Weighted local Hardy spaces associated with operators

Let L be a self-adjoint positive operator on L-2(R-n). Assume that the semigroup e(-tL) generated by -L satisfies the Gaussian kernel bounds on L-2(R-n). In this article, we study weighted local Hardy space h(L,w)(1) (R-n) associated with L in terms of the area function characterization, and prove their atomic characters. Then, we introduce the weighted local BMO space bmo(L,w)(R-n) and prove that the dual of h(L,w)(1) (R-n) is bmoL, w(Rn). Finally a broad class of applications of these results is described.