On the stability of analytic germs under ultradifferentiable perturbations

Let f be a real-analytic function germ at the origin in Rn, whose critical locus contains a given real-analytic set X, and let Y be a germ of a closed subset at the origin. We study the stability of f under perturbations u that are flat on Y and that belong to a given Denjoy–Carleman non-quasianalytic class. We obtain a condition ensuring that f+u=f○Φ where Φ is a germ of diffeomorphism whose components belong to a (generally larger) Denjoy–Carleman class. Roughly speaking, this condition involves a Łojasiewicz-type separation property between Y and the complex zeros of a certain ideal associated with f and X. The relationship between the Denjoy–Carleman classes of u and Φ is controlled precisely by the inequality. This result extends, and simplifies, former work of the author on germs with isolated critical points.