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

Tomography bounds for the Fourier extension operator and applications