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Commutators with fractional integral operators

Volume 233 / 2016

Irina Holmes, Robert Rahm, Scott Spencer Studia Mathematica 233 (2016), 279-291 MSC: Primary 42A05, 42A50, 42B20; Secondary 42A61, 42B25. DOI: 10.4064/sm8419-4-2016 Published online: 24 May 2016

Abstract

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and $\alpha /n+1/q=1/p$, the norm $\|[b,I_\alpha ]:L^p(\mu ^p)\to L^q(\lambda ^q)\| $ is equivalent to the norm of $b$ in the weighted BMO space ${\rm BMO}(\nu )$, where $\nu =\mu \lambda ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

Authors

  • Irina HolmesSchool of Mathematics
    Georgia Institute of Technology
    686 Cherry Street
    Atlanta, GA 30332-0160, U.S.A.
    e-mail
  • Robert RahmSchool of Mathematics
    Washington University in St. Louis
    One Brookings Drive
    St. Louis, MO 63130, U.S.A.
    e-mail
  • Scott SpencerSchool of Mathematics
    Georgia Institute of Technology
    686 Cherry Street
    Atlanta, GA 30332-0160, U.S.A.
    e-mail

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