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Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. (English) Zbl 1161.42014

Summary: We introduce and study weighted anisotropic Hardy spaces ${H}_{w}^{p}\left({ℝ}^{n};A\right)$ associated with general expansive dilations and ${A}_{\infty }$ Muckenhoupt weights. This setting includes the classical isotropic Hardy space theory of Fefferman and Stein, the parabolic theory of Calderón and Torchinsky, and the weighted Hardy spaces of García-Cuerva, Stroemberg, and Torchinsky.

We establish characterizations of these spaces via the grand maximal function and their atomic decompositions for $p\in \left(0,1\right]$. Moreover, we prove the existence of finite atomic decompositions achieving the norm in dense subspaces of ${H}_{w}^{p}\left({ℝ}^{n};A\right)$. As an application, we prove that for a given admissible triplet ${\left(p,q,s\right)}_{w}$, if $T$ is a sublinear operator and maps all ${\left(p,q,s\right)}_{w}$-atoms with $q<\infty$ (or all continuous ${\left(p,q,s\right)}_{w}$-atoms with $q=\infty$) into uniformly bounded elements of some quasi-Banach space $ℬ$, then $T$ uniquely extends to a bounded sublinear operator from ${H}_{w}^{p}\left({ℝ}^{n};A\right)$ to $ℬ$. The last two results are new even for the classical weighted Hardy spaces on ${ℝ}^{n}$.

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