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

Let (X, d, μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a second order self-adjoint positive operator on L²(X). Assume that the semigroup e^-tL generated by -L satisfies the Gaussian upper bounds on L²(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.