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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. 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