Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces

In this paper, the main aim is to consider the boundedness of commutators of multilinear Calder'{o}n-Zygmund operators with Lipschitz functions in the context of the variable exponent Lebesgue spaces. Furthermore, the variable versions of the Lipschitz spaces are also discussed, and with the help of a key tool of a pointwise estimate involving the sharp maximal operator of the multilinear fractional commutator and certain associated maximal operators.