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

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