107 problems

Let … be the graph of a smooth function $h$, and define the extension operator … Write $q'=\frac{q}{q-1}$. Stein's restriction conjecture. If the Gaussian curvature of $\mathcal S$…

Let $d\geq 2$ and let $\vec{v}=(v_1,\ldots,v_{d-1})\in\{-1,1\}^{d-1}$. Consider the hyperbolic paraboloid … For $f$ on $[0,1]^{d-1}$, define its extension operator by … where…

Let $\phi(x,\xi)$ be a Hörmander-type phase function, and let $G(x,\xi)$ denote its associated Gauss map. Suppose that Bourgain's condition holds: … for some scalar function…

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…

Normalized local restriction conjecture. If $p\geq\frac{2n}{n-1}$, then for every $\varepsilon>0$ and $R>1$,

Restriction conjecture. If $p\le p_{d-1}=d(d-1)$, then

Let $d\geq 2$, $N\geq 1$, and define … … Let $\mathcal{M}$ be a real analytic hypersurface in $\mathbb{T}^d$ with nonnegative curvature and surface measure $\sigma$. Paraboloid res…

Consider a family of oscillatory integral operators associated with a phase function satisfying the natural non-degeneracy conditions appropriate to the variable-coefficient settin…

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…

Let $n\geq 2$, let $p=2n/(n+1)$, and let the oscillatory sum and square function be the quantities denoted by and in the source. Endpoint oscillatory sum conjecture. One should be…

Bilinear restriction conjecture. These necessary conditions are sharp: they suffice for the corresponding bilinear restriction estimate. This conjecture is still open except in dim…

Let $S$ be a compact subset of the sphere or paraboloid, or a compact subset of the cone, and let $R^*_S(q' \to p')$ denote the corresponding adjoint restriction estimate. For the…

Let $n\geq 2$, and let the oscillatory expression and square function be the quantities denoted by and in the source. Let $p=2n/(n+1)$, and write $R$ for the scale parameter. Endpo…

The bilinear restriction conjecture. These necessary conditions are sharp. The conjecture is still open except in dimension two, and, up to endpoints, it is equivalent to the usual…

Let $S$ be a compact subset of the sphere, paraboloid, or cone in the relevant dimension, and let $R^*_S(q' \to p')$ denote the corresponding adjoint restriction estimate. For the…

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…

Smooth volume density conjecture. The optimal decay of measures supported on $M$ may be achieved by a smooth volume density on $M$, that is, by a Knapp-type example in restriction…

Let $M$ be a three-dimensional analytic manifold, let $R_{\varepsilon,N}f$ denote the operator defined in the source, and let $p$ be an exponent. Restriction conjecture. For every…

Let $d\geq 4$ be even, and consider the sharp $L^2$ restriction estimate for finite-field spheres in dimension $d$. Write $R_{S_j^{d-1}}(p\to 2)\lesssim 1$ for the corresponding re…

Let $E$ be the extension operator for the truncated parabola, corresponding to $\Gamma(\xi)=\xi^2$, and let $\omega:B_R\to[0,\infty)$ be a one-dimensional weight, so that … for eve…

Let $\Gamma$ be a convex $C^2$ function on $[0,1]$, let $E_\Gamma$ be its extension operator, and let $\omega:B_R\to[0,\infty)$ be a one-dimensional weight, meaning … for every bal…