Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications

Abstract

The main purpose of this paper is to establish a number of results concerning boundedness of multi-linear Calderón–Zygmund operators with kernels of mild regularity. Let T be a multilinear Calderón–Zygmund operator of type ω(t) with ω being nondecreasing and ω∈Dini(1), but without assuming ω to be concave. We obtain the end-point weak-type estimates for multilinear operator T. The multiple-weighted norm inequalities for multilinear operator T and multilinear commutators of T with BMO functions are also established.As applications, multiple-weighted norm estimates for para-products and bilinear pseudo-differential operators with mild regularity and their commutators are obtained.Moreover, some boundedness properties of the multilinear operators are also established on variable exponent Lebesgue spaces.Our results improve most of the earlier ones in the literature by removing the assumption of concavity of ω(t) and weakening the assumption of ω∈Dini(1/2) to ω∈Dini(1).