Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces

TL;DR

This paper establishes that metric quasiconformal homeomorphisms between certain non-Ahlfors regular spaces belong to Sobolev spaces, extending classical results and including new cases like Carnot groups without requiring bounded distortion.

Contribution

It proves Sobolev regularity of quasiconformal maps under weaker assumptions, notably pointwise Ahlfors regularity, broadening applicability to weighted and non-Ahlfors regular spaces.

Findings

Sobolev regularity $f otin L^{ty}$ in Carnot groups

Extension of quasiconformal Sobolev regularity to non-Ahlfors regular spaces

Application to classical Euclidean and weighted spaces

Abstract

Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable 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 of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors $Q$-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the classical Euclidean setting. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity $f\in N_{\rm{loc}}^{1,Q}(X;Y)$ without the strong assumption of the infinitesimal distortion $h_f$ belonging to $L^{\infty}(X)$.