Continuity of Multilinear Operators on Triebel-lizorkin Spaces

Let T be the Calderón-Zygmund singular integral operator, a well-known result of Coifman et al. (see [6]) states that the commutator [b,T]( f ) = T(b f )− bT( f ) (where b ∈ BMO) is bounded on Lp(Rn) (1 < p <∞); Chanillo (see [1]) proves a similar result when T is replaced by the fractional integral operator; in [8, 9], these results on the TriebelLizorkin spaces and the case b ∈ Lipβ (where Lipβ is the homogeneous Lipschitz space) are obtained. The main purpose of this paper is to study the continuity of some multilinear operators related to certain convolution operators on the Triebel-Lizorkin spaces. In fact, we will obtain the continuity on the Triebel-Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As the applications, the continuity of the multilinear operators related to the Littlewood-Paley operator and Marcinkiewicz operator on the Triebel-Lizorkin spaces are obtained.