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

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 TT be a multilinear Calderón–Zygmund operator of type ω(t)ω(t) with ωω being nondecreasing and ω∈Dini(1), but without assuming ωω to be concave. We obtain the end-point weak-type estimates for multilinear operator TT. The multiple-weighted norm inequalities for multilinear operator TT and multilinear commutators of TT with BMOBMO 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)ω(t) and weakening the assumption of ω∈Dini(1/2) to ω∈Dini(1).