# $C^m(\mathbb{R}^d)|_U$ versus $C^m(\bar{U})$ for $U \subset \mathbb{R}^d$ -- restriction versus definition on subset

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.

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
• 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. Commented May 28, 2021 at 5:46