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 $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ 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 Calder\'on-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p>1$ or the Hardy space $H^p({{\mathbb R}^n\times{\mathbb R}^m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^q({{\mathbb R}^n\times{\mathbb R}^m})$ with some $q>1$.