Lipschitz-type spaces and harmonic mappings in the space

Abstract

We obtain a sharp estimate of the derivatives of harmonic quasiconformal extension u = P[phi] of a Lipschitz map phi: Sn-1 -> R-n. We also consider additional conditions which provide that u is Lipschitz on the unit ball; in particular, we give characterizations of Lipschitz continuity of u in the planar case and in the upper half space setting. We also answer a question posed by Martio in [OM] and extend this to the case of several variables.