Calderón-Zygmund operators on product Hardy spaces

Let T be a product Calderón-Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of H^p (Rⁿ × R^m) and Journé's geometric covering lemma, R. Fefferman proved the remarkable H^p (Rⁿ × R^m) - L^p (Rⁿ × R^m) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood-Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journé's covering lemma to show that T is bounded on product H^p (Rⁿ × R^m) for max {frac(n, n + ε), frac(m, m + ε)} < p ≤ 1 if and only if T₁^* (1) = T₂^* (1) = 0, where ε is the regularity exponent of the kernel of T.