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We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A 2 constant of the weight, then its commutator [b, T ] with a function b in BM O will obey a quadratic bound with respect to the A 2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a(More)
We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m, n), for m and n natural numbers. We use the ideas developed by Nazarov and Volberg in [NV] to prove that the weighted L 2 (w)-norm of a paraproduct with complexity (m, n), associated to a function b ∈ BM O d , depends linearly on the(More)
The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp(w) if and only if the weight w belongs to the Muckenhoupt class A d p. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2(w) case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space(More)
We show that if a weight w ∈ C d 2t and there is q > 1 such that w 2t ∈ A d q , then the L 2-norm of the t-Haar multiplier of complexity (m, n) associated to w depends on the square root of the C d 2t-characteristic of w times the square root A d q-characteristic of w 2t times a constant that depends polynomially on the complexity. In particular, if w ∈ C d(More)
We survey the recent solution of the so-called A 2 conjecture, all Calderón-Zygmund singular integral operators are bounded on L 2 (w) with a bound that depends linearly on the A 2 characteristic of the weight w, as well as corresponding results for commutators. We highlight the interplay of dyadic harmonic analysis in the solution of the A 2 conjecture,(More)