The multilinear restriction conjecture

Equivalent formulations 4

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Multilinear restriction conjecture

   Let $U_j$ be compact neighbourhoods of the origin in $\mathbb{R}^{d-1}$, let $\Sigma_j:U_j\to\mathbb{R}^d$ be $C^{1,\beta}$ parametrisations of codimension-one submanifolds $S_j$, and let $\mathcal{E}_j$ be the associated Fourier extension operators

   $$\mathcal{E}_jg(\xi)=\int_{U_j}g(x)e^{i\langle \xi,\Sigma_j(x)\rangle}\,\mathrm{d}x.$$

   Suppose that the submanifolds $S_1,\ldots,S_d$ are transversal in a neighbourhood of the origin, $q\geq\frac{2d}{d-1}$, and $p'\leq\frac{d-1}{d}q$. Multilinear restriction conjecture. There exists a constant $C$ such that

   $$\left\|\prod_{j=1}^{d}\mathcal{E}_{j}g_{j}\right\|_{L^{q/d}(\mathbb{R}^d)}\leq C\prod_{j=1}^{d}\|g_{j}\|_{L^p(U_j)}$$

   for all $g_1,\ldots,g_d$ supported in a sufficiently small neighbourhood of the origin. This conjecture is part of the multilinear restriction theory for the Fourier transform; the cited work of Bennett, Carbery and Wright and Bennett, Carbery and Tao concerns this family of estimates. The supplied material does not establish whether the stated range is resolved, so its status remains open.

   source: Jonathan Bennett and Neal Bez, “Some nonlinear Brascamp-Lieb inequalities and applications to harmonic analysis”, arXiv:0906.2064 (2010).

2. The multilinear restriction conjecture

   Let $2\leq k\leq d$. A $k$-tuple $S_1,\ldots,S_k$ of smooth codimension-one submanifolds of $\mathbb{R}^d$ is transversal if the wedge product of any choice of unit normal vectors has a positive uniform lower bound. Let $d\sigma_j$ be the induced measures and let $p'$ be the Hölder conjugate of $p$. Multilinear restriction conjecture. If $S_1,\ldots,S_k$ are transversal with everywhere positive principal curvatures, $\frac{1}{q}<\frac{d-1}{2d}$, $\frac{1}{q}\leq\frac{d+k-2}{d+k}\frac{1}{p'}$, and $\frac{1}{q}\leq\frac{d-k}{d+k}\frac{1}{p'}+\frac{k-1}{k+d}$, then

   $$\left\|\prod_{j=1}^{k}\widehat{f_jd\sigma_j}\right\|_{L^{q/k}(\mathbb{R}^d)}\lesssim\prod_{j=1}^{k}\|f_j\|_{L^p(d\sigma_j)}.$$

   This generalizes the bilinear restriction perspective and is intended to clarify the roles of curvature and transversality.

   source: Jonathan Bennett, “Aspects of Multilinear Harmonic Analysis Related to Transversality”, arXiv:1405.5369 (2014).

3. The multilinear restriction conjecture

   Let $\{U_j:j\in[d]\}$ be a collection of $C^2$ hypersurfaces in $\mathbb{R}^d$, with the normal to $U_j$ at every point within $\frac{1}{100}$ of the $x_j$-axis. Let $\mathcal E_j$ be the corresponding extension operators. For $\varepsilon>0$, $q\geq\frac{2d}{d-1}$, and $p'\leq\frac{q(d-1)}{d}$, the multilinear restriction conjecture asserts that

   $$\left\|\prod_{j=1}^d\mathcal E_jg_j\right\|_{L^{q/d}(B(0,R))}\lesssim\prod_{j=1}^d\|g_j\|_{L^p(U_j)}$$

   for all $g_j\in L^p(U_j)$ and all $R\geq1$. It is known away from the endpoint and at the endpoint with $R^\varepsilon$ losses; the supplied counterexample shows only that the Mizohata–Takeuchi route cannot prove the lossless endpoint estimate, not that this conjecture itself is refuted.

   source: Hannah Cairo, “A Counterexample to the Mizohata-Takeuchi Conjecture”, arXiv:2502.06137 (2025).

4. The multilinear restriction conjecture

   For each $1\leq j\leq d$, let $U_j$ be a parameter domain, let $\Sigma_j:U_j\to\mathbb{R}^d$ be smooth, and let $\mathcal{E}_j$ be the associated extension operator. Define

   $$Y_j(x)=\bigwedge_{k=1}^{d-1}\frac{\partial}{\partial x_k}\Sigma_j(x).$$

   Assume the transversality condition $\det(Y_1(x^{(1)}),\ldots,Y_d(x^{(d)}))\geq\nu$ for all $x^{(j)}\in U_j$, and the smoothness condition $\|\Sigma_j\|_{C^2(U_j)}\leq A$ for all $j$. Multilinear restriction conjecture. If $q\geq\frac{2d}{d-1}$ and $p'\leq\frac{d-1}{d}q$, then there exists a constant $C$, depending only on $A$, $\nu$, $d$, and $U_1,\ldots,U_d$, such that

   $$\left\|\prod_{j=1}^d\mathcal{E}_jg_j\right\|_{L^{q/d}(\mathbb{R}^d)}\leq C\prod_{j=1}^d\|g_j\|_{L^p(U_j)}$$

   for all $g_j\in L^p(U_j)$. This is the multilinear analogue of the restriction conjecture and is designed to retain only the transversality needed among the normals; its full asserted range is open in general.

   source: Jonathan Bennett, Anthony Carbery and Terence Tao, “On the Multilinear Restriction and Kakeya conjectures”, arXiv:math/0509262 (2005).