Besov Spaces, Triebel–Lizorkin Spaces and Modulation Spaces

  • Yoshihiro Sawano
Part of the Developments in Mathematics book series (DEVM, volume 56)


Having set down elementary facts in the previous chapter, we take a detailed look at Besov spaces and Triebel–Lizorkin spaces, which are the main theme of this book. Chapter 2 is devoted to the introduction of elementary definitions together with some fundamental properties. First we define the Besov space \(B^s_{p q}({\mathbb R}^n)\) with 1 ≤ p, q ≤ and \(s \in {\mathbb R}\) in the spirit of Peetre, although Besov introduced Besov spaces in [171, 172]. After the Besov space \(B^s_{p q}({\mathbb R}^n)\) for such a restricted case we define \(A^s_{p q}({\mathbb R}^n)\), which unifies the Besov space \(B^s_{p q}({\mathbb R}^n)\) with 0 < p, q ≤ and \(s \in {\mathbb R}\) and the Triebel–Lizorkin space \(F^s_{p q}({\mathbb R}^n)\) with 0 < p < , 0 < q ≤ and \(s \in {\mathbb R}\).


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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Yoshihiro Sawano
    • 1
  1. 1.Department of Mathematics and Information ScienceTokyo Metropolitan UniversityTokyoJapan

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