Sharp norm inequalities for commutators of classical operators. (English) Zbl 1255.42015

Authors’ abstract: 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 \([b,T]\), \(b\in BMO\) 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 \([b,I_{\alpha}]\), \(b\in BMO\), 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 that our bounds are sharp.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42A50 Conjugate functions, conjugate series, singular integrals
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