2011/10/07 by Nick Haber, Haber, Nick, András Vasy +1
Mathematics · #35A21 #35P25 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP) #math.AP #math.SP #msc:35A21 #msc:35P25
paper · pdf · doi:10.48550/arxiv.1110.1419
39 pages, 4 figures
arxiv created 2011/10/07 · openalex publication_date 2011/10/07 · arxiv updated 2016/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 contained in the characteristic set of P. The standard propagation of singularities theorem of Duistermaat-Hormander gives no information at the Lagrangian submanifold. 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.