An inverse Sobolev lemma

Abstract

We establish an inverse Sobolev lemma for quasiconformal mappings and extend a weaker version of the Sobolev lemma for quasiconformal mappings of the unit ball of ${\mathbb{R}}^n$ to the full range $0<p<n$. As an application we obtain sharp integrability theorems for the derivative of a quasiconformal mapping of the unit ball of ${\mathbb{R}}^n$ in terms of the growth of the mapping.