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arxiv: 1907.09953 · v2 · pith:GS5E2YGLnew · submitted 2019-07-23 · 🧮 math.FA

Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions

Zunwei Fu , Elodie Pozzi , Qingyan Wu This is my paper

Pith reviewed 2026-05-24 16:59 UTC · model grok-4.3

classification 🧮 math.FA
keywords commutatorsmaximal functionsspaces of homogeneous typeweighted BMOlocal versionsinterpolation
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The pith

Commutators of maximal functions on spaces of homogeneous type are characterized by BMO membership.

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

The paper establishes characterizations showing that commutators of maximal functions are bounded on L^p precisely when the symbol function lies in BMO, and this holds in the setting of spaces of homogeneous type. It further derives weighted versions of these commutator theorems through interpolation by supplying fresh characterizations of weighted BMO spaces. A concrete example is given to demonstrate that local versions of the commutators merit separate study. These results extend classical boundedness statements from Euclidean settings to abstract spaces equipped with a quasi-metric and doubling measure.

Core claim

We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest.

What carries the argument

The commutator [M, b] formed by a maximal function M and a symbol b, studied via boundedness on Lebesgue spaces over spaces of homogeneous type.

If this is right

  • Boundedness of the commutator on L^p is equivalent to the symbol belonging to BMO.
  • The weighted boundedness results follow directly from the new characterizations of weighted BMO.
  • Local versions of the commutators possess independent interest as illustrated by the given example.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The interpolation technique used for weighted BMO might extend to commutators with other operators such as singular integrals on the same spaces.
  • The local commutator results could be checked explicitly on standard examples like the real line to see whether they recover or differ from global versions.

Load-bearing premise

The space satisfies the doubling and quasi-metric conditions of homogeneous type and standard interpolation applies directly to the commutator operators.

What would settle it

A concrete homogeneous type space together with a weight where the commutator is bounded on weighted L^p yet the symbol fails to satisfy the corresponding weighted BMO condition would disprove the claimed equivalence.

read the original abstract

We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to establish characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. It further obtains weighted versions of these results via interpolation theory together with new characterizations of weighted BMO spaces, and supplies a concrete example illustrating that the local versions of the commutators are of independent interest.

Significance. If the derivations are complete and the constants are controlled uniformly, the results would extend commutator characterizations from Euclidean or doubling-measure settings to general spaces of homogeneous type, a direction of interest in harmonic analysis. The explicit use of interpolation to reach weighted BMO characterizations and the inclusion of a local-version example are concrete strengths that would be useful for subsequent work.

major comments (1)
  1. [weighted-version section / interpolation argument] The passage from unweighted bounds to weighted commutator characterizations via interpolation (described in the abstract and the weighted-version section) invokes standard real or complex interpolation without an explicit verification that the commutator norm remains bounded independently of the quasi-triangle constant K and the doubling constant C_μ. In spaces of homogeneous type the weak-type constants for M and [b,M] typically depend on these structural constants; if the interpolation functor is applied without uniform control, the resulting weighted BMO characterization may fail to hold for the A_p weights defined with respect to μ.
minor comments (1)
  1. The abstract states that 'a concrete example shows the local version of commutators also has an independent interest,' but the manuscript should clarify whether this example is used to motivate the local theory or merely to illustrate it after the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. Below we respond point-by-point to the single major comment concerning the interpolation argument used for the weighted results.

read point-by-point responses

  1. Referee: The passage from unweighted bounds to weighted commutator characterizations via interpolation (described in the abstract and the weighted-version section) invokes standard real or complex interpolation without an explicit verification that the commutator norm remains bounded independently of the quasi-triangle constant K and the doubling constant C_μ. In spaces of homogeneous type the weak-type constants for M and [b,M] typically depend on these structural constants; if the interpolation functor is applied without uniform control, the resulting weighted BMO characterization may fail to hold for the A_p weights defined with respect to μ.

    Authors: We agree that explicit control of constants is important. In the unweighted theorems the operator norms of M and [b,M] are established with explicit dependence on the fixed structural constants K and C_μ of the given space of homogeneous type. The real-interpolation argument applied in the weighted section is the standard one between the unweighted endpoint bounds; consequently the interpolated weighted bounds inherit the same dependence on K and C_μ. Because the underlying space (hence K and C_μ) is fixed for the entire paper and the A_p weights are defined with respect to the same measure μ, the resulting weighted BMO characterizations remain valid with constants that may depend on K, C_μ, p and the A_p characteristic of the weight. We will insert a short clarifying paragraph in the weighted-version section stating this dependence explicitly and confirming that no further uniformity with respect to varying K or C_μ is claimed or needed. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on external standard interpolation theory.

full rationale

The abstract and description establish commutator characterizations on homogeneous spaces then invoke standard interpolation theory for weighted BMO versions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The chain is self-contained against external benchmarks (Marcinkiewicz/Riesz-Thorin interpolation) with no reduction of claims to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

