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

Abstract

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 $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$ vanishes to second order at $(x_0,\xi_0)\in T^*\Om\setminus 0,$ and denote by $Q_\H$ the Hessian form associated to $p_k$ on $T_{(x_0,\xi_0)}T^*\Om.$ 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 $\theta\in\RR$ such that $\Re (e^{i\theta}Q_\H)\ge 0.$