Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces

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.