Lipschitz-type spaces and harmonic mappings in the space

Abstract

We obtain a sharp estimate of the derivatives of harmonic quasiconformal extension u = P[ø] of a Lipschitz map ø: Sn-1 → Rn. 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.