Let be an RD-space, which means that is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in . A positive function on is called admissible if there exist positive constants and such that for all ,
where is the metric on . A nontrivial class of admissible function is the well-known reverse Hölder class. Let be the completion of the set which is composed of test functions with the additional property. The Hardy space associated to is defined as follows:
where , and is the grand maximal function associated to .
At first, the authors obtain an atomic decomposition characterization of . They show that with equivalent norms, where is the atomic Hardy space associated to . Secondly, they establish a radial maximal function characterization of and obtain another characterization of via a variant of the radial maximal function, where the radial maximal function is associated to the admissible function . Moreover, they prove the boundedness of certain localized singular integrals on via a finite atomic decomposition characterization of some dense subspace of . The theory in this paper can be applied, respectively, to the Schrödinger operator or degenerate Schrödinger operator on , or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups and some new results are also obtained.