The X-ray transform analogue of the endpoint restriction conjecture

Let $n\geq 2$, let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ be the unit sphere with surface measure $d\sigma$, and let $B_R\subset\mathbb{R}^n$ be the ball of radius $R$. For a function $f$ on $\mathbb{R}^n$, define the X-ray transform by

$$Xf(\omega,v)=\int_{\mathbb{R}}f(v+s\omega)\,ds,$$

where $\omega\in\mathbb{S}^{n-1}$ and $v\in\langle\omega\rangle^\perp$. Define the Fourier extension operator by

$$\widehat{gd\sigma}(x)=\int_{\mathbb{S}^{n-1}}e^{ix\cdot\xi}g(\xi)\,d\sigma(\xi).$$

X-ray endpoint conjecture. For every $\varepsilon>0$, there is a constant $C_\varepsilon<\infty$ such that

$$\big\|X\big(\mathbf{1}_{B_R}|\widehat{gd\sigma}|^{\frac{2}{n-1}}\big)\big\|_{L^n_\omega L^\infty_v}\leq C_\varepsilon R^\varepsilon\|g\|_{L^{\frac{2n}{n-1}}(\mathbb{S}^{n-1})}^{\frac{2}{n-1}}$$

for all $R>0$. This is described as a conjectural analogue of the endpoint restriction estimate, lying between the restriction and Kakeya conjectures; its general validity remains open.