Sharp norm inequalities for commutators of classical operators

We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We found suffcient Ap-bump conditions on pairs of weights (u; v) such that [b; T], b 2 BMO and T a singular integral operator (such as the Hilbert or Riesz transforms), maps Lp(v) into Lp(u...

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Bibliographic Details
Published in:Publicacions matemàtiques 2012, Vol.56 (1), p.147-190
Main Authors: Cruz-Uribe, David, SFO, Moen, Kabe
Format: Article
Language:English
Subjects:
Commutators
Cubes
Dyadics
Integers
Logarithms
Mathematical functions
Mathematical inequalities
Mathematical integrals
Mathematics
Sufficient conditions
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Summary:We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We found suffcient Ap-bump conditions on pairs of weights (u; v) such that [b; T], b 2 BMO and T a singular integral operator (such as the Hilbert or Riesz transforms), maps Lp(v) into Lp(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 we nd the sharp one-weight bound on [b; I ], b 2 BMO, in terms of the Ap;q constant of the weight. We also prove sharp two-weight bounds for [b; I ] analogous to those of singular integrals. We prove two-weight weak type inequalities for [b; T] and [b; I ] for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.
ISSN:0214-1493
2014-4350
DOI:10.5565/publmat_56112_06