  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.
