Weak differentiability for commutators of multilinear maximal functions of Sobolev functions on domains

A systematic study is given for weak differentiability for the commutators of multilinear maximal operators and multilinear maximal commutators associated with a vector-valued function $\overrightarrow{b}=(b_1,\dots ,b_m)$ as well as their fractional variants on domains, where each $b_i$ belongs to Lipschitz space. The bounds and continuity for the above commutators are established on the first order Sobolev spaces. The bounds for the above commutators are also proved on the Sobolev spaces with zero boundary values.