Unicité de Cauchy pour des opérateurs de type principal. (Cauchy uniqueness for operators of principal type) (Q1063775)

Let P be a second order differential operator, with smooth principal symbol p(x,\(\xi)\) and bounded lower order term, in a neighborhood V of a point \(x_ 0\in {\mathbb{R}}^ n\). Let S be a smooth non characteristic hypersurface through \(x_ 0:\) \(S=\{x\in V:\) \(\phi (x)=\phi (x_ 0)\}\). We denote by \(H_ p\) the Hamiltonian field of p. The main result of this paper is the following remarkable extension of the well known Hörmander's theorem: Suppose P of principal type and that, for every \((x,\xi)\in V\times {\mathbb{R}}^ n,\quad p(x,\xi)=0,\quad H_ p(\phi)(x,\xi)=0\Rightarrow H^ 2_ p(\phi)(x,\xi)\leq 0,\) then every solution \(u\in C^{\infty}(V)\) of \(Pu=0\), whose support is contained in \(\{\) \(x: \phi (x)\geq \phi (x_ 0)\}\) and supp \(u\cap \{x:\) \(\phi (x)=\phi (x_ 0)\}\) is compact, must vanish near \(x_ 0.\) Some corollary concerning the usual Cauchy problem is derived from this result. Let us note that an improvement of the above theorem has been given by \textit{L. Hörmander} [''The analysis of linear partial differential operators. IV: Fourier integral operators'', Chapter 28, Berlin etc.: Springer-Verlag (1985)].