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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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