The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L (2)(R (n) ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A (r) weight on R (n) x R (n) , 1 < r < a. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H (L,w) (p) (R (n) xR (n) ), 0 < p 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H (L,w) (1)(R (n) x R (n) ) and BMO (L,w) (R (n) x R (n) ).