New oscillation classes and two weight bump conditions for commutators

In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calder'on-Zygmund operator $T$, define the commutator $T_b^{1}f=[T,b]f=bTf-T(bf)$, and for $m\ge 2$ define the iterated commutator $T_b^{m}f=[b,T_b^{m-1}]f$. Traditionally, commutators are defined for functions $b\in BMO$, but we show that if we replace $BMO$ by an oscillation class first introduced by P'erez [31], we can give a range of sufficient conditions on a pair of weights $(u,v)$ for $T^m_b : L^p(v)\rightarrow L^p(u)$ to be bounded. Our results generalize work of the first two authors in [10], and more recent work by Lerner, et al. [28]. We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz, et al.[20].