Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms

TL;DR

This paper investigates the regularity properties of linear partial differential operators with polynomial coefficients using a Wigner type transform, extending results to Schwartz and ultradifferentiable spaces with various weight functions.

Contribution

It introduces a novel approach to analyze PDE regularity in ultradifferentiable spaces via Wigner type transforms, expanding the understanding of operator behavior in these function spaces.

Findings

Regularity results in Schwartz space $\\mathcal S$

Regularity results in ultradifferentiable space $\mathcal S_\omega$

Discussion of several examples in the new setting

Abstract

We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space $\mathcal S$ and in the space ${\mathcal S}_\omega$ as introduced by Bj\"orck for weight functions $\omega$. Several examples are discussed in this new setting.