Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces

Abstract

Given a homeomorphism f:X→Yf:X\to Y between QQ-dimensional spaces X,YX,Y, we show that ff satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that ff belongs to the Sobolev class Nloc1,p(X;Y){N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y), where 1≤p≤Q1\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 QQ-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∈Nloc1,Q(X;Y)f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion hf{h}_{f} belonging to L∞(X){L}^{\infty }\left(X).

Keywords

• quasiconformal mapping
• newton-sobolev mapping
• modulus of a curve family
• absolute continuity
• ahlfors regularity
• weighted space
• 30l10
• 30c65
• 46e36