Arc-smooth functions on closed sets

Abstract

By an influential theorem of Boman, a function f on an open set U in R d is smooth (C ∞ ) if and only if it is arc-smooth, i.e., f •c is smooth for every smooth curve c : R → 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 Hölder 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 ⊆ R d is any such set and f : X → R is arc-smooth, then f extends to a smooth function defined on R d . We also get a version of the Bochnak-Siciak theorem on all closed fat subanalytic and all closed sets with Hölder boundary: if f : X → R is the restriction of a smooth function on 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 C d . Similar results hold for non-quasianalytic Denjoy-Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

“…Since E = U is quasiconvex, (f α ) is a Whitney jet of class C ∞ and hence extends to a smooth function on R n ; cf. [34,Proposition 1.10]. That (f α ) ∈ B [M] (E) follows from [34, Lemma 10.1] (which is only formulated for Roumieu classes, but its proof also shows the Beurling case).…”

Functions with ultradifferentiable powers

“…where g j,ε is defined by g j,ε (ζ) = g(z j,ε + 2η ε ζ). Property (18) and the assumptions on g ensure that the function g j,ε is holomorphic in a neighborhood of D(0, 1). Set…”

“…As pointed out in [4,13], the result also holds for complex-valued functions. Various generalizations were subsequently established around the notion of pseudo-immersion [4,13,18].…”

Nonlinear conditions for ultradifferentiability

“…(1 Q − 1 R\Q ) 2 = 1, where 1 A is the indicator function of a set A.) It was soon realized that the statement also holds for complex valued functions and it led to the study of so-called pseudoimmersions [7,12,13,19]. A simple proof based on ring theory was given by [1].…”