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Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$. In this paper, the authors first introduce the notion of admissible functions $\rho$ and then develop a theory of localized Hardy spaces $H^1_\rho ({\mathcal X})$ associated with $\rho$, which includes several maximal function characterizations of $H^1_\rho ({\mathcal X})$, the relations between $H^1_\rho ({\mathcal X})$ and the classical Hardy space $H^1({\mathcal X})$ via constructing a kernel function related to $\rho$, the atomic decomposition characterization of $H^1_\rho ({\mathcal X})$, and the boundedness of certain localized singular integrals in $H^1_\rho({\mathcal X})$ via a finite atomic decomposition characterization of some dense subspace of $H^1_\rho ({\mathcal X})$. This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schr\"odinger operator or the degenerate Schr\"odinger operator on $\rn$, or the sub-Laplace Schr\"odinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schr\"odinger operators considered here are associated with nonnegative potentials satisfying the reverse H\"older inequality.

Localized Hardy spaces 𝐻¹ related to admissible functions on RD-spaces and applications to Schrödinger operators