Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non--quasianalytic classes

We prove the following inclusion \[ WF_* (u)\subset WF_*(Pu)\cup \Sigma, \quad u\in\E^\prime_\ast(\Omega), \] where $WF_*$ denotes the non--quasianalytic Beurling or Roumieu wave front set, $\Omega$ is an open subset of $\R^n$, $P$ is a linear partial differential operator with suitable ultradifferentiable coefficients, and $\Sigma$ is the characteristic set of $P$. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.