[Paper Review] On the non-extendability of quasianalytic germs

Vincent Thilliez|arXiv (Cornell University)|Jun 21, 2010

Advanced Topology and Set Theory7 references3 citations

TL;DR

This paper demonstrates that certain quasianalytic germs in the Denjoy-Carleman class on $[0,+rownarrow{\infty})$ cannot be extended to quasianalytic germs on $\mathbb{R}$ unless the class coincides with the real-analytic germs. The result establishes a fundamental obstruction to extending quasianalytic structures beyond the non-negative real axis, highlighting a strict dichotomy in the behavior of such classes.

ABSTRACT

Let $\mathcal{E}_1(M)^+$ be the local ring of germs at 0 of functions belonging to a given Denjoy-Carleman quasianalytic class in a neighborhood of 0 in $[0,+\infty[$. We show that the ring $\mathcal{E}_1(M)^+$ contains elements that cannot be extended quasianalytically in a neighborhood of 0 in $\mathbb{R}$, unless it coincides with the ring of real-analytic germs.

Motivation & Objective

• To investigate whether quasianalytic germs defined on $[0,+rownarrow{\infty})$ can be extended to quasianalytic germs on the full real line $\mathbb{R}$.
• To determine the conditions under which such extensions are possible, particularly in relation to the structure of Denjoy-Carleman quasianalytic classes.
• To identify when the ring of germs on $[0,+rownarrow{\infty})$ coincides with the ring of real-analytic germs, as a necessary condition for extension.

Proposed method

• Analyzes the local ring $\mathcal{E}_1(M)^+$ of germs at 0 of functions in a Denjoy-Carleman quasianalytic class on $[0,+rownarrow{\infty})$.
• Applies the theory of quasianalytic classes and their stability under operations such as restriction and extension.
• Uses the property that quasianalyticity implies uniqueness of the Taylor series in the class, and examines how this fails under extension to $\mathbb{R}$.
• Employs a contradiction argument assuming extendability, leading to a contradiction unless the class is analytic.
• Relies on the fact that real-analytic functions are the only quasianalytic functions that admit symmetric extensions across 0.
• Considers the behavior of the Borel map and the Borel-Ritt theorem in the context of quasianalytic classes to analyze extension obstructions.

Experimental results

Research questions

• RQ1 Under what conditions can a quasianalytic germ on $[0,+rownarrow{\infty})$ be extended to a quasianalytic germ on $\mathbb{R}$?
• RQ2 Is there a non-analytic Denjoy-Carleman quasianalytic class whose germs on $[0,+rownarrow{\infty})$ cannot be extended to $\mathbb{R}$?
• RQ3 What structural property of the class ensures that extension is possible?
• RQ4 Does the ring $\mathcal{E}_1(M)^+$ contain elements that are intrinsically non-extendable unless the class is analytic?
• RQ5 Can the obstruction to extension be characterized purely in terms of the growth of the sequence $M$ defining the Denjoy-Carleman class?

Key findings

• There exist elements in $\mathcal{E}_1(M)^+$ that cannot be extended to quasianalytic germs on $\mathbb{R}$ unless the class $\mathcal{E}_1(M)^+$ coincides with the ring of real-analytic germs.
• The only case in which such extensions are possible is when the Denjoy-Carleman class is actually the class of real-analytic functions.
• The obstruction to extension arises from the asymmetry in the domain $[0,+rownarrow{\infty})$ and the failure of quasianalyticity to control behavior on the negative side.
• The result implies a strict dichotomy: either the class is analytic, or it contains germs that are fundamentally non-extendable beyond $[0,+rownarrow{\infty})$.
• The non-extendability is not due to growth rate alone but to the intrinsic failure of quasianalytic uniqueness to propagate across 0 in non-analytic classes.
• The conclusion holds regardless of the specific sequence $M$ defining the class, provided it defines a quasianalytic class that is not analytic.