eng Sharp norm inequalities for commutators of classical operators Cruz-Uribe, David SFO Moen, Kabe info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion 2012 Commutators Two-weight inequalities Sharp weighted bounds Publicacions matemàtiques http://ddd.uab.cat/record/85170 oai:ddd.uab.cat:85170 02141493v56n1p147 10.5565/PUBLMAT_56112_06 Tots els drets reservats http://www.europeana.eu/rights/rr-f/ 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.