Variable Hardy Spaces

By D. Cruz-Uribe, Sfo, L. Wang. 2012

212 citations recorded by the source corpus.

We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on variable Hardy spaces with minimal regularity assumptions on the exponent function.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/bc9ff6be50174ea35d90c5bc7707e3f57d158e9c

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Ciqiang Zhuo
4. Jun Liu
5. Yiyu Liang
6. Yong Jiao
7. E. Nakai
8. Ferenc Weisz

Citations in the corpus

• Variable weak Hardy spaces and their applications (2016, methodology): “On the other hand, based on the variable Lebesgue space, several variable function spaces are developed rapidly in recent years (see, for example, [3, 4, 12, 14, 32, 41, 44, 45]).”
• Extrapolation and weighted norm inequalities in the variable Lebesgue spaces (2014, background): “Vector-valued inequalities for the maximal operator play an important role in studying functions spaces in the variable exponent setting; see, for example, [22,26].”
• Martingale Hardy spaces with variable exponents (2014, methodology): “p(·). For p(·) ∈ P and p− > 1, we define the conjugate exponent p′(·) by the equation 1 p(x) + 1 p′(x) = 1. We collect some useful lemmas as follows, which will be used in the paper. Lemma 2.1 (see [7], page 5) Let p(·) ∈ P, and p− ≥ 1 then for all r > 0, we have k|f|rk p(·) = kfk r rp(·). Lemm”
• Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents (2014, methodology): “3 (i) Independently, Cruz-Uribe and Wang in [7] introduced the variable exponent Hardy space, denoted by H̃ p(·), in the following way: Let p(·) ∈ P(Rn) satisfy that there exist p0 ∈ (0, p−) and a positive constant C , only depending on n, p(·) and p0, such that ‖M( f )‖L p(·)/p0 (Rn) ≤ C‖ f ‖L p(·)”
• Variable Anisotropic Hardy Spaces and Their Applications (2017, background): “[19] D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math.”
• Characterizations of variable exponent Hardy spaces via Riesz transforms (2015, methodology): “1 to the variable exponent Hardy space Hp(·)(Rn) which was first studied by Nakai and Sawano [14] and, independently, by Cruz-Uribe and Wang [5].”
• Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation (2019, unknown): “7 (see, for instance, [23, 24]) and Assumption 2.”
• Variable Hardy–Lorentz spaces Hp(·),q(Rn) (2018, background): “It should be mentioned that Cruz-Uribe and Wang [18] independently introduced the variable Hardy space 𝐻𝑝(⋅)(ℝ𝑛), investigated its atomic decomposition and discussed the boundedness of operators on it with the variable exponents 𝑝(⋅) satisfying some condition slightly weaker than that used in [4”
• Commutators of multilinear Calderón-Zygmund operators with kernels of Dini's type and applications (2019, unknown): “( [5]) Given p(·) ∈ P0 , then for all s > 0 , we have ∥∥| f |s∥∥Lp(·)(Rn) = ‖ f‖Lsp(·)(Rn).”
• New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators (2011, background): “ak-Orlicz functions, BMOmultipliers, Hardy spaces, atomic decompositions, Hardy-Orlicz spaces, quasi-Banach spaces, sublinear operators. 1 NEW HARDY SPACES OF MUSIELAK-ORLICZ TYPE 2 see, for example, [17, 16, 47]. A key feature of the classical Hardy spaces is their atomic decomposition characteriza”