Arc-quasianalytic functions

TL;DR

This paper establishes a characterization of arc-quasianalytic functions within quasianalytic classes, showing they become quasianalytic after finite local blow-ups, thus generalizing previous arc-analytic function results.

Contribution

It provides a new criterion for arc-quasianalytic functions based on local blow-ups, extending known theorems from arc-analytic to quasianalytic functions.

Findings

Arc-quasianalytic functions are characterized by their quasianalyticity after finite local blow-ups.

The result generalizes a theorem on arc-analytic functions to quasianalytic classes.

The paper links quasianalytic equations to the arc-quasianalytic property.

Abstract

We work with quasianalytic classes of functions. Consider a real-valued function y = f(x) on an open subset U of Euclidean space, which satisfies a quasianalytic equation G(x, y) = 0. We prove that f is arc-quasianalytic (i.e., its restriction to every quasianalytic arc is quasianalytic) if and only if f becomes quasianalytic after (a locally finite covering of U by) finite sequences of local blowing-ups. This generalizes a theorem of the first two authors on arc-analytic functions.