Radial extensions in fractional Sobolev spaces

TL;DR

This paper corrects a previous proof and extends results on the boundedness of radial extension operators in fractional Sobolev spaces, including higher order derivatives and more general operators.

Contribution

It provides a correct proof of a key boundedness result for radial extensions in fractional Sobolev spaces and generalizes it to higher order derivatives and broader classes of operators.

Findings

Corrected proof of the boundedness of radial extension operators

Extended results to higher order derivatives

Established boundedness for more general radial operators

Abstract

Given $f:\partial (-1,1)^n\to{\mathbb R}$, consider its radial extension $Tf(X):=f(X/\|X\|_{\infty})$, $\forall\, X\in [-1,1]^n\setminus\{0\}$. In "On some questions of topology for $S^1$-valued fractional Sobolev spaces" (RACSAM 2001), the first two authors (HB and PM) stated the following auxiliary result (Lemma D.1). If $0<s<1$, $1< p<\infty$ and $n\ge 2$ are such that $1<sp<n$, then $f\mapsto Tf$ is a bounded linear operator from $W^{s,p}(\partial (-1,1)^n)$ into $W^{s,p}((-1,1)^n)$. The proof of this result contained a flaw detected by the third author (IS). We present a correct proof. We also establish a variant of this result involving higher order derivatives and more general radial extension operators. More specifically, let $B$ be the unit ball for the standard Euclidean norm $|\ |$ in ${\mathbb R}^n$, and set $U_af(X):=|X|^a\, f(X/|X|)$, $\forall\, X\in \overline B\setminus\{0\}$, ${\forall\,} f:\partial B\to{\mathbb R}$. Let $a\in{\mathbb R}$, $s>0$, $1\le p<\infty$ and $n\ge 2$ be such that $(s-a)p<n$. Then $f\mapsto U_af$ is a bounded linear operator from $W^{s,p}(\partial B)$ into $W^{s,p}(B)$.