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discussion (0)

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Agcayazi, A

    M. Agcayazi, A. Gogatishvili, K. Koca and R. Mustafayev , A note on maximal commutators and commutators of maximal functions. J. Math. Soc. Japan 67 (2) (2015), 581–593

    work page 2015

  2. [2]

    Aimar, A

    H. Aimar, A. Bernardis, and B. Iaffei , Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type, J. Approx. Theory 148 (1) (2007), 12–34

    work page 2007

  3. [3]

    A. M. Alphonse , An end point estimate for maximal commutators, J. Fourier Anal. Appl. 6 (4) (2000), 449–456

    work page 2000

  4. [4]

    Auscher and T

    P. Auscher and T. Hyt ¨onen, Orthonormal bases of regular wavelets in spaces of homogen eous type, Appl. Comput. Harmon. Anal. 34 (2) (2013), 266–296

    work page 2013

  5. [5]

    Bastero, M

    J. Bastero, M. Milman and F. J. Ruiz , Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc. 128 (2000), 3329–3334. 20 ZUNWEI FU, ELODIE POZZI AND QINGYAN WU

    work page 2000

  6. [6]

    Boutet de Monvel and J

    L. Boutet de Monvel and J. Sj ¨ostrand, Sur la singularit´ e des noyaux de Bergman et de Szeg¨ o, Ast´ erisque, 34-35 (1976), 123–164

    work page 1976

  7. [7]

    P. Chen, J. Li, and L.A. W ard , BMO from dyadic BMO via expectations on product spaces of homogeneous type, J. Funct. Anal. 265 (2013), 2420–2451

    work page 2013

  8. [8]

    Chen and E

    W. Chen and E. Sawyer , Endpoint estimates for commutators of singular integrals on spaces of homogeneous type, J. Math. Anal. Appl. 282 (2) (2003), 553–566

    work page 2003

  9. [9]

    Coifman, R

    R. Coifman, R. Rochberg and G. Weiss , Factorization theorems for Hardy spaces in several variab les, Ann. Math. 103 (1976), 611–635

    work page 1976

  10. [10]

    Coifman and G

    R.R. Coifman and G. Weiss , Analyse harmonique non-commutative sur certains espaces h omog` enes. ´Etude de certaines int´ egrales singuli` eres. (French), Lecture Notes in Mathematics, Vol. 242, Springer – Verlag, Berlin, 1971

    work page 1971

  11. [11]

    Coifman and G

    R.R. Coifman and G. Weiss , Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645

    work page 1977

  12. [12]

    David, J.-L

    G. David, J.-L. Journ ´e, and S. Semmes , Calder´ on–Zygmund operators, para-accretive functions and interpolation, Rev. Mat. Iberoamericana 1 (4) (1985), 1–56

    work page 1985

  13. [13]

    Deng and Y

    D.G. Deng and Y. Han , Harmonic analysis on spaces of homogeneous type, with a pre face by Yves Meyer, Lecture Notes in Mathematics, Vol. 1966, Springer-V erlag, Berlin, 2009

    work page 1966

  14. [14]

    X. T. Duong, M. Lacey, J. Li, B. D. Wick and Q. Y. Wu , Commutators of Cauchy type integrals for domains in Cn with minimal smoothness, Indiana University Math. J. in press

    work page

  15. [15]

    D. E. Edmunds, V. Kokilashvili and A. Meskhi , Weight inequalities for singular integrals defined on spaces of homogeneous and nonhomogeneous type, Georgian Math. J. 8 (1) (2001), 33–59

    work page 2001

  16. [16]

    Fefferman, The Bergman kernel and biholomorphic mappi ngs of pseudoconvex domains

    C. Fefferman, The Bergman kernel and biholomorphic mappi ngs of pseudoconvex domains. Invent. Math., 26 (1974), 1–65

    work page 1974

  17. [17]

    Garc´ıa-Cuerva and J

    J. Garc´ıa-Cuerva and J. L. Rubio de Francia , Weighted norm inequalities and related topics, North Holland Math. Studies 116, North Holland, Amsterdam, (1985 )

    work page 1985

  18. [18]

    Garc´ıa-Cuerva, E

    J. Garc´ıa-Cuerva, E. Harboure, C. Segovia, J.L. Torrea , Weighted norm inequalities for com- mutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), 1397–1420

    work page 1991

  19. [19]

    Grafakos, L

    L. Grafakos, L. G. Liu and D. C. Yang , Vector-valued singular integrals and maximal functions o n spaces of homogeneous type, Math. Scand. 104 (2009), no. 2, 296–310

    work page 2009

  20. [20]

    Han , Calder´ on-type reproducing formula and the Tb theorem, Rev

    Y. Han , Calder´ on-type reproducing formula and the Tb theorem, Rev. Mat. Iberoamericana 10 (1994), 51–91

    work page 1994

  21. [21]

