Local solvability of second order 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 second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that $Q_A$ and $Q_B$ are two real quadratic forms on a finite dimensional symplectic vector space, and let $Q_C:=\{Q_A,Q_B\}$ be given by the Poisson bracket of $Q_A$ and $Q_B.$ Then $Q_C$ is again a quadratic form, and we may ask: When can we find a common zero of $Q_A$ and $Q_B$ at which $Q_C$ does not vanish? The second paper, in combination with the first one, will give a fairly comprehensive picture of what rules local solvability of invariant second order operators on the Heisenberg group.