Weighted Anisotropic Product Hardy Spaces and Boundedness of Sublinear Operators

Let A1 and A2 be expansive dilations, respectively, on R n and R. Let ~ A ≡ (A1, A2) and Ap( ~ A) be the class of product Muckenhoupt weights on R × R for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ Ap( ~ A), the authors characterize the weighted Lebesgue space L w (R × R) via the anisotropic Lusin-area function associated with ~ A. When p ∈ (0, 1], w ∈ A∞( ~ A), the authors introduce the weighted anisotropic product Hardy space H w (R × R; ~ A) via the anisotropic Lusin-area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of H w (R × R; ~ A) is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space B, then T uniquely extends to a bounded sublinear operator from H w (R × R; ~ A) to B. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.