Resumen de New oscillation classes and two weight bump conditions for commutators

David Cruz-Uribe , Kabe Moen, Quan Minh Tran

• In this paper we consider two weight bump conditions for higher order commutators. Given b and a Calderón–Zygmund operator T, define the commutator T1bf=[T,b]f=bTf−T(bf), and for m≥2 define the iterated commutator Tmbf=[b,Tm−1b]f. Traditionally, commutators are defined for functions b∈BMO, but we show that if we replace BMO by an oscillation class first introduced by Pérez (J Funct Anal 128(1):163–185, 1995), we can give a range of sufficient conditions on a pair of weights (u, v) for Tmb:Lp(v)→Lp(u) to be bounded. Our results generalize work of Cruz-Uribe and Moen (Publ Mat 56(1):147–190, 2012), and more recent work by Lerner et al. (J Funct Anal 281(8):46, 2021). We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz et al. (Commutators in the two scalar and matrix weighted setting. Preprint, 2020. http://arxiv.org/abs/2001.11182).