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 \(\Omega \subset \mathbb{R}^n.\) Suppose the principal symbol pk of L vanishes to second order at \((x_0, \xi_0) \in T^*\Omega \setminus 0\), and denote by \(Q_\mathcal{H}\) the Hessian form associated to pk on \(T_{(x_0,\xi_0)}T^*\Omega\). 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 x0 is the existence of some \(\theta \in \mathbb{R}\) such that \({\rm Re}\,(e^{i\theta} Q_\mathcal{H}) \ge 0\). We apply this result in particular to operators of the form $$ L=\sum\limits^m_{j,k=1}\alpha_{jk}(x) X_j X_k+\,{\rm lower}\,{\rm order}\,{\rm terms},\quad (0.1) $$ where the Xj are smooth real vector fields and the αjk are smooth complex coefficients forming a symmetric matrix \(\mathcal{A}(x) := \{\alpha_{jk}(x)\}_{j,k}\). We say that L is essentially dissipative at x0, if there is some \(\theta \in \mathbb{R}\) such that eiθL is dissipative at x0, in the sense that \({\rm Re}\,\big(e^{i\theta}\mathcal{A}(x_0)\big) \ge 0\). For a large class of doubly characteristic operators L of this form, our main result implies that a necessary condition for local solvability at x0 is essential dissipativity of L at x0. By means of Hormander’s classical necessary condition for local solvability, the proof of the main result can be reduced to the following question: suppose that QA and QB are two real quadratic forms on a finite dimensional symplectic vector space, and let QC: = {QA,QB} be given by the Poisson bracket of QA and QB. Then QC is again a quadratic form, and we may ask: when can we find a common zero of QA and QB at which QC does not vanish? The study of this question occupies most of the paper, and the answers may be of independent interest. In the second paper of this series, building on joint work with F. Ricci, M. Peloso and others, we shall study local solvability of essentially dissipative left-invariant operators of the form (0.1) on Heisenberg groups in a fairly comprehensive way. Various examples exhibiting a kind of exceptional behaviour from previous joint works, e.g., with G. Karadzhov, have shown that there is little hope for a complete characterization of locally solvable operators on Heisenberg groups. However, the “generic” scheme of what rules local solvability of second order operators on Heisenberg groups becomes evident from our work.