Some BMO estimates for vector-valued multilinear singular integral operators

Let b ∈ BMO(Rn) and T be the Calderón–Zygmund singular integral operator. The commutator [b,T ] generated by b and T is defined as [b,T ]( f )(x) = b(x)T ( f )(x)−T (b f )(x). By using a classical result of Coifman et al [8], we know that the commutator [b,T ] is bounded on Lp(Rn) for 1 < p < ∞. Chanillo [1] proves a similar result when T is replaced by the fractional integral operator. However, it was observed that the commutator is not bounded, in general, from H p(Rn) to Lp(Rn) for 0 < p ≤ 1 [13,14,15]. In [11], the boundedness properties of the commutator for the extreme values of p are obtained. Also, in [2], Chanillo studies some commutators generated by a very general class of pseudodifferential operators and proves the boundedness on Lp(Rn) (1 < p < ∞) for the commutators, and note that the conditions on the kernel of the singular integral operator arise from a pseudo-differential operator. As the development of singular integral operators and their commutators, multilinear singular integral operators have been well-studied. It is known that multilinear operator, as a non-trivial extension of the commutator, is of great interest in harmonic analysis and has been widely studied by many authors [3,4,5,6,7]. In [9], the weighted Lp(p > 1)-boundedness of the multilinear operator related to some singular integral operators is obtained and in [3], the weak (H1, L1)-boundedness of the multilinear operator related to some singular integral operators is obtained. The main purpose of this paper is to establish the BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators. First, let us introduce some notations [10,16]. Throughout this paper, Q = Q(x,r) will denote a cube of Rn with sides parallel to the axes and centered at x and having side length. For a locally integrable function f and non-negative weight function w, let w(Q) = ∫