Injectivity and stability for a generic class of generalized Radon transforms

TL;DR

This paper investigates a broad class of generalized Radon transforms on analytic Riemannian manifolds, establishing their injectivity and stability properties under certain conditions using microlocal analysis.

Contribution

It proves microlocal regularity, injectivity, and stability for generalized Radon transforms satisfying the Bolker condition on analytic manifolds.

Findings

Proved microlocal regularity for the transforms.

Established injectivity for an open dense subset.

Demonstrated stability for the class of transforms.

Abstract

Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for generalized Radon transforms on analytic manifolds defined on an analytic family of hypersurfaces. We then show injectivity and stability for an open, dense subset of smooth generalized Radon transforms satisfying the Bolker condition, including the analytic ones.