On function spaces for radial functions

TL;DR

This paper develops a framework for function spaces of radial functions, including classical, Sobolev, distribution, and Hankel spaces, and applies it to boundary-value problems in cylindrical domains.

Contribution

It introduces a systematic construction of function spaces for radial functions and extends the theory to distributions and Hankel spaces, with applications to PDE boundary problems.

Findings

Constructed classical and Sobolev spaces for radial functions from standard spaces.

Developed a theory of radial distributions.

Introduced Hankel spaces as radial counterparts of Bessel-potential spaces.

Abstract

This paper is concerned with complex Banach-space valued functions of the form $$ \hat{f}_k(r\cos\theta,r\sin\theta,z)=\mathrm{e}^{\mathrm{i} k \theta}f_k(r,z), \qquad r \in [0,\infty), \theta \in \mathbb{T}^1, z \in \mathbb{R}, $$ 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.