2009/03/23 by Marcin Bownik, Baode Li, Bownik, Marcin +5 · 1 citation
Mathematics · #42B20 (Secondary) #42B25 #42B30 (Primary) #42B35 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.CA #math.FA #msc:42B20 #msc:42B25 #msc:42B30 #msc:42B35
paper · pdf · doi:10.48550/arxiv.0903.3775
Math. Nachr. (to appear)
openalex publication_date 2009/03/23 · arxiv created 2009/11/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A1 and A2 be expansive dilations, respectively, on \mathbb Rn and \mathbb Rm. Let A≡(A1, A2) and \mathcal Ap( A) be the class of product Muckenhoupt weights on \mathbb Rn×\mathbb Rm for p∈(1, ∞]. When p∈(1, ∞) and w∈\mathcal Ap( A), the authors characterize the weighted Lebesgue space Lpw(\mathbb Rn×\mathbb Rm) via the anisotropic Lusin-area function associated with A. When p∈(0, 1], w∈ \mathcal A_∞( A), the authors introduce the weighted anisotropic product Hardy space Hpw(\mathbb Rn×\mathbb 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 Hpw(\mathbb Rn×\mathbb 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 \mathcal B , then T uniquely extends to a bounded sublinear operator from Hpw(\mathbb Rn×\mathbb Rm; A) to \mathcal B. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.