The restriction–Brascamp–Lieb conjecture for smooth submanifolds

For each $1\leq j\leq m$, let $\Sigma_j:U_j\to\mathbb{R}^n$ be a smooth parametrization of an $n_j$-dimensional submanifold $S_j$ by a neighborhood $U_j$ of the origin in $\mathbb{R}^{n_j}$. Define the associated extension operator by

$$\mathcal{E}_jg_j(\xi):=\int_{U_j}e^{-2\pi i\xi\cdot\Sigma_j(x)}g_j(x)\,\mathrm{d}x.$$

Let $L_j:=(\mathrm{d}\Sigma_j(0))^\ast:\mathbb{R}^n\to\mathbb{R}^{n_j}$ and let $\operatorname{BL}(\mathbf{L},\mathbf{p})$ denote the Brascamp–Lieb constant. The restriction–Brascamp–Lieb conjecture. If $\operatorname{BL}(\mathbf{L},\mathbf{p})$ is finite, then, provided the neighborhoods $U_j$ of $0$ are chosen small enough,

$$\int_{\mathbb{R}^n}\prod_{j=1}^{m}|\mathcal{E}_jg_j|^{2p_j}\lesssim\prod_{j=1}^{m}\|g_j\|_{L^2(U_j)}^{2p_j}$$

holds for all $g_j\in L^2(U_j)$, $1\leq j\leq m$. This conjecture seeks a nonlinear extension of the Brascamp–Lieb finiteness condition; the supplied text gives no resolution status.