On commutators of vector BMO functions and multilinear singular integrals with non-smooth kernels

Let T be a bounded multilinear operator on some product of Lq(Rn) spaces. Assume that T has a non-smooth associated kernel which satisfies certain weak regularity conditions but not regular enough to fall under the scope of the standard multilinear Calderón-Zygmund theory. The main aim of this paper is to establish a sufficient condition on the kernel of T so that the commutator of a vector BMO function and T is bounded on certain product Lp(Rn) spaces. We obtain boundedness of the commutator of and T by first proving certain pointwise estimates on the Fefferman-Stein sharp maximal operator. An important example of multilinear operators which satisfy our kernel conditions is the maximal mth order Calderón commutator.     

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

In this article, we obtain some weighted estimates for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type. The weight $\omega$ considered here belongs to the Muckenhoupt’s class $A_∞.$ Moreover, weighted estimates for commutators of BMO functions and Marcinkiewicz integrals are also given.