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Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$,we show that $f$ satisfying the metric definition of quasiconformality outsidesuitable exceptional sets implies that $f$ belongs to the Sobolev class$N_{\rm{loc}}^{1,p}(X;Y)$, where $1< p\le Q$, and also implies one direction ofthe geometric definition of quasiconformality. Unlike previous results, we onlyassume a pointwise version of Ahlfors $Q$-regularity, which particularlyenables various weighted spaces to be included in the theory. Unexpectedly, wecan apply this to obtain results that are new even in the classical Euclideansetting. In particular, in spaces including the Carnot groups, we are able toprove the Sobolev regularity $f\in N_{\rm{loc}}^{1,Q}(X;Y)$ without the strongassumption of the infinitesimal distortion $h_f$ belonging to $L^{\infty}(X)$.

Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces