Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

TL;DR

This paper investigates the regularity of solutions to PDEs with polynomial coefficients within non-isotropic ultradifferentiable spaces, utilizing time-frequency analysis methods like Gabor and Wigner transforms to extend regularity results beyond classical hypoellipticity.

Contribution

It introduces a novel approach using Wigner-type representations to establish regularity for PDE operators lacking classical hypoelliptic properties in non-isotropic ultradifferentiable spaces.

Findings

Regularity results for PDEs with polynomial coefficients in non-isotropic ultradifferentiable spaces.

Application of Gabor and Wigner transforms to analyze PDE regularity.

Extension of regularity theory beyond classical hypoelliptic operators.

Abstract

In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.