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Do the Sobolev inequalities as they are stated on Wikipedia also hold on bounded open subsets of $\mathbb R^d$ whose boundary is not $C^1$ but Lipschitz?

If so, what is a good reference for a proof?

Actually, I'm looking for a reference where the inequalities are proved for any bounded, open set which has the "extension property" (see, for example, [Renardy, Definition 7.11]).

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    Check Adam's book (Sobolev Spaces). – 2017-02-03
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    @levap Please take note of my edit. Does the book you mention provide proofs which show that the inequalities hold for any bounded, open set which has the extension property. – 2017-02-03
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    I have no idea what is your "extension property" nor which Sobolev inequalities you care about. The book proves certain Sobolev embeddings assuming the cone condition and certain embeddings assuming the strong Lipschitz condition and have some discussion about when it is possible to weaken the assumptions. You can check it out and see if it gives you what you want (it also discusses extension operations). – 2017-02-03
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    @levap I've added a link. $\Lambda$ has the $k$-extension property, if there is a bounded linear operator $E:H^k(\Lambda)\to H^k(\mathbb R^d)$ with $$\left.Eu\right|_\Lambda=u\;\;\;\text{for all }u\in H^k(\Lambda)\;.$$ – 2017-02-03
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    @levap So, the crucial question is shifted to the exitence of $E$. I only know books which mention that, but can't find a book with rigorous (and elegant) proofs. – 2017-02-03

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Yes, those inequalities still hold. You only need the construction of the extension operator, which can be found, for example, in Stein's Singular Integrals.

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    Do you have a reference where the statement is explicitly made and proved? – 2017-02-03
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    Yes; this is theorem 5 on page 181 in Stein's book. This discusses the "extension property" you mention, for Lipschitz domains. – 2017-02-04
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    Maybe you can take a look at my [related question](http://math.stackexchange.com/questions/2132010/if-there-is-a-bounded-linear-extension-wk-p%ce%9b-to-wk-p%e2%84%9dd-are-we). – 2017-02-06