A non uniqueness result for operators of principal type (Q1909564)

We present in this paper a detailed proof of a result announced in [the first author, Ann. Math., II. Ser. 117, 77-108 (1983; Zbl 0516.35018)], with only a sketch of the proof. A typical example of our main theorem is the wave equation for a time-like surface; though Holmgren's theorem applies to yield Cauchy uniqueness with respect to such a surface, our result shows that a zero order smooth perturbation may cause uniqueness to fail. In other words, there is no ``stable'' uniqueness. This theorem improves a previous result due to Hörmander, based on Cohen's counterexample, and shows the optimality of the Hörmander-Lerner-Robbiano theorem in the weakly pseudoconvex case [see \textit{N. Lerner} and \textit{L. Robbiano}, J. Anal. Math. 44, 32-66 (1985; Zbl 0574.35003)]. The method of proof is based on geometrical optic techniques.