© 2018

Theory of Besov Spaces


  • Comprises a self-contained description of all contents, accessible to readers familiar with integration theory

  • Provides the shortest available introduction to the theory of Besov spaces, beginning with Chapter 1

  • Covers and summarizes two volumes by Hans Triebel, Theory of Function Spaces I, II


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

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Yoshihiro Sawano
    Pages 1-204
  3. Yoshihiro Sawano
    Pages 321-428
  4. Yoshihiro Sawano
    Pages 709-889
  5. Back Matter
    Pages 891-945

About this book


This is a self-contained textbook of the theory of Besov spaces and Triebel–Lizorkin spaces oriented toward applications to partial differential equations and problems of harmonic analysis. These include a priori estimates of elliptic differential equations, the T1 theorem, pseudo-differential operators, the generator of semi-group and spaces on domains, and the Kato problem.  Various function spaces are introduced to overcome the shortcomings of Besov spaces and Triebel–Lizorkin spaces as well. The only prior knowledge required of readers is familiarity with integration theory and some elementary functional analysis.
Illustrations are included to show the complicated way in which spaces are defined. Owing to that complexity, many definitions are required. The necessary terminology is provided at the outset, and the theory of distributions, Lp spaces, the Hardy–Littlewood maximal operator, and the singular integral operators are called upon. One of the highlights is that the proof of the Sobolev embedding theorem is extremely simple. There are two types for each function space: a homogeneous one and an inhomogeneous one. The theory of function spaces, which readers usually learn in a standard course, can be readily applied to the inhomogeneous one. However, that theory is not sufficient for a homogeneous space; it needs to be reinforced with some knowledge of the theory of distributions. This topic, however subtle, is also covered within this volume. Additionally, related function spaces—Hardy spaces, bounded mean oscillation spaces, and Hölder continuous spaces—are defined and discussed, and it is shown that they are special cases of Besov spaces and Triebel–Lizorkin spaces.


Harmonic analysis Function spaces Differential equations Besov spaces Triebel-Lizorkin spaces

Authors and affiliations

  1. 1.Department of Mathematics and Information ScienceTokyo Metropolitan UniversityTokyoJapan

About the authors

Yoshihiro Sawano is a professor in Tokyo Metropolitan University.

Bibliographic information


“This monograph is an impressive piece of work dealing with a complicated and deep subject, Besov spaces, which surely enjoys less popularity than it deserves. … the monograph can really be recommended as a very good, elaborate and up-to-date compendium on Besov spaces, in particular for scientists who have some experience and sound knowledge working in the field of function spaces.” (Dorothee D. Haroske, Mathematical Reviews, June, 2020)

“This voluminous book provides an exhaustive and self-contained treatment of several spaces related to Besov spaces. The useful applications of Besov spaces and Triebel-Lizorkin spaces to partial differential equations allow the reader to examine in detail many properties of the solutions of the equations.” (Maria Alessandra Ragusa, zbMATH 1414.46004, 2019)