Quasiconformal mappings and H\"older continuity

We establish that every $K$-quasiconformal mapping $w$ of the unit ball $\IB$ onto a $C^2$-Jordan domain $\Omega$ is H\"older continuous with constant $\alpha= 2-\frac{n}{p}$, provided that its weak Laplacean $\Delta w$ is in $ L^p(\IB)$ for some $n/2<p<n$. In particular it is H\"older continuous for every $0<\alpha<1$ provided that $\Delta w\in L^n(\IB)$.