Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces

Abstract

Given a homeomorphism f : X → Y f:X\to Y between Q Q -dimensional spaces X , Y X,Y , we show that f f satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that f f belongs to the Sobolev class N loc 1 , p ( X ; Y ) {N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y) , where 1 ≤ p ≤ Q 1\le p\le Q , and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors Q Q -regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f ∈ N loc 1 , Q ( X ; Y ) f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion h f {h}_{f} belonging to L ∞ ( X ) {L}^{\infty }\left(X) .