Note on wave front set of solutions to non effectively hyperbolic operators (Q1109224)

We study the propagation of wave front set of solutions to non effectively hyperbolic operators of second order which satisfy the strict Ivrij-Petkov-Hörmander condition at a double characteristic. We follow \textit{V. Ya. Ivrij} [(*) Trans. Mosc. Math. Soc. 34, 149-168 (1978; Zbl 0403.35060)] and employ the microlocal energy estimates as in (*), but the presence of symplectic characteristics obey us to impose some additional conditions to ``symbols of spatial type'' in (*) (see Lemma 2.1 below due to C. Parenti in a work to appear). Our main concern in this note is then how to construct such symbols which control well the wave front set.