Arc-smooth functions and cuspidality of sets

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, Hölder 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 Hölder sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.

31 pages, 3 figures; minor corrections and additions; 32 pages, final version, accepted for publication in Journal d'Analyse Mathématique