Function spaces of variable smoothness and integrability

By Lars Diening, P. Hasto, Svetlana Roudenko. 2007

279 citations recorded by the source corpus.

Abstract In this article we introduce Triebel–Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bc57e788eebcff37b2d6db78357f3c1f268bd556

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Ciqiang Zhuo
4. Yoshihiro Sawano
5. Yiyu Liang
6. Jun Liu
7. Kwok-Pun Ho
8. E. Nakai

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Ciqiang Zhuo
4. Yoshihiro Sawano
5. Kwok-Pun Ho
6. Jun Liu
7. Yiyu Liang
8. E. Nakai

Citations in the corpus

• Besov spaces with variable smoothness and integrability (2010, methodology): “We next present some useful lemmas from [15].”
• Littlewood-Paley spaces (2011, background): “) quasi-Banach function spaces [5, 42, 46], the variable Lebesgue spaces [14, 15] and the Morrey type spaces [12, 32, 39, 52, 53].”
• Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces (2012, methodology): “When u ≡ 1, the variable Triebel–Lizorkin–Morrey spaces reduce to the Triebel– Lizorkin spaces F α(·) p(·),q(·) defined in [20].”
• Besov-Type Spaces with Variable Smoothness and Integrability (2015, background): “The concept of function spaces with variable smoothness and variable integrability was firstly mixed up by Diening, Hästö and Roudenko in [16], they introduced Triebel-Lizorkin spaces with variable exponents F s(·) p(·),q(·)(R ) and proved a trace theorem as follows:”
• 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, background): “grals with nonstandard growth conditions. We also note that in the recent years more attention was turned to the study of function spaces with variable exponent in harmonic analysis; see for instance [2, 4, 5, 8, 22, 26, 30]. Let Ω ⊂ Rn. We say that a function p(·) : Ω → Ris locally log-Ho¨lder cont”
• Musielak-Orlicz Campanato Spaces and Applications (2013, background): “The motivation to study function spaces of Musielak-Orlicz type comes from applications to elasticity, fluid dynamics, image processing, nonlinear partial differential equations and the calculus of variation; see, for example, [4, 5, 6, 8, 9, 22] and their references.”
• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates (2013, background): “It is worth pointing out that Musielak-Orlicz functions are the natural generalization of Orlicz functions (see, for example, [28, 29, 49, 59]) and the motivation to study function spaces of Musielak-Orlicz type is attributed to their extensive applications to many fields of mathematics (see, for ex”
• Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications (2012, background): “Recall that Musielak-Orlicz functions are the natural generalization of Orlicz functions that may vary in the spatial variables (see, for example, [17, 18, 34, 41]).”
• New real-variable characterizations of Musielak-Orlicz Hardy spaces (2012, background): “Recall that the motivation to study function spaces of Musielak-Orlicz type comes from applications to elasticity, fluid dynamics, image processing, nonlinear partial differential equations and the calculus of variation; see, for example, [6, 7, 8, 10, 11, 24].”
• A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (2012, unknown): “In recent years, it turned out to be necessary to generalize even further and replace the fundamental space Lp(Rn) by something more general, like a Lebesgue space with variable exponents ([11, 12]) or, more generally, an Orlicz space.”
• A note on the spaces of variable integrability and summability of Almeida and Hästö (2011, background): “Spaces of Triebel-Lizorkin type with variable indices have recently been considered in [2].”