Boundedness of sublinear operators on product Hardy spaces and its application

Abstract

Let $p\in (0,1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({\mathit{R}}^n\times {\mathit{R}}^m)$ to some quasi-Banach space $\mathcal{B}$ if and only if $T$ maps all $(p,2,s_1,s_2)$-atoms into uniformly bounded elements of $\mathcal{B}$. Here $s_1\ge \lfloor n(1/p-1)\rfloor$ and $s_2\ge \lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $p>1$ or the Hardy space $H^p({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $p\le 1$ but near 1 to the Lebesgue space $L^q({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $q>1$.