A characterization of the subspace of radially symmetric functions in Sobolev spaces

TL;DR

This paper characterizes the subspace of radially symmetric functions within Sobolev spaces by establishing an equivalence with weighted Sobolev norms of their radial profiles, resolving an open problem.

Contribution

It provides a complete characterization of radial Sobolev spaces using weighted Sobolev spaces, including applications to corotational maps.

Findings

Equivalence of Sobolev norms and weighted Sobolev norms for radial functions

Complete characterization of radial Sobolev spaces

Application to Sobolev norms of corotational maps

Abstract

In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.