Regularity of class of differential operators

TL;DR

This paper investigates the regularity of certain non hypo-elliptic differential operators with polynomial coefficients, using advanced mathematical tools to relate their properties to hypo-elliptic operators like the harmonic oscillator.

Contribution

It introduces a novel approach using tensor products and a Wigner transform to analyze the regularity of non hypo-elliptic operators by connecting them to hypo-elliptic operators.

Findings

Reduced the study of regularity of non hypo-elliptic operators to hypo-elliptic ones.

Applied tensor products and Wigner transform techniques in the analysis.

Provided insights into the regularity properties of operators like the twisted Laplacian.

Abstract

In this paper we deal with the problem of regularity for non hypo-elliptic partial differential equations with polynomial coefficients. An operator $A$ on on the space of tempered distributions $\mathcal{S}^\prime$ is regular if $u$ belongs to the Schwartz class $\mathcal S$ whenever $Au\in \mathcal S$. By using tensor products of topological vector spaces and a kind of Wigner transform we reduce the study of regularity of a non hypo-elliptic differential operator, like the twisted laplacian, to a hypo-elliptic one, like the harmonic oscillator.