Quasiconformal maps with controlled Laplacian

We establish that every K-quasiconformal mapping w of the unit disk D onto a C-2-Jordan domain is Lipschitz provided that w L-p(D) for some p > 2. We also prove that if in this situation K 1 with ||w||L-p(D) 0, and D in C-1,C--sense with > 1/2, then the bound for the Lipschitz constant tends to 1. In addition, we provide a quasiconformal analogue of the Smirnov theorem on absolute continuity over the boundary.Peer reviewe