Radial extensions in fractional Sobolev spaces

Abstract

Given $f:\partial (-1,1{)}^n\to \mathbb{R}$, consider its radial extension $Tf(X):=f(X/\Vert X{\Vert }_{\infty })$, $\forall X\in [-1,1{]}^n\setminus \{0\}$. 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 $U_{a}f(X):=|X{|}^{a}f(X/|X|)$, $\forall X\in \overline{B}\setminus \{0\}$, $\forall f:\partial B\to \mathbb{R}$. Let $a\in \mathbb{R}$, $s>0$, $1\le p<\infty$ and $n\ge 2$ be such that $(s-a)p<n$. Then $f\mapsto U_{a}f$ is a bounded linear operator from $W^{s,p}(\partial B)$ into $W^{s,p}(B)$.