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For $U \subset \mathbb{R}^d$ open, let $\|u\|_{C^m(\bar{U})} = \max_{|\alpha| \leq m} \sup_{x\in U}|D^\alpha u|$ and $C^m(\bar{U}) = \{u \colon U \mapsto \mathbb{R} \mid \|u\|_{C^m(\bar{U})} < \infty\}$.

Also, using $f|_U$ to denote the restriction of $f$ to $U$, define the quotient norm to be $\|u\|_{C^m(\mathbb{R}^d)|_U} = \inf \{\|g\|_{C^m(\mathbb{R}^d)} \mid g|_U = u \}$, and let $C^m(\mathbb{R}^d)|_U = \{u \colon U \mapsto \mathbb{R} \mid \|u\|_{C^m(\mathbb{R}^d)|_U} < \infty \}$.

Are $C^m(\bar{U})$ and $C^m(\mathbb{R}^d)|_U$ norm equivalent? There's a result by Charles Fefferman 2007 that shows that for any $E \subset \mathbb{R}^d$ there is a bounded linear map from $C^m(\mathbb{R}^d)|_E$ to $C^m(\mathbb{R}^d)$, and I've seen it suggested that this suffices, but I don't quite see it myself.

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No, these spaces are different (and by general functional analysis, the smaller one is much smaller). The difference is that elements of the restriction space $C^m(\mathbb R^d)|_U$ have to satisfy a local Lipschitz condition implied by the mean value theorem on lines connecting two points $x,y\in U$ which does not follow from the differentiability only on $\overline U$ because smooth curves in $U$ connecting the points are much longer than the euclidean distance. Typical examples are inward directed cusps like $$U=\mathbb R^2\setminus \{(x,y): x\ge 0, 0\le y\le e^{-1/x}\}.$$ This question was thoroughly investigated by Whitney in the 30's and by many others later on. A related problem with yet another seemingly natural definition was studied recently here: https://link.springer.com/article/10.1007/s00025-020-01303-3

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  • $\begingroup$ Thanks, yes of course the nature of the boundary will matter a great deal. Retrospectively I've noticed that "extension theorem" in the context of Sobolev spaces per textbooks typically refers to bounded linear extensions from $W^{k,p}(U) \mapsto W^{k,p}(\mathbb{R}^d)$, rather than extensions from restricted spaces as in Fefferman's papers. $\endgroup$
    – Oxonon
    Commented May 28, 2021 at 5:46

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