  We extend the recently much-studied Hardy factorization theorems to the
weight case. The key point of this paper is to establish the factorization
theorems without individual condition on the weight functions. As a direct
application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and
Lipschitz spaces via the weighted boundedness of commutators of multilinear
Calder\'{o}n-Zygmund operators with the genuinely multilinear weights.
