Finely quasiconformal mappings

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}({\mathbb{R}}^n;{\mathbb{R}}^n)$-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.