Local solvability of linear differential operators with double characteristics I: Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\text{Om}\subset {\text{RR}}^n\mathrm{.}$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,{\xi }_0)\in T^{\ast }\text{Om}\setminus 0,$ and denote by Q_\H the Hessian form associated to $p_k$ on $T_{(x_0,{\xi }_0)}{T}^{\ast }\text{Om}\mathrm{.}$ As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of $L$ at $x_0$ is the existence of some $\theta \in \text{RR}$ such that \Re (e^{i\theta}Q_\H)\ge 0.