Microlocal analysis of Radon transforms over quadric surfaces

TL;DR

This paper analyzes the microlocal properties of Radon transforms over quadric surfaces, revealing how the signature of the defining matrix and the geometry of the hypersurface influence the singularities of the transform.

Contribution

It provides a detailed microlocal analysis of Radon transforms over various quadric surfaces, characterizing the types of singularities based on matrix signature and hypersurface geometry.

Findings

Singularities are folds for positive/negative definite matrices with convex hypersurfaces.

Indefinite matrices lead to cusp, fold, or blowdown singularities depending on the hypersurface.

The Bolker condition is satisfied for paraboloid surfaces.

Abstract

We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.