Quasianalytic wave front sets for solutions of linear partial differential operators
The authors study the wave front set \(WF_*(u)\) of a distribution \(u\), where \(*\) stands for a quasianalytic or nonquasianalytic class of ultradifferentiable functions of Beurling or Roumieu type [in the sense of \textit{R.\,W.\thinspace Braun, R.\,Meise} and \textit{B.\,A.\thinspace Taylor}, Result.\ Math.\ 17, No.\, 3--4, 206--237 (1990; Zbl 0735.46022)]. It is proved that Hörmander's classical inclusion \(WF_*(u)\subset \Sigma\cup WF_*(Pu)\) also holds in this frame. Here, \(P\) is a differential operator with coefficients slightly more regular than the class \(*\) and \(\Sigma\) is the characteristic set of \(P\). The Beurling case is treated first and the Roumieu case is deduced from that, since the Roumieu wave front set \(WF_{\{\omega\}}(u)\) is the closure of the union of all Beurling wave front sets \(WF_{(\sigma)}(u)\) for all weights \(\sigma=o(\omega)\).
scientific article; zbMATH DE number 5774247
scientific article