A characterization of the wave front set defined by the iterates of an operator with constant coefficients

TL;DR

This paper characterizes the wave front set related to iterates of constant coefficient differential operators using advanced Paley-Wiener theorems, providing new insights into distribution regularity and wave front set construction.

Contribution

It introduces a new characterization of the wave front set for distributions via operator iterates, employing recent ultradifferentiable class theorems and offering applications and examples.

Findings

New characterization of wave front set using Paley-Wiener theorems

Examples illustrating the regularity of variable coefficient operators

Construction of distributions with prescribed wave front sets

Abstract

We characterize the wave front set $WF^P_\ast(u)$ with respect to the iterates of a linear partial differential operator with constant coefficients of a classical distribution $u\in{\mathcal D}'(\Omega)$, $\Omega$ an open subset in ${\mathbb R}^n$. We use recent Paley-Wiener theorems for generalized ultradifferentiable classes in the sense of Braun, Meise and Taylor. We also give several examples and applications to the regularity of operators with variable coefficients and constant strength. Finally, we construct a distribution with prescribed wave front set of this type.