Radial extensions in fractional Sobolev spaces

Given f: ∂(- 1 , 1) ⁿ→ R, consider its radial extension Tf(X) : = f(X/ ‖ X‖ _∞) , ∀X∈[-1,1]n\{0}. Brezis and Mironescu (RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95:121–143, 2001), stated the following auxiliary result (Lemma D.1). If 0 < s< 1 , 1 < p< ∞ and n≥ 2 are such that 1 < sp< n, then f↦ Tf is a bounded linear operator from W^s,p(∂(- 1 , 1) ⁿ) into W^s,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 || in Rⁿ, and set Uaf(X):=|X|af(X/|X|), ∀X∈B¯\{0}, ∀f:∂B→R. Let a∈ R, s> 0 , 1 ≤ p< ∞ and n≥ 2 be such that (s- a) p< n. Then f↦ U_af is a bounded linear operator from W^s,p(∂B) into W^s,p(B).