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

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.