Local Hardy spaces of Musielak-Orlicz type and their applications

By Dachun Yang, Sibei Yang. 2011

85 citations recorded by the source corpus.

Let φ: ℝn × [0,∞) → [0,∞) be a function such that φ(x, ·) is an Orlicz function and $$\phi ( \cdot ,t) \in \mathbb{A}_\infty ^{loc} \left( {\mathbb{R}^n } \right)$$ (the class of local weights introduced by Rychkov). In this paper, the authors introduce a local Musielak-Orlicz Hardy space hφ(ℝn) by the local grand maximal function, and a local BMO-type space bmoφ(ℝn) which is further proved to be the dual space of hφ(ℝn). As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmoφ(ℝn), characterized by Nakai and Yabuta, is just the dual of $$L^1 \left( {\mathbb{R}^n } \right) + h_{\Phi _0 } \left( {\mathbb{R}^n } \right)$$, where ϕ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ0 a Musielak-Orlicz function induced by ϕ. Characterizations of hφ(ℝn), including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hφ(ℝn), from which, the authors further deduce some criterions for the boundedness on hφ(ℝn) of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on hφ(ℝn).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4072ad4323d4865b3751bbe42bc917ae52d2485c

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In Applied Mathematics

1. Dachun Yang
2. Yiyu Liang
3. Sibei Yang
4. Luong Dang Ky
5. Wen Yuan
6. Jizheng Huang
7. The Anh Bui
8. Jun Cao

In Mathematical Physics

1. Dachun Yang
2. Luong Dang Ky
3. Sibei Yang
4. Yiyu Liang
5. The Anh Bui
6. Wen Yuan
7. Jizheng Huang
8. Jun Cao

Citations in the corpus

• Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications (2012, background): “ates for the div-curl lemma and the commutators of singular integral operators (see [3, 5, 28]). Moreover, the local Musielak-Orlicz Hardy space, hϕ(Rn), and its dual space, bmoϕ(Rn), were studied in [48] and some applications of hϕ(Rn) and bmoϕ(Rn), to pointwise multipliers of BMO-type spaces and t”
• New real-variable characterizations of Musielak-Orlicz Hardy spaces (2012, background): “We remark that the Littlewood-Paley function characterizations of Hφ(Rn) have local variants, which will be studied in a forthcoming paper; see [37] for the definition of local Musielak-Orlicz Hardy spaces hφ(Rn).”
• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates (2013, background): “Moreover, the local Hardy space of MusielakOrlicz type was studied in [73].”
• New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators (2011, background): “Recently, new Hardy spaces of Musielak–Orlicz type have developed in many directions (see [11,47,50,56,57]), some their characterizations and applications were also established and studied in [11,27,38,50].”