Arc-smooth functions on closed sets

TL;DR

This paper extends Boman's theorem to closed sets with specific boundary properties, showing arc-smooth functions can be extended to smooth or holomorphic functions under certain conditions.

Contribution

It generalizes the characterization of smooth functions via arc-smoothness to fat closed sets with boundary conditions and explores extensions to holomorphic and Denjoy-Carleman classes.

Findings

Arc-smooth functions on certain closed sets extend to smooth functions.

Extension of arc-smooth functions to holomorphic functions on neighborhoods.

Results apply to subanalytic sets and sets with H"older boundary.

Abstract

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb R^d$ is smooth ($\mathcal C^\infty$) if and only if it is arc-smooth, i.e., $f\circ c$ is smooth for every smooth curve $c : \mathbb R \to U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman's theorem on fat closed sets with H\"older boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X \subseteq \mathbb R^d$ is any such set and $f : X \to \mathbb R$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb R^d$. We also get a version of the Bochnak-Siciak theorem on all closed fat subanalytic and all closed sets with H\"older boundary: if $f : X \to \mathbb R$ is the restriction of a smooth function on $\mathbb R^d$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb C^d$. Similar results hold for non-quasianalytic Denjoy-Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.