A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions

TL;DR

This paper provides a simplified proof of the Kotake-Narasimhan theorem for elliptic operators within ultradifferentiable function classes, clarifying previous proof issues and extending the theorem's applicability.

Contribution

The authors present a more straightforward proof of the Kotake-Narasimhan theorem in ultradifferentiable settings, addressing prior proof gaps and removing restrictive conditions.

Findings

Ellipticity is necessary for the theorem to hold.

The proof is simplified and clarified, especially regarding the induction hypothesis.

The theorem applies without the previous condition on the weight function.

Abstract

We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of M\'etivier, we also show that the ellipticity is a necessary condition for the theorem to be true. The present new version of the paper modifies the proof of Theorem 1.4 for an observation by Hoepfner and Rampazo who pointed out that an induction hypothesis depends on a constant $C_q$ that changes in the induction process, and hence the argument might not work as it was written. However, the statement of the result was originally correct and modifying $C_q$ with a more concrete expression in the induction hypothesis, the induction procedure is easily clarified with almost the same proof. Moreover, we eliminate the condition that the weight is identically zero in the interval [0,1], showing that the statements hold true with very similar arguments.