Paraproducts in One and Several Parameters

For multiparameter bilinear paraproduct operators B we prove the estimate B : L × L 7→ L, 1 < p, q ≤ ∞. Here, 1/p+ 1/q = 1/r and special attention is paid to the case of 0 < r < 1. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele [10,11]. We offer a different proof of these inequalities.