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Multilinear Restriction Conjecture

Establish that for any collection {U_j : j ∈ [d]} of C^2 hypersurfaces in R^d whose normals at all points are within 1/100 of the coordinate axis e_j, with corresponding extension operators E_j, the bound ||∏_{j=1}^d E_j g_j||_{L^{q/d}(B(0,R))} ≲ ∏_{j=1}^d ||g_j||_{L^p(U_j)} holds for each ε>0, q ≥ 2d/(d−1), and p′ ≤ q(d−1)/d, for all g_j ∈ L^p(U_j) and all R ≥ 1.

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Background

This conjecture, formulated by Bennett–Carbery–Tao (2006), is a multilinear form of the Fourier restriction conjecture and is central to modern harmonic analysis. It has been proved away from the endpoint and with Rε losses at the endpoint.

The paper notes that, via functional-analytic duality, the Mizohata–Takeuchi conjecture would imply this multilinear restriction bound without Rε losses, but the counterexample here shows that route cannot yield endpoint estimates directly.

References

In 2006, the following multilinear form of the restriction conjecture was formulated by : \begin{conjecture}[Multilinear Restriction]\label{conjecture-multilinear-restriction} Let ${U_j:j\in[d]}$ be a collection of $C2$ hypersurfaces in $Rd$, so that the normal to $U_j$ at any point is within $\frac1{100}$ of the $x_j$-axis. Let $j$ denote the corresponding extension operators. Then, for each $\eps>0, q \geq \frac{2d}{d-1}$ and $p{\prime}\leq \frac{q(d-1)}d$, \left|\prod{j=1}d \mathcal{E}j g_j\right|{L{q / d}(B(0, R))} \lesssim\prod_{j=1}d\left|g_j\right|_{Lp\left(U_j\right)} for all $g_j \in Lp\left(U_j\right), 1 \leqslant j \leqslant d$, and all $R \geqslant 1$. \end{conjecture}

— A Counterexample to the Mizohata-Takeuchi Conjecture (2502.06137 - Cairo, 10 Feb 2025) in Conjecture [Multilinear Restriction], Section 1.1 (Multilinear Restriction Estimates)