Bilipschitz extensions of maps having quasiconformal extensions (Q789567)

We consider bilipschitz maps \(f: A\to \mathbb R^ n\), \(X\subset \mathbb R^ n\). We show that if \(n\neq 4\) and if \(f\) has a quasiconformal extension to \(\mathbb R^ n\), then \(f\) has also a bilipschitz extension to \(\mathbb R^ n\). Thus, for example, \(f\) has such an extension whenever \(X\) and \(fX\) are quasiconformal spheres. An application to the flattening theory of Lipschitz manifolds is given.