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

TL;DR

This paper establishes necessary conditions for the local solvability of doubly characteristic linear differential operators, focusing on the behavior of the principal symbol and its Hessian form at critical points.

Contribution

It introduces a new necessary condition involving the Hessian form for local solvability of doubly characteristic operators, extending previous understanding.

Findings

Necessary condition involving the Hessian form for local solvability

Conditions on the principal symbol's second-order vanishing

Analysis under rank and mild additional 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 $\Re (e^{i\theta}Q_\H)\ge 0.$