30 problems

Bilinear restriction range conjecture. If $n\geq2$, then $R^*(p\times p\to q)$ holds whenever

The sharp restriction conjecture. $R^*(p\to q)$ holds whenever

Let $d\geq 2$, let $U$ be a compact neighbourhood of the origin in $\mathbb{R}^{d-1}$, and let $\Sigma:U\to\mathbb{R}^d$ be a smooth parametrisation of a $(d-1)$-dimensional subman…

Mizohata–Takeuchi conjecture. If $\Sigma$ is convex, then for all $f\in L^2(\Sigma)$ and measurable weights $w\colon\mathbb{R}^n\to[0,\infty)$,

Let $k\geq 2$. Let $S_1,\ldots,S_k\subseteq\mathbb{R}^{d}$ be smooth hypersurfaces equipped with measures $\sigma_1,\ldots,\sigma_k$, respectively. They are transversal if there is…

Gaussian maximizer conjecture. The maximizers for the paraboloid Strichartz inequality are exactly the Gaussian-type functions $g\in\mathcal{G}$. This conjecture has been confirmed…

Hörmander dichotomy conjecture. The operators $T_\phi^\lambda$ satisfy the operator boundedness estimate for $p>\frac{2n}{n-1}$ if and only if $\phi$ satisfies Bourgain's condition…

Sharp $L^2\to L^6$ extension conjecture. For all $f\in L^2(\mathbb{S}^1)\setminus\{0\}$,

Let $\mathbf{Q}$ be a quadratic manifold, and let “well-curved” and “best Stein–Tomas” denote the geometric and Fourier-analytic conditions defined in the paper. Best Stein–Tomas c…

Hyperbolic-paraboloid decoupling conjecture. For $p>3$, for every $\varepsilon>0$ there exists $C_\varepsilon>0$ such that for all $R\geq1$,

The restriction conjecture for the sphere. This estimate should hold when

Lee–Rogers–Seeger local space-time conjecture. The estimate

Restriction conjecture. The estimate

Let $d\geq 3$ be an integer, let $\Gamma=\{(\xi,\xi^d)\in\mathbb R^2:1\leq|\xi|\leq2\}$, and let $P_j=(\xi_j,\xi_j^d)$ for $j=1,\ldots,N$ be equally distributed in $\Gamma$. For ea…

Let $d\geq 3$ be an integer, let $N\in\mathbb N$, let $a_1,\ldots,a_N\in\mathbb C$, and write $\mathbb T^2$ for the two-dimensional torus. For $p\geq 2(d+1)$ and every…

The family of fractal Fourier restriction estimates. For every $\epsilon>0$ there is a constant $C_\epsilon$ such that

Fourier restriction conjecture for Frostman measures on the parabola. For every $0\leq s\leq 1$ and $\epsilon>0$, there exists $p=p(\epsilon,s)\geq 1$ such that

Let $S$ be a smooth compact surface in $\mathbb{R}^3$ with positive definite second fundamental form, and let $d\sigma$ be its induced Lebesgue measure. Define the Fourier extensio…

X-ray endpoint conjecture. For every $\varepsilon>0$, there is a constant $C_\varepsilon<\infty$ such that

Elliptic extension conjecture. For $N$ sufficiently large, $\varepsilon_0>0$ sufficiently large, and $1\leq p,q\leq\infty$ satisfying

Monotonicity conjecture. Then

Let $n\geq 2$, let $1\leq r,s\leq\infty$, and write $r'$ for the Hölder conjugate of $r$. Let the discrete restriction estimate refer to the estimate defined earlier in the source.…

Let $n\geq 2$, let $N$ be an odd positive integer, and let $\mathcal{E}$ denote the extension operator for the paraboloid over $\mathbb Z/N\mathbb Z$, acting on functions $H$ on…

Let $q=|\mathbb F_q|$, and let $P\subset\mathbb F_q^d$ be the paraboloid … Write $R_P^*(p\to r)\lesssim1$ for the uniform finite-field extension estimate. Critical endpoint conject…

Let $q=|\mathbb F_q|$, and let $P\subset \mathbb F_q^d$ be the paraboloid … Write $R_P^*(p\to r)\lesssim 1$ when the finite-field extension operator for $P$ is bounded from…