New weighted norm inequalities and applications for multilinear Calderón-Zygmund operators with fractional kernels of Dini’s type and their commutators

Let T be a multilinear Calderón-Zygmund operator with fractional kernel of Dini’s type. In this article, we introduce the multilinear fractional new maximal operator M ϕ,β to control the dyadic fractional sharp maximal functions of T and its com-mutators. Meanwhile, a new class of multiple weights A p,β (ϕ) is also introduced to characterize the boundedness of M ϕ,β. With the aid of the properties of M ϕ,β and A p,β (ϕ) condition, the strong and weak type multiple weighted estimates for T and its commutators with BMO ϕ,β functions are established. As applications, the weighted norm inequalities for the commutators of M ϕ,β will be obtained. Mathematics Subject Classification 42B25 · 42B20 · 42B35 · 47B47