    Han , Plancherel-Pˆ olya type inequality on spaces of homogeneo us type and its applications, Proc

    Y. Han , Plancherel-Pˆ olya type inequality on spaces of homogeneo us type and its applications, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3315–3327

    work page 1998

  22. [22]

    Y. Han, J. Li, and C.-C. Lin , Criterions of the L2 boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneo us type, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (3) (2016), 845–907

    work page 2016

  23. [23]

    Han and E.T

    Y. Han and E.T. Sawyer , Littlewood–Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 530 (110) (1994), vi + 126 pp

    work page 1994

  24. [24]

    Hart and R

    J. Hart and R. H. Torres , John–Nirenberg inequalities and weight invariant BMO spa ces, J. Geom. Anal. 29 (2) (2019), 1608-1648

    work page 2019

  25. [25]

    Z. He, Y. Han, J. Li, L. Liu, D. Yang and W. Yuan , A complete real-variable theory of Hardy spaces on spaces of homogeneous type, J. Fourier Anal. Appl. in press

    work page

  26. [26]

    Kwok-Pun Ho , Characterizations of BMO spaces by Ap weights and p-convexity, Hiroshima Math. J. 41 (2011), 153-165

    work page 2011

  27. [27]

    G. Hu, H. Lin and D. Yang , Commutators of the Hardy-Littlewood maximal operator wit h BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal. 2008 (2008), Art. ID 237937

    work page 2008

  28. [28]

    Hu and D

    G. Hu and D. Yang Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type, J. Math. Anal. Appl. 354 (1) (2009), 249–262

    work page 2009

  29. [29]

    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

    work page 2012

  30. [30]

    S. G. Krantz and S.-Y. Li , Boundedness and compactness of integral operators on spac es of homoge- neous type and applications, I I, J. Math. Anal. Appl. , 258 (2001), 642–657

    work page 2001

  31. [31]

    Kronz , Some function spaces on spaces of homogeneous type, Manuscripta Math

    M. Kronz , Some function spaces on spaces of homogeneous type, Manuscripta Math. 106 (2) (2001), 219–248

    work page 2001

  32. [32]

    Lanzani and E

    L. Lanzani and E. Stein , The Cauchy–Szeg¨ o projection for domains in Cn with minimal smoothness, Duke Math. J. 166 (2017), 125–176

    work page 2017

  33. [33]

    A. K. Lerner , On weighted estimates of non-increasing rearrangements, East J. Approx. 4 (1998), 277–290

    work page 1998

  34. [34]

    Mac ´ıas and C

    R.A. Mac ´ıas and C. Segovia , Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257–270. COMMUTATORS OF MAXIMAL FUNCTIONS ON SPACES OF HOMOGENEOUS T YPE 21

    work page 1979

  35. [35]

    Milman and T

    M. Milman and T. Schonbek , Second order estimates in interpolation theory and applic ations, Proc. Amer. Math. Soc. 110 (4) (1990), 961–969

    work page 1990

  36. [36]

    Nagel, J.-P

    A. Nagel, J.-P. Rosay , E. M. Stein and S. Wainger, Estimates for the Bergman and Sze g¨ o kernels in C2, Ann. Math. 129 (2) (1989), 113–149

    work page 1989

  37. [37]

    Nagel and E.M

    A. Nagel and E.M. Stein , On the product theory of singular integrals, Rev. Mat. Iberoamericana 20 (2004), 531–561

    work page 2004

  38. [38]

    Nagel and E.M

    A. Nagel and E.M. Stein , The ¯∂b-complex on decoupled boundaries in Cn, Ann. Math. 164 (2) (2006), 649–713

    work page 2006

  39. [39]

    Pradolini, O

    G. Pradolini, O. Salinas , Commutators of singular integrals on spaces of homogeneou s type, Czechoslovak Math. J. 57 (1) (2007) 75–93

    work page 2007

  40. [40]

    R. M. Range , Holomorphic functions and integral representations in sev eral complex variables, Graduate Texts in Mathematics, 108, Springer-Verlag, New York, 1986

    work page 1986

  41. [41]

    M. M. Rao and Z. D. Ren , Theory of Orlicz Spaces , vol. 146 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1991

    work page 1991

  42. [42]

    Stein , Boundary Behavior of Holomorphic Functions of Several Comp lex Variables , Princeton Uni- versity Press, Princeton, 1972

    E. Stein , Boundary Behavior of Holomorphic Functions of Several Comp lex Variables , Princeton Uni- versity Press, Princeton, 1972. Zunwei Fu, Department of Mathematics, Linyi University, Sh andong, 276005, China E-mail address : fuzunwei@lyu.edu.cn Elodie Pozzi, Department of Mathematics and Statistics, Sa int Louis University, 220 N. Grand Bl vd, 63103...

    work page 1972