An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators

Let w   be an ApAp weight on RnRn, 1⩽p<∞1⩽p<∞. Let L   be a non-negative self-adjoint operator acting on L2(Rn)L2(Rn) satisfying a pointwise Gaussian estimate for its heat kernel. In this article we obtain a weighted q  -atomic decomposition with q>pq>p for the weighted Hardy space HL,w1(Rn) associated to L, which was introduced in [28]. The main contribution of this article is to weaken conditions on the weight w   of the weighted Hardy spaces HL,w1(Rn) and improve the result in [28].