Tomographic Fourier Extension Identities for Submanifolds of ${\mathbb{R}}^n$

Abstract

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

10 pages, minor revision including some updated and additional references