Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

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AbstractLet A1 and A2 be expansive dilations, respectively, on ℝn and ℝm. Let A ≡ (A1, A2) and 𝒜p(A) be the class of product Muckenhoupt weights on ℝn × ℝm for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ 𝒜p(A), the authors characterize the weighted Lebesgue space Lp w(ℝn × ℝm) 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 Hp w (ℝn × ℝm; 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 Hp w (ℝn×ℝm; 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 ℬ︁, then T uniquely extends to a bounded sublinear operator from Hp w (ℝn × ℝm; A) to ℬ︁. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)