article

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

Authors: Dachun Yang, Marcin Bownik, Baode Li and Yuan Zhou

Issue: Volume 57 (2008), Issue 7, 3065-3100

Abstract:

$In this paper we introduce and study weighted anisotropic Hardy spaces $H^p_w(\mathbb{R}^n;A)$ 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 Calder\'on and Torchinsky, and the weighted Hardy spaces of Garc\'ia-Cuerva, Str\"omberg, 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 $H^p_w(\mathbb{R}^n;A)$. 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 $\mathcal{B}$, then $T$ uniquely extends to a bounded sublinear operator from $H^p_w(\mathbb{R}^n;A) $ to $\mathcal{B}$. The last two results are new even for the classical weighted Hardy spaces on $\mathbb{R}^n$.$