Homogeneous Besov spaces

TL;DR

This paper provides a comprehensive survey of homogeneous Besov spaces on Euclidean spaces, detailing their fundamental properties and discussing potential extensions of these function spaces.

Contribution

It offers a self-contained overview of the properties of homogeneous Besov spaces and related function spaces, including extensions and nonhomogeneous counterparts.

Findings

Fundamental properties of $ ext{Homogeneous Besov spaces}$ are established.

Extensions to related function spaces are briefly discussed.

The survey covers both homogeneous and nonhomogeneous Besov and Triebel-Lizorkin spaces.

Abstract

This note is based on a series of lectures delivered in Kyoto University. This note surveys the homogeneous Besov space $\dot{B}^s_{pq}$ on ${\mathbb R}^n$ with $1 \le p,q \le \infty$ and $s \in {\mathbb R}$ in a rather self-contained manner. Possible extensions of this type of function spaces are breifly discussed in the end of this article. In particular, the fundamental properties are stated for the spaces $\dot{B}^s_{pq}$ with $0<p,q \le \infty$ and $s \in {\mathbb R}$ and $\dot{F}^s_{pq}$ with $0<p<\infty$, $0<q \le \infty$ and $s \in {\mathbb R}$ as well as nonhomogeneous coupterparts $B^s_{pq}$ with $0<p,q \le \infty$ and $s \in {\mathbb R}$ and $F^s_{pq}$ with $0<p<\infty$, $0<q \le \infty$ and $s \in {\mathbb R}$.