A local version of Hardy spaces associated with operators on metric spaces

Let (X, d, A mu) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure A mu. Let L be a second order self-adjoint positive operator on L (2)(X). Assume that the semigroup e (-tL) generated by -L satisfies the Gaussian upper bounds on L (2)(X). In this article we study a local version of Hardy space h (L) (1) (X) associated with L in terms of the area function characterization, and prove their atomic characters. Furthermore, we introduce a Moser type local boundedness condition for L, and then we apply this condition to show that the space h (L) (1) (X) can be characterized in terms of the Littlewood-Paley function. Finally, a broad class of applications of these results is described.