Boundary differentiability of Douady-Earle extensions of diffeomorphisms of Sⁿ

Let f be a homeomorphism of the unit sphere Sⁿ and Φ(f) be the Douady-Earle extension of f to the closed unit ball Bⁿ⁺¹ bounded by Sⁿ. We show that if f is C¹ in a neighborhood of a point p ∈ Sⁿ and $D f(p)$ is non-singular, then Φ(f) is C¹ in a neighborhood of p in Bⁿ⁺¹. Moreover, when $n=1$, if f is orientation-preserving, then the complex derivative of Φ(f) at p is equal to f^(p), and ∂/∂ z Φ(f)(z) and ∂∂ z̄ Φ(f)(z) converge to f^(p) and 0 , respectively, as z → p; if f is orientation-reversing, then the anticomplex derivative of Φ(f) at p is equal to f^(p), and ∂/∂ z Φ(f)(z) and ∂/∂ z Φ(f)(z) converge to 0 and f^(p), respectively, as z → p.