Sharp Function Estimate and Boundedness on Morrey Spaces for Multilinear Commutator of Marcinkiewicz Operator

As the development of singular integral operators, their commutators have been well studied(see [1][3-5][10-12]). Let T be the Calderón-Zygmund singular integral operator. A classical result of Coifman, Rocherberg and Weiss (see [3]) state that commutator [b, T ](f) = T (bf) − bT (f)(where b ∈ BMO(Rn)) is bounded on Lp(Rn) for 1 < p < ∞. In [10-12], the sharp estimates for some multilinear commutators of the Calderón-Zygmund singular integral operators are obtained. The Marcinkiewicz operator is an important operator in harmonic analysis(see [6-8]). As the Morrey space may be considered as an extension of Lebesgue space, it is natural and important to study the boundedness of multilinear commutator related to the Marcinkiewicz operator on the Morrey Spaces. The purpose of this paper has two, first, we establish a sharp estimate for the multilinear commutator related to the Marcinkiewicz operator, and second, we prove the boundedness for the multilinear commutator related to the Marcinkiewicz operator on the generalized Morrey spaces by using the sharp estimate.