Bejenaru's multilinear restriction conjecture for curved transversal hypersurfaces

Let $n\geq 1$, let $U_i\subset\mathbb{R}^n$ be bounded parameter domains, and let $S_i\subset\mathbb{R}^{n+1}$ be smooth, compact hypersurfaces parametrized by smooth maps $\Sigma_i:U_i\to\mathbb{R}^{n+1}$. For functions $f_i$ on $U_i$, write

$$\mathcal E_i f_i(x)=\int_{U_i}e^{i x\cdot\Sigma_i(\xi)}f_i(\xi)\,d\xi$$

and denote the $k$-linear restriction estimate by $\mathcal R^*(2\times\cdots\times2\to p)$ when

$$\left\|\prod_{i=1}^k\mathcal E_i f_i\right\|_{L^p(\mathbb{R}^{n+1})}\lesssim\prod_{i=1}^k\|f_i\|_{L^2(U_i)}.$$

Under appropriate transversality and curvature conditions on the surfaces $S_i$, Bejenaru's conjecture. $\mathcal R^*(2\times\cdots\times2\to p)$ holds for every

$$p\geq p(k)=\frac{2(n+1+k)}{k(n+k-1)}.$$

The conjecture predicts the curvature-improved optimal range of exponents for multilinear restriction estimates, extending the known transversality-based exponent $2/(k-1)$; the paper notes that the optimal estimate in the general setting remains open.