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Localized Hardy spaces $H^1$ related to admissible functions on RD-spaces and applications to Schrödinger operators. (English) Zbl 1217.42044
Let $\mathcal {X}$ be an RD-space, which means that ${\cal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\cal X}$. A positive function $\rho$ on ${\cal X}$ is called admissible if there exist positive constants $C_3$ and $k_0$ such that for all $x,y\in{\cal X}$, $$\rho(y)\le C_3[\rho(x)]^{\frac{1}{1+k_0}}[\rho(x)+d(x,y)]^{\frac{k_0}{1+k_0}},$$ where $d$ is the metric on ${\cal X}$. A nontrivial class of admissible function is the well-known reverse Hölder class. Let ${\cal G}^\varepsilon_0(\beta,\gamma)$ be the completion of the set which is composed of test functions with the additional property. The Hardy space $H^1_{\rho}({\cal X})$ associated to $\rho$ is defined as follows: $$H^1_{\rho}({\cal X})=\{f\in {\cal G}^\varepsilon_0(\beta,\gamma)': \|f\|_{H^1_{\rho}({\cal X})}=\|G_{\rho}(f)\|_{L^1({\cal X})}<\infty \},$$ where $\varepsilon\in (0,1)$, $\beta,\gamma\in (0,\varepsilon)$ and $G_\rho(f)$ is the grand maximal function associated to $\rho$. At first, the authors obtain an atomic decomposition characterization of $H^1_\rho({\cal X})$. They show that $H^1_{\rho}({\cal X})=H^{1,q}_{\rho}({\cal X})$ with equivalent norms, where $H^{1,q}_{\rho}({\cal X})$ is the atomic Hardy space associated to $\rho$. Secondly, they establish a radial maximal function characterization of $H^1_{\rho}({\cal X})$ and obtain another characterization of $H^1_{\rho}({\cal X})$ via a variant of the radial maximal function, where the radial maximal function is associated to the admissible function $\rho$. Moreover, they prove the boundedness of certain localized singular integrals on $H^1_{\rho}({\cal X})$ via a finite atomic decomposition characterization of some dense subspace of $H^1_{\rho}({\cal X})$. The theory in this paper can be applied, respectively, to the Schrödinger operator or degenerate Schrödinger operator on ${\cal R}^n$, 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.
Reviewer: Liu Yu (Beijing)

42B30$H^p$-spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
42B35Function spaces arising in harmonic analysis
42B37Harmonic analysis and PDE
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