The Kotake-Narasimhan Theorem in general ultradifferentiable classes

TL;DR

This work extends the Kotake–Narasimhan theorem to general ultradifferentiable classes defined by weight matrices, unifying results across Denjoy–Carleman and Braun–Meise–Taylor frameworks and providing sharp Beurling-type statements. The authors develop a fundamental $L^2$-estimate and an iterative scheme to prove a local Kotake–Narasimhan theorem for elliptic systems with coefficients in $\mathcal{E}^{[\mathfrak{M}]}(\Omega)$, handling Roumieu and Beurling cases (with Beurling reducible to Roumieu) and establishing corresponding results for weight functions. They also discuss elliptic regularity, microlocal hypoellipticity, and global variants on Lipschitz domains, thereby strengthening the link between iterates, ultradifferentiable structure theory, and elliptic PDE regularity. The results yield new instances (e.g., $N_k^q=q^{k^2}$) where Kotake–Narasimhan holds and open avenues for conjectures about broader ultradifferentiable settings and closure properties under derivation and composition.

Abstract

We prove a Kotake-Narasimhan type theorem in general ultradifferentiable classes given by weight matrices. In doing so we simultaneously recover and partially generalize the known results for classes given by weight sequences and weight functions. In particular, we obtain a sharp Kotake-Narasimhan theorem for Beurling classes.

Theorems & Definitions (38)

• Definition 1.1
• Example 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• Remark 2.2
• Lemma 2.3