On the uniqueness of the Norton–Sullivan quasiconformal extension

Abstract

We show that the extension map   \(\mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\)   for all \(z=x+iy\in\mathbb{H}\), defined by Norton and Sullivan in 1996, is the only locally linear extension map taking bi-Lipschitz functions on \(\mathbb{R}\) to quasiconformal functions on \(\mathbb{H}\), modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extensions like this one (lying in the orbit of such group action), such as \(f(x)\mapsto f(x)+i(f(x)-f(x-y))\).