In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators T-b(m). Our results are new even for the first order commutator T-b(1). A new bump type necessary condition for the two-weighted boundedness of T-b(m) is obtained as well. We also provide some results related to a converse to Bloom's theorem. (C) 2021 Elsevier Inc. All rights reserved.
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