Boundedness of Multilinear Operators in Herz-type Hardy Space

Abstract

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝ n are bounded from $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{1} ,p_{1} }}_{{q1}} {\left( {\mathbb{R}^{n} } \right)}\times \cdots \times{\kern 1pt} H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha _{k} ,p_{k} }}_{{q_{k} }} {\left( {\mathbb{R}^{n} } \right)} $$ into $$ H\ifmmode\expandafter\dot\else\expandafter\.\fi{K}^{{\alpha ,p}}_{q} {\left( {\mathbb{R}^{n} } \right)} $$ 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 Calderon-Zygmund singular integrals and the fractional integrals of any orders.