Sharp Estimates for Blowing Down Functions in a Denjoy–Carleman Class

Abstract

Abstract If $F$ is a $C^{\infty }$ function whose composition $F \circ \sigma $ with a blowing-up ${\sigma }$ belongs to a Denjoy–Carleman class $C_M$, then $F$, in general, belongs to a larger class $C_{M^{(2)}}$; that is, there is a loss of regularity. We show that this loss of regularity is sharp. In particular, the loss of regularity of Denjoy–Carleman classes is intrinsic to arguments involving resolution of singularities.