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 $\mathbb {R}^{n}$ are bounded from $H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}}$ into $H\dot K_{q}^{\alpha ,p}$ if and only if they have vanishing moments up to a certain order dictated by the target space. Here $H\dot K_{q}^{\alpha ,p}$ are homogeneous Herz-type Hardy spaces with $1/p=1/p_{1}+1/p_{2},$ $0<p_{i}\le \infty ,$ $1/q=1/q_{1}+1/q_{2},$ $1<q_{1},q_{2}<\infty ,$ $1\le q<\infty ,$ $\alpha =\alpha _{1}+\alpha _{2}$ and $-n/q_{i}<\alpha _{i}<\infty$. As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.