Let T be a product Calderon-Zygmund singular integral introduced by Journe. Using an elegant rectangle atomic decomposition of H(p) (R(n) x R(m)) and Journe's geometric covering lemma, R. Fefferman proved the remarkable H(p)(R(n) x R(m)) - L(p)(R(n) x R(m)) boundedness of T. In this paper we apply vector-valued singular integral, Calderon's identity, Littlewood-Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journe's covering lemma to show that T is bounded on product H(p)(R(n) x R(m)) for max{n/n+epsilon, m/m+epsilon} < p <= 1 if and only if T(1)*(1) = T(2)*(1) = 0, where epsilon is the regularity exponent of the kernel of T. (C) 2009 Elsevier Inc. All rights reserved.
