Finely quasiconformal mappings

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}({\mathbb R}^n;{\mathbb R}^n)$ -mappings. Then we show on the plane that this relaxed definition can be used to prove Sobolev regularity, and that these ‘finely quasiconformal’ mappings are in fact quasiconformal.