85 problems

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 $n\geq 3$, and let … with $x\in\mathbb{R}^{n-1}$, $t\in\mathbb{R}$ and $y\in\mathbb{R}^{n-1}$. For a smooth amplitude $a$ supported near the origin, define the associated Hörma…

Mixed-norm boundedness conjecture. If $\Sigma(Q_1,Q_2)$ is well-curved, then these necessary conditions are sufficient for the mixed-norm estimate, with the possible exception of t…

Constant-extremizer conjecture. Constant functions are the only extremizers, modulo symmetries, for the Tomas–Stein inequality for every $d\geq 2$. The conjecture concerns the clas…

Cone restriction conjecture. The estimate

Guo–Wang–Zhang conjecture. If $\phi$ satisfies Bourgain's condition, then $\mathbf{H}_{p\to p}(\phi)$ holds for every

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…

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

Carleson–Sj"olin model-phase conjecture. The operator norm should satisfy

Folding-operator extension conjecture. This behavior should remain true for general oscillatory integral operators with folding canonical relations satisfying the elliptical curvat…

Let $d\geq 2$, let $S^{d-1}\subseteq\mathbb{R}^d$ be the unit sphere, and let $\sigma$ be its surface measure. For a continuous function $f:S^{d-1}\to\mathbb{C}$ with…

Restriction conjecture. For every bounded function $f$ on the sphere, $\widehat{f\,d\sigma}$ belongs to $L^p(\mathbb{R}^n)$ for every $p>\frac{2n}{n-1}$.

Let $\mathbb{F}$ be the finite field under consideration, let $j\in\mathbb{F}^\times$, and let $S_j$ be the sphere centered at the origin of radius $j$. Write $R_{S_j}^*(2\to r)$ f…

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 $\phi$ be a H"ormander phase function satisfying Bourgain's condition at every point. Let $p$ be the exponent in the restriction estimate … should hold for all … The conjecture…

Let $d\geq n$, let $\Sigma\subset\mathbb{R}^{d+n}$ be a smooth $d$-dimensional submanifold, and let $\mathcal{T}(\Sigma)$ be its Fourier restriction trapezoid. Let $\mathcal{T}_0$…

Let $\Sigma\subset\mathbb{R}^{d+n}$ be a $d$-dimensional submanifold, and let $\mathcal{T}(\Sigma)$ be the right-angled trapezoid of exponents $(1/p,1/q)$ for which the correspondi…

Let $d$ and $n$ be the dimensions appearing in the paper, and let the (CM) condition denote the stated integrability condition for the Hessian determinant. Banner's endpoint Stein–…

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 $R\geq1$,

Let $S\subset\mathbb{R}^n$ be a smooth compact hypersurface with non-vanishing Gaussian curvature, and consider the restriction estimate in Stein's restriction conjecture. Wang–Wu'…