Continuity of the local multilinear maximal commutators in Sobolev spaces

This paper investigates the continuity mapping properties of local multilinear maximal commutator and its fractional variant on the first order Sobolev spaces. The continuity of local multilinear maximal commutator and its fractional variant on the first order Sobolev spaces are proved under the symbols in the first order Sobolev spaces. The main results not only extend the main result of 18 to a multilinear version and fractional version, but also essentially address some questions motivated by Liu and Xi in 17 and Zhang and Liu in 26. It should be pointed out that the Sobolev continuity result for the multilinear fractional maximal commutator is new even in the linear case m = 1.