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

TL;DR

This paper establishes sharp estimates for the loss of regularity when blowing down functions in Denjoy-Carleman classes, revealing an intrinsic limitation in resolution of singularities.

Contribution

It proves that the loss of regularity in Denjoy-Carleman classes under blowing down is sharp and unavoidable, clarifying the intrinsic nature of this regularity loss.

Findings

Loss of regularity is sharp in Denjoy-Carleman classes.

Regularity loss is intrinsic to resolution of singularities.

Provides precise estimates for regularity loss in blow-down functions.

Abstract

If F is an infinitely differentiable function whose composition with a blowing-up belongs to a Denjoy-Carleman class C_M (determined by a log convex sequence M=(M_k)), then F, in general, belongs to a larger shifted class C_N, where N_k = M_2k; i.e., there is a loss of regularity. We show that this loss of regularity is sharp. In particular, loss of regularity of Denjoy-Carleman classes is intrinsic to arguments involving resolution of singularities.