Homogeneous Besov spaces

Abstract

This note is based on a series of lectures delivered at Kyoto University in 2015. This note surveys the homogeneous Besov space ${\dot{B}}_{pq}^s$ on ${\mathbb{R}}^n$ with $1\le p$, $q\le \infty$ and $s\in \mathbb{R}$ in a rather self-contained manner. Among other results, we show that $\mathcal{S}{\text{'}}_{\infty }$ and $\mathcal{S}\text{'}/\mathcal{P}$ are isomorphic, and we also discuss the realizations in ${\dot{B}}_{pq}^s$. The fact that $\mathcal{S}{\text{'}}_{\infty }$ and $\mathcal{S}\text{'}/\mathcal{P}$ are isomorphic can be found in textbooks. The realization of ${\dot{B}}_{pq}^s$ can be found in works by Bahouri, Chemin, and Danchin and by Bourdaud for example. Here, we prove these facts using fundamental results in functional analysis such as the Hahn–Banach extension theorem.