the present derivation goes through; if not, Theorems 1.1 and 1.2 are conditional. This is self-citation load-bearing, not definitional equivalence, so the score is moderate. The exact conjugation identities (2.1)–(2.5) are also cited from [21], but they are elementary algebra and do not by themselves create circularity. No fitted-input-called-prediction or renamed-known-result pattern is present.
Assumptions & free parameters
assumptions (4)
- domain assumption The companion results in [21] are correct: [U_w]_{A_p(ρ_α)}=[w]_{A_{p,α}}, the norm identities (2.3)-(2.5), and the quotient-kernel estimates in Propositions 4.1 and 4.2.
- standard math Sparse domination of commutators on spaces of homogeneous type as in [7, Theorem 3.7], and the dyadic Bloom equivalences of [11].
- standard math The weighted VMO density theorem [14, Theorem 4.1] and its half-line adaptation.
- domain assumption The maximal truncation of R_α is of weak type (1,1) with respect to ρ_α.
Cite this review
Pith. "Pith review of Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights." pith.science (2026). https://pith.science/paper/IGCHI6N7
@misc{pith2026260821104,
author = {Pith},
title = {Pith review of: Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights},
year = {2026},
howpublished = {\url{https://pith.science/paper/IGCHI6N7}},
note = {Machine review of arXiv:2608.21104}
}
abstract
Let $\alpha>-1/2$, $\alpha\ne0$, and let $$ \Delta_\alpha=-\frac{d^2}{dx^2}-\frac{2\alpha}{x}\frac{d}{dx} $$ be the Bessel operator on $\mathbb R_+=(0,\infty)$. We characterize boundedness and compactness of commutators of $R_\alpha=\frac{d}{dx}\Delta_\alpha^{-1/2}$ on the Andersen--Kerman two-weight setting. For $1<p<\infty$ and $\mu,\lambda\in A_{p,\alpha}$, put $ \nu=\left(\frac{\mu}{\lambda}\right)^{1/p}, \ d\rho_\alpha(x)=x^{2\alpha+1}\,dx. $ For real-valued symbols, $$ \|[b,R_\alpha]\|_{L^p(\mu\,dx)\to L^p(\lambda\,dx)} \simeq \|b\|_{\rm{BMO}_{\nu,\alpha}}, $$ where the Bloom oscillation is computed with respect to $d\rho_\alpha$. Moreover $$ [b,R_\alpha]:L^p(\mu\,dx)\to L^p(\lambda\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{\nu,\alpha}. $$ The proof uses the exact conjugation $ U_w(x)=x^{p-2\alpha-1}w(x), \ [U_w]_{A_p(\rho_\alpha)}=[w]_{A_{p,\alpha}}, $ which transfers the problem to the quotient Bessel kernel on $(\mathbb R_+,|x-y|,\rho_\alpha)$. We isolate the required Calder\'on--Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $d\rho_\alpha$ is distinct from the usual Bessel measure $dm_\alpha=x^{2\alpha}\,dx$. It is the Bloom base measure selected by the Andersen--Kerman conjugation and is used throughout the two-weight theory below.
Reference graph
Works this paper leans on
-
[21]
C. Wen,Sharp weighted estimates for the Bessel Riesz transform and its commutator with Andersen–Kerman weights, arXiv: 2608.15986. 2, 5, 25, 27 JiLi, School ofMathematical andPhysicalSciences, MacquarieUniversity, NSW 2109, Australia Email address:ji.li@mq.edu.au Chong-WeiLiang, Department ofMathematics, NationalTaiw anUniversity, Taiw an Email address:d1...
