We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators that include Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.

The authors would like to thank Michael Lacey for having made this collaboration possible. The authors are also grateful to Rodolfo Torres for kindly pointing out several inaccuracies of references in an earlier version of the paper. The second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester when this work was carried out, and was supported by the National Science Foundation under Grant No. DMS-1440140.