Linear extension operators for smooth functions: from compact sets to compact sets

Question

I'm considering a situation where I have the linear restriction map of Fréchet spaces $$ C^\infty(C_1) \to C^\infty(C_2) $$ where $C_2 \hookrightarrow C_1$ are a pair of compact, connected subsets of $\mathbb{R}^n$ homeomorphic to closed balls, and interiors diffeomorphic to open balls. I believe I can assume that $C_2$ is a manifold with at most codimension 3 corners and $C_2$ with at most codimension 2 corners.

What I'm interested in is whether this has a (Edit: continuous!) linear section.

The case of $n=1$, restriction along $[a,b] \hookrightarrow [c,d]$, I believe I have the requisite understanding to extract as a corollary from a theorem of Seeley (use $n=0$ in the result that the restriction $C^\infty(\mathbb{R}^{n+1}) \to C^\infty(\mathbb{R}^n\times\mathbb{R}_{\geq 0})$ has a linear section, using the usual Fréchet topologies -- thanks to Andrew Stacey for pointing this out), but I don't know how one would go about the more general case.

In looking around I find a lot of work by Fefferman on the case of $C^k$ maps, and also a lot of work by people considering general extension problems for inclusions $A \hookrightarrow \mathbb{R}^n$ and arbitrary functions $A \to \mathbb{R}$ for all different sorts of subsets $A$, including very diverse examples.

Nothing I've found though seems to be the sort of thing I'd need, but that may be my unfamiliarity with this sort of analysis. Ideally, the necessary result is right under my nose, and it just needs someone to say "oh, that clearly follows from so-and-so's theorem".

Answer 1

In general, there are several candidates for the definition of $C^\infty(K)$: One is the space $\lbrace f|_K: f\in C^\infty(\mathbb R^n)\rbrace$ of all restrictions (endowed with the quotient topology), another is the intersection $\bigcap\limits_{k\in\mathbb N_0} \lbrace f|_K: f\in C^k(\mathbb R^n)\rbrace$ (which is equal to the former for $n=1$ due to Merrien but different in general — an elementary example is in Wieslaw Pawlucki, On the algebra of functions $\scr C^k$-extendable for each $k$ finite, Proc. Amer. Math. Soc. 133 (2005), no. 2, 481–484), and finally the probably best understood definition is that of Whitney jets, i.e. families $(f^{(\alpha)})_{\alpha \in \mathbb N_0^d}$ of continuous functions which satisfy the correct Taylor approximations on $K$ as if $f^{(\alpha)}=\partial^{\alpha} f$ for some $f\in C^\infty(\mathbb R^n)$.

If $K$ is the closure of its interior the definitions coincide and you should consult the literature about extension of Whitney jets. The article Leonhard Frerick, Extension operators for spaces of infinite differentiable Whitney jets, J. Reine Angew. Math. 602 (2007), 123–154, contains a lot of information. As mentioned by Deane Yang. Lipschitz boundary is enough for having a continuous linear extension operator (this is due to E.M. Stein). However, a sharp cusp like $K=\lbrace (x,y)\in [0,1]^2: y\le \exp(-1/x)\rbrace$ does not have such an extension. For general $K$ and the space of all restrictions, the question is wide open, besides the examples of Fefferman and Ricci mentioned by David Roberts there are some results of Dietmar Vogt, Restriction spaces of $A^\infty$, Rev. Mat. Iberoam. 30 (2014), no. 1, 65–78.