Radial extensions in fractional Sobolev spaces

Given f : partial derivative (-1, 1)(n) -> R, consider its radial extension Tf(X) := f (X / parallel to X parallel to infinity), for all X is an element of [-1, 1](n)backslash {0}. Brezis and Mironescu (RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 95: 121-143, 2001), stated the following auxiliary result (Lemma D.1). If 0 < s < 1, 1 < p < infinity and n >= 2 are such 1 < sp < n, then f vertical bar -> Tf is a bounded linear operator from W-s,W-p (partial derivative(-1, 1)(n)) into W-s,W-p ((-1,1)(n)). The proof of this result contained a flaw detected by Shafrir. 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 vertical bar vertical bar in R-n, and set U(a)f(X) := vertical bar X vertical bar(a) f (X/vertical bar X vertical bar), for all X is an element of (B) over bar backslash{0}, for all f : partial derivative B -> R. Let a is an element of R, s > 0, 1 <= p < infinity and n >= 2 be such that (s - a) p < n. Then F vertical bar -> U(a)f is a bounded linear operator from W-s,W-p (partial derivative B) into W-s,W-p (B).