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Sharp norm inequalities for commutators of classical operators

David Cruz-Uribe, SFO and Kabe Moen

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We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in \mathit{BMO}$ and $T$ a singular integral operator (such as the Hilbert or Riesz transforms), maps $L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the commutators require a "double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator $I_\alpha$ we find the sharp one-weight bound on $[b,I_\alpha]$, $b\in \mathit{BMO}$, in terms of the $A_{p,q}$ constant of the weight. We also prove sharp two-weight bounds for $[b,I_\alpha]$ analogous to those of singular integrals. We prove two-weight weak type inequalities for $[b,T]$ and $[b,I_\alpha]$ for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.

Article information

Publ. Mat., Volume 56, Number 1 (2012), 147-190.

First available in Project Euclid: 15 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42A50: Conjugate functions, conjugate series, singular integrals

Commutators two-weight inequalities sharp weighted bounds


Cruz-Uribe, David; Moen, Kabe. Sharp norm inequalities for commutators of classical operators. Publ. Mat. 56 (2012), no. 1, 147--190.

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