Mizohata–Takeuchi conjecture for Fourier extension operators

Fix $n\geq 2$, and let $\Sigma\subseteq\mathbb{R}^n$ be a compact $C^2$ hypersurface with surface measure $\mathrm{d}\sigma$. Define the Fourier extension operator by

$$\mathcal{E}_{\Sigma}f(x)=\int_{\Sigma}e^{2\pi i x\cdot y}f(y)\,\mathrm{d}\sigma(y).$$

For a function $w\colon\mathbb{R}^n\to\mathbb{C}$ and an affine line $\ell\in\operatorname{AGr}_1(\mathbb{R}^n)$, define the X-ray transform by

$$Xw(\ell)=\int_{\ell}w.$$

Mizohata–Takeuchi conjecture. If $\Sigma$ is convex, then for all $f\in L^2(\Sigma)$ and measurable weights $w\colon\mathbb{R}^n\to[0,\infty)$,

$$\|\mathcal{E}_{\Sigma}f\|_{L^2(\mathbb{R}^n,w\,\mathrm{d}x)}^2\lesssim\|Xw\|_{L^\infty(\operatorname{AGr}_1(\mathbb{R}^n))}\|f\|_{L^2(\Sigma)}^2.$$

This conjecture predicts the weighted $L^2$ behavior of Fourier extension operators for convex hypersurfaces and connects restriction theory with X-ray transform estimates. Its general status is not established by the supplied source context.