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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 $${\mathcal X}$$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $${\mathcal X}$$. A positive function $$\rho$$ on $${\mathcal X}$$ is called admissible if there exist positive constants $$C_3$$ and $$k_0$$ such that for all $$x,y\in{\mathcal X}$$,
$\rho(y)\leq C_3[\rho(x)]^{\frac{1}{1+k_0}}[\rho(x)+d(x,y)]^{\frac{k_0}{1+k_0}},$
where $$d$$ is the metric on $${\mathcal X}$$. A nontrivial class of admissible function is the well-known reverse Hölder class. Let $${\mathcal G}^\varepsilon_0(\beta,\gamma)$$ be the completion of the set which is composed of test functions with the additional property. The Hardy space $$H^1_{\rho}({\mathcal X})$$ associated to $$\rho$$ is defined as follows:
$H^1_{\rho}({\mathcal X})=\{f\in {\mathcal G}^\varepsilon_0(\beta,\gamma)': \|f\|_{H^1_{\rho}({\mathcal X})}=\|G_{\rho}(f)\|_{L^1({\mathcal X})}<\infty \},$
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 $$H^1_\rho({\mathcal X})$$. They show that $$H^1_{\rho}({\mathcal X})=H^{1,q}_{\rho}({\mathcal X})$$ with equivalent norms, where $$H^{1,q}_{\rho}({\mathcal X})$$ is the atomic Hardy space associated to $$\rho$$. Secondly, they establish a radial maximal function characterization of $$H^1_{\rho}({\mathcal X})$$ and obtain another characterization of $$H^1_{\rho}({\mathcal X})$$ 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 $$H^1_{\rho}({\mathcal X})$$ via a finite atomic decomposition characterization of some dense subspace of $$H^1_{\rho}({\mathcal X})$$. The theory in this paper can be applied, respectively, to the Schrödinger operator or degenerate Schrödinger operator on $${\mathcal R}^n$$, 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)

##### MSC:
 42B30 $$H^p$$-spaces 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 42B37 Harmonic analysis and PDEs
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