Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators

Abstract

In this paper we introduce and study weighted anisotropic Hardy spaces H p w (R 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 Calderon and Torchinsky, and the weighted Hardy spaces of Garcia-Cuerva, Stromberg, 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 p w (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 < oo (or all continuous (p, q, s) w -atoms with q = ∞) into uniformly bounded elements of some quasi-Banach space B, then T uniquely extends to a bounded sublinear operator from H p w (R n ;A) to B. The last two results are new even for the classical weighted Hardy spaces on R n .