15 problems

Small cap decoupling conjecture. The inequality in Theorem 11 should hold in the larger range $|\vec{\alpha}|\leq n-1$.

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

Let $R$ be a large parameter, let $B_R$ be a ball or cube of radius or side length $R$ in ${\mathbb R}^3$, and let $E_SF$ be the extension operator for the two-dimensional parabolo…

Let $R$ be a large parameter, let $B_R$ be a ball or cube of radius or side length $R$ in ${\mathbb R}^3$, and let $E_SF$ denote the Fourier extension operator associated to the tw…

Restriction conjecture. The estimate

Sharp adjoint restriction conjecture. It is conjectured that

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…

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…

Let $E$ be the extension operator associated to the paraboloid in $\mathbb{R}^3$, defined by … where $e(a):=e^{2\pi ia}$ for $a\in\mathbb{R}$. The restriction conjecture. The estim…

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 $\tau_1,\tau_2,\tau_3\subset[-1,1]^3$ be cubes of side length comparable to $1$ satisfying … where $n(\xi)=(-2\xi_1,-2\xi_2,-2\xi_3,1)$ is the normal vector to the paraboloid.…

$k$-linear restriction conjecture. The estimate

Let $S$ be a non-empty smooth compact subset of the paraboloid … where $n\geq1$, and let $\mathrm{d}\sigma$ be the pull-back of $n$-dimensional Lebesgue measure under the projectio…

For a finite field $\mathbb{F}$, let … be the three-dimensional paraboloid, and let $\mathcal{R}^{*}(p\rightarrow q)$ denote the best constant in the extension estimate … Assume th…