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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
About this Title
Óscar Domínguez and Sergey Tikhonov
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1393
ISBNs: 978-1-4704-5538-5 (print); 978-1-4704-7349-5 (online)
DOI: https://doi.org/10.1090/memo/1393
Published electronically: January 3, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Embeddings between Besov, Sobolev and Triebel-Lizorkin spaces with logarithmic smoothness
- 4. Characterizations and embedding theorems for general monotone functions
- 5. Characterizations and embedding theorems for lacunary Fourier series
- 6. Optimality of Propositions 1.2 and 1.3
- 7. Optimality of embeddings between Sobolev and Besov spaces with smoothness close to zero
- 8. Comparison between different kinds of smoothness spaces involving only logarithmic smoothness
- 9. Optimality of embeddings between Besov spaces
- 10. Various characterizations of Besov spaces
- 11. Besov and Bianchini norms
- 12. Functions and their derivatives in Besov spaces
- 13. Lifting operators in Besov spaces
- 14. Regularity estimates of the fractional Laplace operator
- A. List of symbols
Abstract
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results:
- Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces;
- Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms;
- Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian.
The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
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