Endpoint multilinear $k_j$-restriction conjecture

For $1\leq j\leq n$, let $\Omega_j\subseteq\mathbb{R}^d$ be smooth submanifolds with compact closure, let $\Sigma_j:U_j\to\mathbb{R}^d$ be smooth parametrizations of subsets of $\Omega_j$, where $U_j$ is a neighborhood of $0$, and define

$$E_jg_j(\xi)=\int_{U_j}e^{2\pi i\xi\cdot\Sigma_j(x)}g_j(x)\,\mathrm{d}x.$$

Assume $\sum_{j=1}^n\dim\Omega_j=d$ and

$$T_{\Sigma_1(0)}\Omega_1\wedge\cdots\wedge T_{\Sigma_n(0)}\Omega_n\neq0.$$

Endpoint multilinear $k_j$-restriction conjecture. When the $U_j$ are sufficiently small,

$$\int_{\mathbb{R}^d}\prod_{j=1}^n|E_jg_j|^{\frac{2}{n-1}}\,\mathrm{d}\xi\lesssim_d\prod_{j=1}^n\|g_j\|_{L^2(U_j)}^{\frac{2}{n-1}}.$$

This is an endpoint multilinear restriction estimate for smooth submanifolds whose tangent spaces satisfy the stated transversality condition. The paper notes that the cited multilinear restriction methods establish a local version with an $R^\varepsilon$ loss, while the global endpoint estimate is presented as a conjecture.