In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [b, T ] with a locally integrable function b. This result is applied into two directions. If b ∈ BM O, we improve several weighted weak type bounds for [b, T ]. If b belongs to the weighted BM O, we obtain a quantitative form of the two-weighted bound for [b, T ] due to Bloom-Holmes-Lacey-Wick.2010 Mathematics Subject Classification. 42B20, 42B25.
We obtain a Bloom‐type characterization of the two‐weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals T Ω with Ω ∈ L ∞ (S n−1 ) and the Bochner-Riesz multiplier at the critical index B (n−1)/2 . More precisely, we prove qualitative and quantitative versions of Coifman-Fefferman type inequalities and their vector-valued extensions, weighted A p − A ∞ strong and weak type inequalities for 1 < p < ∞, and A 1 − A ∞ type weak (1, 1) estimates. Moreover, Fefferman-Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 90's. As a corollary, we obtain the weighted A 1 − A ∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function Ω ∈ L q (S n−1 ), 1 < q < ∞, and provide Fefferman-Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by , results by the first author in [37], suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for A ∞ weights [13,16] and ideas contained in previous works by A. Seeger in [50] and D. Fan and S. Sato [22].
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