15 problems

Local Mizohata-Takeuchi conjecture. For every $\alpha>0$, one has

Let $S$ be the relevant two-dimensional paraboloid and let $E_Sf$ denote its extension operator. For $p,q\in[1,\infty]$, use the mixed norm $L^q_{x_1}L^\infty_{x_2}$. Maximal exten…

Let $0<\delta<1$ and $\kappa\in\mathbb{N}$ with $\kappa>\frac{20}{\delta}$. For $1<q<\infty$, $0<\varepsilon,\nu<1$, let…

Let $U_1,U_2,U_3\subset\mathbb{R}^2$ be caps for the two-dimensional paraboloid satisfying the stated transversality hypothesis, and let $\mathcal{E}_{U_j}$ be the corresponding ex…

For each $1\leq j\leq m$, let $\Sigma_j:U_j\to\mathbb{R}^n$ be a smooth parametrization of an $n_j$-dimensional submanifold $S_j$ by a neighborhood $U_j$ of the origin in…

Let $k\geq2$, and let $U_1,\ldots,U_k$ parametrize transversal caps of the paraboloid $x\mapsto|x|^2$ in $\mathbb{R}^{d+1}$. For functions $g_j$ on $U_j$, write $p'$ for the Hölder…

Let $2\leq k\leq d+1$, let $U_j\subset\mathbb{R}^d$ be compact connected domains, and let … Assume that the caps parametrized by $U_j$ are transversal. The multilinear extension co…

Endpoint restriction conjecture. For every $\varepsilon>0$,

Restriction conjecture. The estimate

For $n\geq 2$, let $U$ be an open subset of $\mathbb{R}^{n-1}$, let $\phi:\mathbb{R}^{n-1}\to\mathbb{R}$ be smooth, and let $S=\{(\xi,\phi(\xi)):\xi\in U\}$ be the corresponding hy…

Let $d\ge 2$ be even, and let $S_j\subset\mathbb F_q^d$ be the sphere of radius $j\ne 0$ centered at the origin. Even-dimensional sphere extension conjecture. One has … This estima…

Let $S_j\subset\mathbb F_q^d$ be a sphere of nonzero radius $j$. Odd-dimensional sphere extension conjecture. The following estimates should hold: 1. If $d=4k+1$, $k\in\mathbb N$,…

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…