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