Boundedness of Calderón-Zygmund operators on weighted product hardy spaces

Let T be a singular integral operator in Journé's class with regularity exponent ε, w ∈ A_q, 1 ≤ q < 1 + ε, and q/(1 + ε) < p ≤ 1. We obtain the H^p_w(R×R)-L^p_w (R²) boundedness of T by using R. Fefferman's "trivial lemma" and Journé's covering lemma. Also, using the vector-valued version of the "trivial lemma" and Littlewood-Paley theory, we prove the H^p_w (R×R)-boundedness of T provided T*₁(1) = T*₂(1) = 0; that is, the reduced T1 theorem on H^p_w(R×R). In order to show these two results, we demonstrate a new atomic decomposition of H^p_w(R×R) ∩ L²_w(R²), for which the series converges in L²_w. Moreover, a fundamental principle that the boundedness of operators on the weighted product Hardy space can be obtained simply by the actions of such operators on all atoms is given.