Duality theory of weighted Hardy spaces with arbitrary number of parameters

In this paper, we use the discrete Littlewood-Paley-Stein analysis to get the duality result of the weighted product Hardy space for arbitrary number of parameters under a rather weak condition on the product weight w is an element of A(infinity)(R-n1 x ... x R-nk). We will show that for any k >= 2, (H-w(p) (R-n1 x ... x R-nk))* = CMOwp (R-n1 x ... x R-nk) (a generalized Carleson measure), and obtain the boundedness of singular integral operators on BMOw. Our theorems even when the weight function w = 1 extend the H-1-BMO duality of Chang-R. Fefferman for the non-weighted two-parameter Hardy space H-1(R-n x R-m) to H-p (R-n1 x ... x R-nk)for all 0 < p <= 1 and our weighted theory extends the duality result of Krug-Torchinsky on weighted Hardy spaces H-w(p) (R-n x R-m) for w is an element of A(r) (R-n x R-m) with 1 <= r <= 2 and r/2 < p <= 1 to H-w(p) (R-n1 x ... x R-nk) with w is an element of A(infinity) (R-n1 x ... x R-nk) for all 0 < p <= 1.