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 Hp(Rn × Rm) and Journé’s geometric covering lemma, R. Fefferman proved the remarkable Hp(Rn × Rm) − Lp(Rn × Rm) 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 Hp(Rn × Rm) for max{ n n+ε , m m+ε } < p 1 if and only if T ∗ 1 (1) = T ∗ 2 (1) = 0, where ε is the regularity exponent of the kernel of T . © 2009 Elsevier Inc. All rights reserved