Regularity of a Distribution and of the Boundary of Its Support

Abstract

AbstractIn two recent papers, Boman (J Geom Anal 31:2726–2741, 2020, https://doi.org/10.1007/s12220-020-00372-8, J Ill-posed Inverse Probl 2021, https://doi.org/10.1515/jiip-2020-0139), we proved that the Radon transform of a compactly supported distribution can be supported in the set of supporting planes to a bounded, convex domain $$D\subset {\mathbb {R}}^n$$ D ⊂ R n only if the boundary of D is an ellipsoid. Using closely related methods we study here the relationship between the analytic wave front set for the characteristic function, $$\chi _D$$ χ D , of a domain $$D \subset {\mathbb {R}}^n$$ D ⊂ R n and singularities of the boundary $$\partial D$$ ∂ D of the domain. For instance we prove that the boundary surface must be real analytic in a neighborhood of a point $$z \in \partial D \in C^1$$ z ∈ ∂ D ∈ C 1 , if the analytic wave front set of $$\chi _D$$ χ D at z contains no other elements than the conormals to $$\partial D$$ ∂ D at z.