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 F. Liu, Q. Xue and K. Yabuta [Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function, Ann. Fenn. Math. 47 (2022) 203–235] to a multilinear version and fractional version, but also essentially address some questions motivated by Liu and Xi [Sobolev regularity for commutators of the fractional maximal functions, Banach J. Math. Anal. 15 (2021) 1–36] and Zhang and Liu [Regularity for commutators of the local multilinear fractional maximal operators, Adv. Nonlinear Anal. 10 (2021) 849–876]. It should be pointed out that the Sobolev continuity result for the multilinear fractional maximal commutator is new even in the linear case [Formula: see text].