Tomography bounds for the Fourier extension operator and applications

We explore the extent to which the Fourier transform of an $L^p$ density supported on the sphere in $\mathbb{R}^n$ can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $X(|\widehat{gd\sigma}|^2)$ and $\mathcal{R}(|\widehat{gd\sigma}|^2)$, where $X$ and $\mathcal{R}$ denote the X-ray and Radon transforms respectively; here $d\sigma$ denotes Lebesgue measure on the unit sphere $\mathbb{S}^{n-1}$, and $g\in L^p(\mathbb{S}^{n-1})$. We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $\widehat{gd\sigma}$, establishing some natural variants of conjectures of Stein and Mizohata--Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel's theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.