Propagation of singularities around a Lagrangian submanifold of radial points

Haber, Nick; Vasy, András

Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 4, pp. 679-726

In this work we study the wavefront set of a solution u to Pu=f, where P is a pseudodifferential operator on a manifold with real-valued homogeneous principal symbol p, when the Hamilton vector field corresponding to p is radial on a Lagrangian submanifold Λ contained in the characteristic set of P. The standard propagation of singularities theorem of Duistermaat-Hörmander gives no information at Λ. By adapting the standard positive-commutator estimate proof of this theorem, we are able to conclude additional regularity at a point q in this radial set, assuming some regularity around this point. That is, the a priori assumption is either a weaker regularity assumption at q, or a regularity assumption near but not at q. Earlier results of Melrose and Vasy give a more global version of such analysis. Given some regularity assumptions around the Lagrangian submanifold, they obtain some regularity at the Lagrangian submanifold. This paper microlocalizes these results, assuming and concluding regularity only at a particular point of interest. We then proceed to prove an analogous result, useful in scattering theory, followed by analogous results in the context of Lagrangian regularity.

Dans cet article on étudie le spectre singulier, WF (u), pour les solutions de l'équation Pu=f, où P est un operateur pseudo-différentiel sur une variété de la classe C∞, X, avec symbole principal homogène p, si le champ Hamiltonien de p est radial sur une sous-variété lagrangienne, Λ, contenue dans l'ensemble caractéristique de P. Le théorème classique de Duistermaat et Hörmander ne fournit aucune information sur Λ. Nous adaptons la preuve de ce théorème utilisant des commutateurs positifs, et prouvons que la solution possède d'une régularité additionelle près d'un point q si on suppose certaine régularité au fond. C'est à dire, l'hypothèse a priori est soit une hypothèse de régularité plus faible à q, soit une hypothèse de régularité près de, mais pas à q. Les résultats plus anciens de Melrose et Vasy donnent une version plus globale de cette analyse. Cet article fournit une version microlocale des résultats de ces auteurs; on suppose et prouve la régularité seulement près du point d'intérêt, q. Nous prouvons aussi un résultat similaire qui est utile dans la théorie de la diffusion, et aussi des résultats de la régularité lagrangienne.

Classification : 35A21, 35P25

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