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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 $𝒳$ be an RD-space, which means that $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $𝒳$. A positive function $\rho$ on $𝒳$ is called admissible if there exist positive constants ${C}_{3}$ and ${k}_{0}$ such that for all $x,y\in 𝒳$,

$\rho \left(y\right)\le {C}_{3}{\left[\rho \left(x\right)\right]}^{\frac{1}{1+{k}_{0}}}{\left[\rho \left(x\right)+d\left(x,y\right)\right]}^{\frac{{k}_{0}}{1+{k}_{0}}},$

where $d$ is the metric on $𝒳$. A nontrivial class of admissible function is the well-known reverse Hölder class. Let ${𝒢}_{0}^{\epsilon }\left(\beta ,\gamma \right)$ be the completion of the set which is composed of test functions with the additional property. The Hardy space ${H}_{\rho }^{1}\left(𝒳\right)$ associated to $\rho$ is defined as follows:

${H}_{\rho }^{1}\left(𝒳\right)=\left\{f\in {𝒢}_{0}^{\epsilon }{\left(\beta ,\gamma \right)}^{\text{'}}{:\parallel f\parallel }_{{H}_{\rho }^{1}\left(𝒳\right)}=\parallel {G}_{\rho }\left(f\right){\parallel }_{{L}^{1}\left(𝒳\right)}<\infty \right\},$

where $\epsilon \in \left(0,1\right)$, $\beta ,\gamma \in \left(0,\epsilon \right)$ and ${G}_{\rho }\left(f\right)$ is the grand maximal function associated to $\rho$.

At first, the authors obtain an atomic decomposition characterization of ${H}_{\rho }^{1}\left(𝒳\right)$. They show that ${H}_{\rho }^{1}\left(𝒳\right)={H}_{\rho }^{1,q}\left(𝒳\right)$ with equivalent norms, where ${H}_{\rho }^{1,q}\left(𝒳\right)$ is the atomic Hardy space associated to $\rho$. Secondly, they establish a radial maximal function characterization of ${H}_{\rho }^{1}\left(𝒳\right)$ and obtain another characterization of ${H}_{\rho }^{1}\left(𝒳\right)$ 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 ${H}_{\rho }^{1}\left(𝒳\right)$ via a finite atomic decomposition characterization of some dense subspace of ${H}_{\rho }^{1}\left(𝒳\right)$. The theory in this paper can be applied, respectively, to the Schrödinger operator or degenerate Schrödinger operator on ${ℛ}^{n}$, 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 (Fourier analysis) 42B20 Singular and oscillatory integrals, several variables 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 42B37 Harmonic analysis and PDE