Endpoint Brascamp–Lieb type restriction conjecture

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 $\Omega_j$, and define

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

Let $k_j=\dim\Omega_j$. Assume there is a vector $\mathbf p=(p_1,\ldots,p_n)$ with $p_j>0$ such that the Brascamp–Lieb constant is finite for

$$\mathbf B(\Sigma)=\bigl(T_{\Sigma_1(0)}\Omega_1,\ldots,T_{\Sigma_n(0)}\Omega_n\bigr),$$

where each component denotes the linear subspace parallel to the corresponding tangent space. No assumption is made that $\sum_jk_j=d$ or that the tangent spaces have nonzero wedge product.

Endpoint Brascamp–Lieb type restriction conjecture. When the $U_j$ are sufficiently small,

$$\int_{\mathbb{R}^d}\prod_{j=1}^n|E_jg_j|^{2p_j}\,\mathrm{d}\xi\lesssim_{d,\mathbf p,BL(\mathbf B(\Sigma),\mathbf p)}\prod_{j=1}^n\|g_j\|_{L^2(U_j)}^{2p_j}.$$

This conjecture extends the endpoint multilinear restriction formulation to general finite Brascamp–Lieb data. The paper proves the corresponding endpoint perturbed Brascamp–Lieb theorem for families of slabs, but the restriction statement itself is presented as a conjecture.