25 problems

Let $S_1$ and $S_2$ be compact, transverse subsets of either the light cone or the paraboloid … Write $R^*_{S_1,S_2}(2\times2\to q)$ for the corresponding bilinear restriction esti…

Let $F$ be a finite field of odd characteristic, and let … be the three-dimensional paraboloid, with normalized counting measure $d\sigma$. Write $R^*(2\to r)$ for the best constan…

Let $S$ be the paraboloid and let $R^*(p\to q)$ denote the linear restriction estimate. In dimension $n=2$, the paper obtains the bound $q>4-\frac{2}{3}$ as an improvement over the…

Extension-surface restriction conjecture. These necessary conditions should also be sufficient, at least when $-1$ is not a square. This is the finite-field analogue of the Euclide…

Let $F$ be a finite field, let $S$ be the paraboloid in $F^3$, and suppose that $-1=i^2$ for some $i\in F$, so that $S$ contains lines. Write $R^*(p\to q)$ for the finite-field res…

Let $F$ be a finite field such that $-1$ is not a square, and let … be the paraboloid. Write $R^*(p\to q)$ for the finite-field restriction estimate from $L^p(S,d\sigma)$ to…

Let $U\subset B_{\mathbb{R}^{2}}(0,\frac{1}{2})$ and let $\mathbb{P}^{2}=\{(x_{1},x_{2},x_{1}^{2}+x_{2}^{2}):(x_{1},x_{2})\in U\}$ be the paraboloid. For $f$ on $U$, define the Fou…

Let $d\ge 3$ be odd, let $P$ be the paraboloid in $\mathbb{F}_q^d$, and suppose that $-1\in\mathbb{F}_q$ is a square. Ko's paraboloid restriction conjecture. … This estimate would…

Sharp exponent conjecture. The sharp exponent for obtaining the corresponding product-set lower bound should be $(d-1)/2$.

Let ${\mathcal P}=\{(\omega,|\omega|^2):|\omega|\leq 1\}\subset\mathbb R^3$ be the unit paraboloid, endowed with the measure $\sigma$ obtained by pushing forward Lebesgue measure o…

Foschi–Klainerman conjecture. For every $(q,r)\in\mathbf{\Gamma}$, there is a constant $C$ depending on $q$, $r$, $\varSigma_1$ and $\varSigma_2$ such that

Fix $d\in\mathbb{N}$ and define the extension operator … Here $p'$ denotes the Hölder conjugate of $p$, and $A_p$ is the optimal operator norm. Adjoint Fourier restriction conjectu…

Let $E$ be an anisotropic paraboloid in $\mathbb R^{2n}$ equipped with its canonical symplectic structure, and let $\mathbf 1_E$ denote its indicator function. Anisotropic-parabolo…

Uniform decoupling conjecture. It was conjectured that $D_{n,p}\leq C_p$ for $1\leq p<\frac{2(n+1)}{n-1}$.

Let $P\subset\mathbb F_q^d$ be the paraboloid. Suppose that $d=4k-1$ for some $k\in\mathbb N$ and that $-1\in\mathbb F_q$ is not a square. Sharp paraboloid extension conjecture. On…

Critical paraboloid extension conjecture. The critical estimate

Let $F$ be the finite field under consideration, let $\mathcal P$ be the three-dimensional discrete paraboloid equipped with normalized surface measure $d\sigma$, and let…

Restriction conjecture for the paraboloid. These necessary conditions are the only obstructions to estimates of the form $R^{*}(p\to r)\lesssim 1$. This conjecture seeks a complete…

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 $q=|\mathbb F_q|$, let $P\subset\mathbb F_q^d$ be the paraboloid … and let $1\leq p,r\leq\infty$. In the $(1/p,1/r)$-plane, define the necessary-condition region as follows: fo…

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…

Let $n\ge 2$, let $P^{n-1}(N)=\{\xi=(\xi_1,\ldots,\xi_n)\in\mathbb Z^n:\xi_n=\xi_1^2+\cdots+\xi_{n-1}^2,\ |\xi_1|,\ldots,|\xi_{n-1}|\le N\}$, and let $a_\xi\in\mathbb C$. The notat…

Full-range adjoint restriction conjecture. This inequality should be valid for all $p\in[1,\frac{2d}{d-1})$.

Let $n\ge 3$, and let $S_1$ and $S_2$ be smooth compact non-empty subsets of the paraboloid in ${\mathbf{R}}\times{\mathbf{R}}^{n-1}$ whose unit normals are separated by at least a…

Linear adjoint restriction conjecture. The inequality above holds with constants depending on $S$, $n$ and $p,q$ if and only if $q>\frac{2n}{n-1}$ and…