Analytic extension of differentiable functions defined in closed sets by means of continuous linear operators (Q2781348)

Let \(F\) be a closed subset of \( {\mathbb R }^{n} \) and \( {\mathcal E}(F)\) the vector space of Whitney jets on \(F\). In the paper ``Analytic extensions of differentiable functions defined on closed sets'' [Trans. Am. Math. Soc. 36, 63-89 (1934; Zbl 0008.24902)], \textit{H. Whitne}y proved that every Whitney jet in \( {\mathcal E}(F)\) can be extended to a function in \( C^{\infty}({\mathbb R}^{n}) \). With standard locally convex topologies, \( {\mathcal E}(F)\) and \( C^{\infty}({\mathbb R}^{n}) \) are Fréchet locally convex spaces. It is well-known that there are closed subsets \(F \subset {\mathbb R}^{n}\) for which there is no continuous linear extension map from \( {\mathcal E}(F) \to C^{\infty}({\mathbb R}^{n}) \). In the paper ``On the existence of continuous analytic extension maps for Whitney jets'' [Bull. Polish. Acad. Sci. Math. 45, No. 4, 359-367 (1997; Zbl 0902.26014)], \textit{J. Schmets} and \textit{M. Valdivia} proved that if \(F\) is compact and there is a continuous linear extension map from \( {\mathcal E}(F) \to C^{\infty}({\mathbb R}^{n}) \), then there is a continuous extension map \(T: {\mathcal E}(F) \to BC^{\infty}({\mathbb R}^{n})\) such that \( T(\phi) \) is analytic in \( {\mathbb R}^{n} \setminus F \) for every \( \phi \in {\mathcal E}(F)\) (here \(BC^{\infty}({\mathbb R}^{n})\) is the vector space of \( C^{\infty}\)-functions on \( {\mathbb R}^{n}\), the derivatives of which are uniformly bounded, and is endowed with the canonical Fréchet structure). NEWLINENEWLINENEWLINEIn the present paper this result is extended to the case when F is any closed set. The main result is: NEWLINENEWLINENEWLINESuppose \(F\) is closed in \( {\mathbb R}^{n}\) and \( {\mathcal E}(F)\) admits a continuous operator. Then the following are equivalent: NEWLINENEWLINENEWLINE(i) \( {\mathcal E}(F)\) admits a continuous operator \(E\) whose values are real analytic outside \(F\). NEWLINENEWLINENEWLINE(ii) \( {\mathcal E}(F)\) admits a continuous operator\(E\) such that \(E(f)\) has an analytic extension to \( ({\mathbb R}^{n} \setminus F)^{*}\) ( here \( ({\mathbb R}^{n} \setminus F)^{*} = \{ (u+iv): u \in ({\mathbb R}^{n} \setminus F)\), \(|v|< \text{dist}(u,F) \})\). NEWLINENEWLINENEWLINE(iii) For every \( \rho > 0 \), the boundary of the union of those components of the complement of \(F\), which have non-empty intersection with the ball \( B_{\rho}= \{ x \in {\mathbb R}^{n}: |x|\leq \rho \} \) of radius \(\rho\), is bounded.