On function spaces for radial functions

Abstract

This paper is concerned with complex Banach-space valued functions of the form $${\hat{f}}_k(r\cos \theta ,r\sin \theta ,z)={\mathrm{e}}^{\mathrm{i}k\theta }{f}_k(r,z),r\in [0,\infty ),\theta \in {\mathbb{T}}^1,z\in \mathbb{R},$$ for some $k\in \mathbb{Z}$. It is demonstrated how classical and Sobolev spaces for the radial function $f_k$ can be constructed in a natural fashion from the corresponding standard function spaces for ${\hat{f}}_k$. A theory of radial distributions is derived in the same spirit. Finally, a new class of \textit{Hankel spaces} for the case $f_k=f_k(r)$ is introduced. These spaces are the radial counterparts of the familiar Bessel-potential spaces for functions defined on ${\mathbb{R}}^d$. The paper concludes with an application of the theory to the Dirichlet boundary-value problem for Poisson's equation in a cylindrical domain.

47 pages, 2 figures