Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

✍ Scribed by Marcin Bownik; Baode Li; Dachun Yang; Yuan Zhou

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 577 KB
• Volume: 283
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910078

✦ Synopsis

Abstract

Let A~1~ and A~2~ be expansive dilations, respectively, on ℝ^n^ and ℝ^m^. Let A ≡ (A~1~, A~2~) and 𝒜~p~(A) be the class of product Muckenhoupt weights on ℝ^n^ × ℝ^m^ for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ 𝒜~p~(A), the authors characterize the weighted Lebesgue space L^p^ ~w~(ℝ^n^ × ℝ^m^) 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^p^ ~w~ (ℝ^n^ × ℝ^m^; 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^p^ ~w~ (ℝ^n^×ℝ^m^; 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 H^p^ ~w~ (ℝ^n^ × ℝ^m^; 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)