Summary: We introduce and study weighted anisotropic Hardy spaces associated with general expansive dilations and 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 . Moreover, we prove the existence of finite atomic decompositions achieving the norm in dense subspaces of . As an application, we prove that for a given admissible triplet , if is a sublinear operator and maps all -atoms with (or all continuous -atoms with ) into uniformly bounded elements of some quasi-Banach space , then uniquely extends to a bounded sublinear operator from to . The last two results are new even for the classical weighted Hardy spaces on .