Homogeneous Besov spaces

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 $s \in {\mathbb R}$.