Three-dimensional multilinear restriction conjecture for transverse hypersurfaces

Let $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ be surfaces in $\mathbb{R}^3$ satisfying the transversality condition above, and let

$$\mathcal E_j f_j(x)=\int_{\Sigma_j}e^{i x\xi}f_j(\xi)\,d\xi, \qquad f_j\in L^1(\Sigma_j).$$

Three-dimensional multilinear restriction conjecture. One has

$$\|\mathcal E_1 f_1\mathcal E_2 f_2\mathcal E_3 f_3\|_{L^1}\leq C\|f_1\|_{L^2(\Sigma_1)}\|f_2\|_{L^2(\Sigma_2)}\|f_3\|_{L^2(\Sigma_3)}.$$

This estimate is a multilinear restriction inequality for three transverse surfaces in three dimensions and has applications to nonlinear dispersive equations. The source notes that an earlier work proves the estimate with an $\epsilon$ loss, while the conjectured lossless bound is presented as the stronger result.