Bilinear operators on Herz-type Hardy spaces

Abstract

The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on R n \mathbb {R}^{n} are bounded from H K ˙ q 1 α 1 , p 1 × H K ˙ q 2 α 2 , p 2 H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}} into H K ˙ q α , p H\dot K_{q}^{\alpha ,p} if and only if they have vanishing moments up to a certain order dictated by the target space. Here H K ˙ q α , p H\dot K_{q}^{\alpha ,p} are homogeneous Herz-type Hardy spaces with 1 / p = 1 / p 1 + 1 / p 2 , 1/p=1/p_{1}+1/p_{2}, 0 > p i ≤ ∞ , 0>p_{i}\le \infty , 1 / q = 1 / q 1 + 1 / q 2 , 1/q=1/q_{1}+1/q_{2}, 1 > q 1 , q 2 > ∞ , 1>q_{1},q_{2}>\infty , 1 ≤ q > ∞ , 1\le q>\infty , α = α 1 + α 2 \alpha =\alpha _{1}+\alpha _{2} and − n / q i > α i > ∞ -n/q_{i}>\alpha _{i}>\infty . As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.