Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

By Marcin Bownik, Baode Li, Dachun Yang, Yuan Zhou. 2009

78 citations recorded by the source corpus.

Let 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)

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4f50a59d32d0c713d5acf68dbaef71cd49faf0b2

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Jun Liu
4. Long Huang
5. Ferenc Weisz
6. Yangyang Zhang
7. Yoshihiro Sawano
8. Songbai Wang

Citations in the corpus

• Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications (2018, methodology): “For more progresses about the theory of anisotropic function spaces on R, we refer the reader to [11, 25, 45, 48, 49, 51, 53, 54, 55] for anisotropic Hardy-type spaces and to [7, 8, 9, 46, 47, 56] for anisotropic Besov and TriebelLizorkin spaces.”
• Summability of Multi-Dimensional Trigonometric Fourier Series (2012, background): “Unfortunately, for a linear operator V , the uniform boundedness of the Lp0-norms of V a is not enough for the boundedness V : H p0(T ) → Lp0(T) (see [13, 64, 65, 14, 70]).”