We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.
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Piotr Hajłasz, Pekka Koskela, Heli Tuominen, Measure density and extendability of Sobolev functions. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 645–669DOI 10.4171/RMI/551