Mean Oscillation and Boundedness of Multilinear Integral Operators with General Kernels

As the development of singular integral operators, their commutators and multilinear operators have been well studied (see [3]–[7], [18]–[20]). Let T be the Calderón-Zygmund singular integral operator and b ∈ BMO(R), a classical result of Coifman, Rochberg and Weiss (see [6]) stated that the commutator [b, T ](f) = T (bf)− bT (f) is bounded on L(R) for 1 < p <∞. The purpose of this paper is to introduce some multilinear operator associated to certain integral operators with general kernels (see [1, 10, 15]) and prove the boundedness properties of the multilinear operators from Lebesgue spaces to Orlicz spaces. In this paper, we are going to consider some integral operators as following (see [1]). Let l and mj be the positive integers (j = 1, . . . , l), m1 + · · ·+ml = m and bj be the functions on R (j = 1, . . . , l). Set, for 1 ≤ j ≤ l,