Tomographic Fourier extension identities for submanifolds of \({\mathbb {R}}^n\)

We establish identities for the composition \(T_{k,n}(|\widehat{gd\sigma }|^2)\) $T_{k,n}(|\widehat{gd\sigma }{|}^2)$ , where \(g\mapsto \widehat{gd\sigma }\) $g\mapsto \widehat{gd\sigma }$ is the Fourier extension operator associated with a general smooth k-dimensional submanifold of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , and \(T_{k,n}\) $T_{k,n}$ is the k-plane transform. Several connections to problems in Fourier restriction theory are presented.