Geodesic
Quasiconformal mappings and Sobolev spaces
Studia Mathematica, Tome 131 (1998) no. 1, pp. 1-17

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DOI

We examine how Poincaré change under quasiconformal maps between appropriate metric spaces having the same Hausdorff dimension. We also show that for many metric spaces the Sobolev functions can be identified with functions satisfying Poincaré, and this allows us to extend to the metric space setting the fact that quasiconformal maps from $ℝ^Q$ onto $ℝ^Q$ preserve the Sobolev space $L^{1,Q}(ℝ^Q)$. 
Pekka Koskela;  . Quasiconformal mappings and Sobolev spaces. Studia Mathematica, Tome 131 (1998) no. 1, pp. 1-17. doi: 10.4064/sm-131-1-1-17
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