Journé's covering lemma and its extension to higher dimensions (Q1103142)

The author extends Journé's covering lemma concerning dyadic coverings which are maximal in one coordinate direction, to the case of product domains \({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m}\) (m\(\geq 3)\) [originally \(m=2\), \textit{J. Journé}, Proc. Am. Math. Soc. 96, 593-598 (1986; Zbl 0594.42015)]. Using it the author establishes the boundedness of Calderón-Zygmund singular integrals (of product type) from the product Hardy space \(H^ p({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m})\) to \(L^ p({\mathbb{R}}^{n_ 1}\times...\times {\mathbb{R}}^{n_ m})\), which extends \textit{R. Fefferman}'s result in the case \(m=2\) [Proc. Nat. Acad. Sci. 83, 840-843 (1986; Zbl 0602.42023)]. It is pointed out that several differences occur in two cases \(m=2\) and \(m\geq 3\).