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Weighted anisotropic product Hardy spaces and boundedness of sublinear operators
Author(s) -
Bownik Marcin,
Li Baode,
Yang Dachun,
Zhou Yuan
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910078
Subject(s) - sublinear function , mathematics , hardy space , bounded function , lp space , norm (philosophy) , product (mathematics) , maximal function , standard probability space , subspace topology , operator (biology) , combinatorics , banach space , lebesgue integration , pure mathematics , mathematical analysis , geometry , biochemistry , chemistry , repressor , political science , law , transcription factor , gene
Let A 1 and A 2 be expansive dilations, respectively, on ℝ n and ℝ m . Let A ≡ ( A 1 , A 2 ) and p ( A ) be the class of product Muckenhoupt weights on ℝ n × ℝ m for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ p ( A ), the authors characterize the weighted Lebesgue space L pw (ℝ n × ℝ m ) via the anisotropic Lusin‐area function associated with A . When p ∈ (0, 1], w ∈ ∞ ( 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 ℬ, then T uniquely extends to a bounded sublinear operator from H p w (ℝ n × ℝ m ; 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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)