Triebel–Lizorkin space estimates for multilinear operators of sublinear operators

Let T be the singular integral operator, a well-known result of Coifman, Rochberg and Weiss [6] which states that the commutator [b,T ] = T (b f )− bT f (where b ∈ BMO) is bounded on Lp(Rn)(1 < p < ∞). Chanillo [1] proves a similar result when T is replaced by the fractional integral operator. In [9,11], these results on the Triebel–Lizorkin 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 for some multilinear operators related to certain convolution operators on the Triebel–Lizorkin spaces. In fact, we shall obtain the continuity on the Triebel–Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As applications, we prove the continuity of the multilinear operators related to the Littlewood–Paley operator and Marcinkiewicz operator on the Triebel–Lizorkin spaces.