Archiv der Mathematik

, Volume 106, Issue 1, pp 53–63 | Cite as

Bloom’s inequality: commutators in a two-weight setting

  • Irina Holmes
  • Michael T. Lacey
  • Brett D. Wick


In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L p spaces, where both weights are in A p . The characterization is in terms of a novel BMO condition. We give a ‘modern’ proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom’s result to all Calderón–Zygmund operators and dimensions.


Commutators Calderón–Zygmund operators BMO Weights Haar multiplier Paraproducts 

Mathematics Subject Classification

Primary 42A 42B Secondary 42A40 42A99 42B35 42B20 42B30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bloom S.: A commutator theorem and weighted BMO. Trans. Amer. Math. Soc. 292, 103–122 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Coifman R. R., Rochberg R., Weiss G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103(2), 611–635 (1976)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Petermichl S., Pott S.: An estimate for weighted Hilbert transform via square functions. Trans. Amer. Math. Soc. 354, 1699–1703 (2002) (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Petermichl S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic. Amer. J. Math. 129, 1355–1375 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Petermichl S.: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol. C. R. Acad. Sci. Paris Sér. I Math. 330, 455–460 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Wittwer J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7, 1–12 (2000)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsWashington University-Saint Louis, One Brookings DriveSaint LouisUSA

Personalised recommendations