Abstract:

This paper establishes a general method for proving the boundedness of a class of product linear operators from weighted product Hardy spaces associated with para-accretive functions to weighted product Lebesgue spaces. Firstly, by using the discrete Calderón reproducing formula, the equivalent characterizations of both the continuous and discrete forms for the weighted product Hardy spaces ${H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)}$ associated with para-accretive functions $b_1$ and $b_2$ are established. Then, combining the density proposition and the norm domination relationship between the $H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$ norm and the $ L^p_w(\mathbb{R} ^{n+m})$ norm for $f \in L^q(\mathbb{R} ^{n+m}) \cap H_{b_1,b_2}^{p,w}(\mathbb{R} ^n\times\mathbb{R}^m)$, the main results are obtained. As an application, the boundedness of generalized product Calderón-Zygmund operators from product Hardy spaces associated with para-accretive functions to product Lebesgue spaces is established. This approach avoids using the atomic decomposition theory of product Hardy spaces associated with para-accretive functions and the Journé covering lemma.

Key words: product linear operator, weighted product Hardy space, para-accretive function, Plancherel-P?lya inequality.