  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 $T^1_bf=[T,b]f= bTf-T(bf)$, and for $m\geq 2$ define the iterated
commutator $T^m_b f = [b,T_b^{m-1}]f$. 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].
