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I've been trying to prove the following assertion:

Assume that $\Omega\in C^{0,1}(\mathbb{R}^d)$. Prove that $W^{d,1}(\Omega)\hookrightarrow C(\overline{\Omega}).$

My approach:

I have proven that $W^{d,1}(\mathbb{R}^d)\hookrightarrow C_b(\mathbb{R}^d)$. Then I was adviced to use this result to prove the assertion. However, I can't figure out how. If I had an extention operator from $W^{d,1}(\Omega)$ to $W^{d,1}(\mathbb{R}^d)$, the rest would be simple. I don't, however, have such an operator, do I? As $\Omega\in C^{0,1}(\Omega)$, I know there is an extension operator from $W^{1,1}(\Omega)$ to $W^{1,1}(\mathbb{R}^d)$ but that doesn't seem to be enough.

I'm rather new to Sobolev spaces so I might be missing something elementary. Still, I'd be really grateful for any advice.

Thank you

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  • $\begingroup$ The Morrey-Sobolev embedding applies to Lipschitz domains. $\endgroup$ – user147263 Nov 15 '15 at 7:39
  • $\begingroup$ I have never heard of it. Could you provide me with some source(s) please? Thank you $\endgroup$ – user1321324 Nov 15 '15 at 11:21
  • $\begingroup$ I recommend Google; it's a popular search engine. $\endgroup$ – user147263 Nov 15 '15 at 15:56
  • $\begingroup$ I did use Google of course. Still, I didn't find anything that would seem to help. I'll try it again, though. $\endgroup$ – user1321324 Nov 15 '15 at 16:16

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