The Kotake-Narasimhan theorem in general ultradifferentiable classes

Area of Science:

• Mathematical Analysis
• Functional Analysis
• Differential Equations

Background:

• The Kotake-Narasimhan theorem is a fundamental result in the theory of partial differential operators.
• Existing theorems apply to specific classes of ultradifferentiable functions, such as those defined by weight sequences or functions.
• Generalizing these results to broader classes is crucial for advancing the field.

Purpose of the Study:

• To prove a Kotake-Narasimhan type theorem for ultradifferentiable classes defined by weight matrices.
• To unify and generalize existing theorems for weight sequences and weight functions.
• To establish a sharp Kotake-Narasimhan theorem for Beurling classes.

Main Methods:

• The study employs techniques from the theory of ultradifferentiable functions and weight matrices.
• A generalized framework is developed to encompass various definitions of ultradifferentiability.
• The proof involves constructing specific examples and utilizing abstract analytical tools.

Main Results:

• A novel Kotake-Narasimhan type theorem is proven for general ultradifferentiable classes characterized by weight matrices.
• The theorem encompasses and generalizes previously known results for weight sequences and weight functions.
• A sharp version of the theorem is obtained for the important case of Beurling classes.

Conclusions:

• The established theorem provides a unified and more general perspective on Kotake-Narasimhan theorems.
• This work deepens the understanding of ultradifferentiable functions and their properties.
• The results have potential implications for the study of partial differential equations and microlocal analysis.