Quasiconformal mappings with controlled Laplacian and Hölder continuity

Abstract

Abstract. We prove that every K-quasiconformal mapping w of the unit ball B ⊂ R, n ≥ 2 onto a C-Jordan domain Ω is Hölder continuous with constant α = 2 − n p , provided its weak Laplacian ∆w is in L(B) for some n/2 < p < n. In particular it is Hölder continuous for every 0 < α < 1 provided that ∆w ∈ L(B). Finally for p > n, we prove that w is Lipschitz continuous, a result, whose proof has been already sketched in [16] by the first author and Saksman. The paper contains the proofs of some results announced in [17].