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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 ( n ;A) associated with general expansive dilations and A 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(0,1]. Moreover, we prove the existence of finite atomic decompositions achieving the norm in dense subspaces of H w p ( 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< (or all continuous (p,q,s) w -atoms with q=) into uniformly bounded elements of some quasi-Banach space , then T uniquely extends to a bounded sublinear operator from H w p ( n ;A) to . The last two results are new even for the classical weighted Hardy spaces on n .

MSC:
42B30H p -spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
42B35Function spaces arising in harmonic analysis