There are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the W m,p -extension domains for 1 < p < ∞. As a corollary we prove that the property of being a W 1,p -extension domain, 1 < p ∞, is invariant under bi-Lipschitz mappings, Theorem 8.

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.

Abstract. The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.

Abstract. We show that a domain is an extension domain for a Haj lasz-Besov or for a Haj lasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case 0 < p < 1. The necessity of the measure density condition is derived from embedding theorems; in the case of Haj lasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj lasz-Besov spaces are intermediate spaces between L p and Haj laszSobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces B s p,q , 0 < s < 1, 0 < p < ∞, 0 < q ≤ ∞, defined via the L p -modulus of smoothness of a function.

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