{\L}ojasiewicz ideals in Denjoy-Carleman classes

TL;DR

This paper extends the concept of { extL}ojasiewicz ideals to non-quasianalytic Denjoy-Carleman classes, providing characterizations and properties analogous to the smooth case, with new estimates differing from classical inequalities.

Contribution

It introduces a characterization of { extL}ojasiewicz ideals in Denjoy-Carleman classes, especially for principal ideals, and establishes their basic properties.

Findings

Characterization of { extL}ojasiewicz ideals via generator properties

Development of estimates different from classical { extL}ojasiewicz inequalities

Extension of properties from smooth functions to Denjoy-Carleman classes

Abstract

The classical notion of {\L}ojasiewicz ideals of smooth functions is studied in the context of non-quasianalytic Denjoy-Carleman classes. In the case of principal ideals, we obtain a characterization of {\L}ojasiewicz ideals in terms of properties of a generator. This characterization involves a certain type of estimates that differ from the usual {\L}ojasiewicz inequality. We then show that basic properties of {\L}ojasiewicz ideals in the $\mathcal{C}^\infty$ case have a Denjoy-Carleman counterpart.