Boundedness of Multilinear Operators in Herz-type Hardy Space

Let k ∈ ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝn are bounded from HKα1, p1q1(ℝn) × ⋯ × HKαk, pkqk(ℝn) into HKα, pq(ℝn) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when αj ≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders.