Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces

TL;DR

This paper establishes that quasiconformal mappings in non-Ahlfors regular metric spaces belong to the Newton-Sobolev class, broadening the scope to include various irregular spaces and revealing new Sobolev mappings even in classical Euclidean contexts.

Contribution

It introduces a novel approach linking quasiconformality to Sobolev regularity under minimal regularity assumptions, extending the theory to non-Ahlfors regular spaces.

Findings

Includes many non-Ahlfors regular spaces like weighted spaces and bowtie.

Detects Sobolev mappings in Euclidean spaces not identified by previous theories.

Shows quasiconformal mappings are in the Newton-Sobolev class under weaker conditions.

Abstract

We show that a mapping $f\colon X\to Y$ satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\textrm{loc}}^{1,1}(X;Y)$. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many non-Ahlfors regular spaces, such as weighted spaces and Fred Gehring's bowtie, to be included in the theory. Unexpectedly, already in the classical setting of unweighted Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.