Sur une variété analytique paracompacte de dimension 2, on considère un opérateur différentiel $P$ à symbole principal $p_m$ analytique vérifiant la condition $\left(𝒫\right)$ de Nirenberg et Treves. En ajoutant une nouvelle variable et en utilisant des estimations a priori de type Carleman, on montre qu’il y a propagation des singularités pour $P$, dans $p_m^{-1}\left(0\right)$, le long des feuilles intégrales du système différentiel engendré par les champs hamiltoniens de Re$p_m$ et Im$p_m$.

Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty }$ manifold $X$. Let $p_m$ be the principal symbol and assume that $\Sigma =p_m^{-1}\left(0\right)$ is an involutive $C^{\infty }$ sub-manifold of $T^{*}X\setminus 0$, satisfying a certain transversality condition. We assume that $p_m$ vanishes exactly to order $M$ on $\Sigma$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M=2$). Suppose that a Levi condition is valid for the lower order symbols. If $u\in {𝒟}^{\text{'}}\left(X\right)$, $Pu\in C^{\infty }\left(X\right)$, then $WF\left(u\right)$ is a union...

In this talk we describe the propagation of ${𝒞}^{\infty }$ and Sobolev singularities for the wave equation on ${𝒞}^{\infty }$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times {ℝ}_t$, $P=D_t^2-{\Delta }_M$, and $u\in H_{\text{loc}}^1\left(X\right)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that ${WF}_{\text{b}}\left(u\right)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${𝒞}^{\infty }$ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary,...

This paper is a continuation of Part I of the same title which has appeared at the last issue of this journal.