-
[6]
R. R. Coifman and G. Weiss,Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977), no. 4, 569–645. 27
work page 1977
-
[1]
K. F. Andersen and R. A. Kerman,Weighted norm inequalities for generalized Hankel conjugate transformations, Studia Math.71(1981), no. 1, 15–26. 1, 2
work page 1981
-
[2]
J. J. Betancor, E. Harboure, A. Nowak and B. Viviani,Mapping properties of fundamental operators in harmonic analysis related to Bessel operators, Studia Math.197(2010), no. 2, 101–140. 1
work page 2010
-
[3]
Bloom,A commutator theorem and weighted BMO, Trans
S. Bloom,A commutator theorem and weighted BMO, Trans. Amer. Math. Soc.292(1985), no. 1, 103–122. 4
work page 1985
-
[4]
A. J. Castro and T. Z. Szarek,Calder´ on–Zygmund operators in the Bessel setting for all possible type indices, Acta Math. Sin. (Engl. Ser.)30(2014), no. 4, 637–648. 1
work page 2014
-
[5]
R. R. Coifman, R. Rochberg and G. Weiss,Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2)103(1976), no. 3, 611–635. 4
work page 1976
-
[7]
X. T. Duong, R. Gong, M.-J. S. Kuffner, J. Li, B. D. Wick and D. Yang,Two weight commutators on spaces of homogeneous type and applications, J. Geom. Anal.31(2021), no. 1, 980–1038. 2, 4, 29
work page 2021
Show all 21 references
-
[8]
X. T. Duong, J. Li, B. D. Wick and D. Yang,Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting, Indiana Univ. Math. J.66(2017), no. 4, 1081–1106. 1
2017
-
[9]
Frey and Z
D. Frey and Z. Nieraeth,Weak and strong type A 1–A∞ estimates for sparsely dominated operators, J. Geom. Anal. 29(2019), no. 1, 247–282
2019
-
[10]
Garc ´ıa-Cuerva and J
J. Garc ´ıa-Cuerva and J. M. Martell,Wavelet characterization of weighted spaces, J. Geom. Anal.11(2001), no. 2, 241–264
2001
-
[11]
Holmes, M
I. Holmes, M. T. Lacey and B. D. Wick,Commutators in the two-weight setting, Math. Ann.367(2017), no. 1–2, 51–80. 4, 9
2017
-
[12]
Hyt ¨onen and A
T. Hyt ¨onen and A. Kairema,Systems of dyadic cubes in a doubling metric space, Colloq. Math.126(2012), no. 1, 1–33
2012
-
[13]
R. A. Kerman,Boundedness criteria for generalized Hankel conjugate transformations, Canad. J. Math.30(1978), no. 1, 147–153. 1
1978
-
[14]
M. T. Lacey and J. Li,Compactness of commutator of Riesz transforms in the two weight setting, J. Math. Anal. Appl.508(2022), no. 1, Paper No. 125869, 11 pp. 4, 16, 33, 34 38 J. LI, C.-W. LIANG, C.J. WEN, AND L. WU
2022
-
[15]
A. K. Lerner, S. Ombrosi and I. P. Rivera-R´ıos,On pointwise and weighted estimates for commutators of Calder´ on– Zygmund operators, Adv. Math.319(2017), 153–181
2017
-
[16]
Lorist,On pointwiseℓ r-sparse domination in a space of homogeneous type, J
E. Lorist,On pointwiseℓ r-sparse domination in a space of homogeneous type, J. Geom. Anal.31(2021), no. 9, 9366–9405
2021
-
[17]
S. Mao, H. Wu and D. Yang,Boundedness and compactness characterizations of Riesz transform commutators on Morrey spaces in the Bessel setting, Anal. Appl. (Singap.)17(2019), no. 1, 145–178
2019
-
[18]
Muckenhoupt and E
B. Muckenhoupt and E. M. Stein,Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc.118(1965), 17–92. 1
1965
-
[19]
Muckenhoupt and R
B. Muckenhoupt and R. L. Wheeden,Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54(1976), no. 3, 221–237
1976
-
[20]
Uchiyama,On the compactness of operators of Hankel type, Tohoku Math
A. Uchiyama,On the compactness of operators of Hankel type, Tohoku Math. J. (2)30(1978), no. 1, 163–171
1978
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