Ellipticity and the problem of iterates in Denjoy-Carleman classes

Area of Science:

• Mathematical Analysis
• Differential Equations
• Harmonic Analysis

Background:

• Métivier's 1978 theorem linked linear differential operator ellipticity to the theorem of iterates in Gevrey classes.
• Gevrey classes are crucial for studying properties of differential operators and their solutions.

Purpose of the Study:

• To extend Métivier's theorem to broader Denjoy-Carleman classes.
• To investigate the applicability of the theorem of iterates for operators with analytic coefficients in these generalized classes.
• To differentiate the behavior of Denjoy-Carleman classes from Braun-Meise-Taylor classes.

Main Methods:

• Extension of Métivier's theorem to Denjoy-Carleman classes defined by strongly non-quasianalytic weight sequences.
• Development of a new technique for constructing optimal functions within Denjoy-Carleman classes using Fourier integrals.
• Comparative analysis of function class properties.

Main Results:

• The theorem of iterates holds for linear differential operators with analytic coefficients in Denjoy-Carleman classes.
• A novel Fourier integral method for constructing optimal functions in these classes was developed.
• The analogous statement was shown to be invalid for Braun-Meise-Taylor classes.

Conclusions:

• The study successfully extends the theorem of iterates to Denjoy-Carleman classes.
• The findings reveal a significant distinction between Denjoy-Carleman and Braun-Meise-Taylor classes regarding the theorem of iterates.
• The new construction method for optimal functions offers potential for further research.