Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

摘要

Let A_1 and A_2 be expansive dilations, respectively, on R~n and R~m. Let A{top}→ ≡ (A_1, A_2) and A_p(A{top}→) be the class of product Muckenhoupt weights on R~n × R~m for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ A_p(A{top}→), the authors characterize the weighted Lebesgue space (L_w)~p(R~n × R~m) via the anisotropic Lusin-area function associated with A{top}→. When p ∈ (0, 1], w ∈ A_∞(A{top}→), the authors introduce the weighted anisotropic product Hardy space (H_w)~p (R~n×R~m; A{top}→) 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)~p (R~n×R~m; A{top}→) 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)~p (R~n × R~m; A{top}→) 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.