Arc-smooth functions and cuspidality of sets

TL;DR

This paper investigates how the boundary shape of sets influences the regularity of arc-smooth functions, extending classical results to more complex sets within o-minimal structures.

Contribution

It establishes a precise link between set cuspidality and the loss of regularity in arc-smooth functions, extending previous results to polynomially bounded o-minimal structures.

Findings

Arc-smooth functions are smooth on open sets, extended to certain tame closed sets.

Cuspidality of a set's boundary affects the derivatives needed to determine function regularity.

Flatness of functions along curves influences their flatness on the set.

Abstract

A function $f$ is arc-smooth if the composite $f\circ c$ with every smooth curve $c$ in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem of Boman. Recently, we extended this result to certain tame closed sets (namely, H\"older sets and simple fat subanalytic sets). In this paper we link, in a precise way, the cuspidality of the (boundary of the) set to the loss of regularity, i.e., how many derivatives of $f\circ c$ are needed in order to determine the derivatives of $f$. We also discuss how flatness of $f \circ c$ affects flatness of $f$. Besides H\"older sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.