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

TL;DR

This paper derives identities involving the Fourier extension operator and the $k$-plane transform for smooth submanifolds in $R^n$, revealing new connections to Fourier restriction problems.

Contribution

It establishes novel identities linking Fourier extension operators and the $k$-plane transform for general submanifolds, advancing understanding in Fourier analysis.

Findings

Identities connecting Fourier extension and $k$-plane transform.

Connections to Fourier restriction theory.

Potential implications for restriction conjectures.

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.