Open Access
2012 Sharp norm inequalities for commutators of classical operators
David Cruz-Uribe SFO, Kabe Moen
Publ. Mat. 56(1): 147-190 (2012).


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.


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David Cruz-Uribe SFO. Kabe Moen. "Sharp norm inequalities for commutators of classical operators." Publ. Mat. 56 (1) 147 - 190, 2012.


Published: 2012
First available in Project Euclid: 15 December 2011

zbMATH: 1255.42015
MathSciNet: MR2918187

Primary: 42A50

Keywords: commutators , sharp weighted bounds , two-weight inequalities

Rights: Copyright © 2012 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.56 • No. 1 • 2012
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