53 problems

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 … 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 $E$ 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…

Let $0<\delta\leq\nu<1$. Three families $\mathbb{T}_{1},\mathbb{T}_{2},\mathbb{T}_{3}$ of $\delta$-tubes are $\nu$-disjoint when their orientations lie in spherical patches of diam…

Let $B_R$ be the ball of radius $R$ centered at the origin, let $\mathbb{T}$ be the relevant family of tubes, and write $w(T)$ for the weight of a tube. Under the hypotheses of the…

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 $d\geq 2$ 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…

The restriction conjecture for the sphere. This estimate should hold when

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

Let $S\subset\mathbb{R}^n$ be a compact $C^2$ hypersurface, possibly with boundary, with strictly positive second fundamental form. Let $E_S$ be its Fourier extension operator. Loc…

Let $\delta\in(0,1)$, and let $(L,Y)_\delta$ be a set of directionally $\delta$-separated lines in $\mathbb{R}^n$ with an $(\varepsilon_1,\varepsilon_2)$ two-ends, $\lambda$-dense…

Let $2\le q<p<\infty$, and let $\Phi$ be a finite, uniformly bounded orthonormal system. For a finite collection $\Theta$ of functions, define $K_p^*(\Theta)$ to be the smallest co…

Let $\mathcal T^\lambda$ be the operator defined earlier in the paper, let $B_\lambda$ be its spatial domain, and let $\{\Gamma_{\theta,v}^\lambda\}_{\theta,v}$ be the associated c…

Large-set planar conjecture. If $|Y|\geq R^{1/2}$, then is true with

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…

Let $fbf{Q}(fxi)=(Q_1(fxi),\ldots,Q_n(fxi))$ be an $n$-tuple of real quadratic forms in $d$ variables, and let … For a rectangle $B\subset[0,1]^d$, call $f$ adapted to $B$ if…

Let $n=2d\ge 4$ be even, let $\mathcal{F}$ be the flat disk in $\mathbb{F}_q^n$, and let $R^*_{\mathcal{F}}(p\to r)$ denote its restriction operator estimate. If $(1/p,1/r)$ lies i…

Sphere extension conjecture. The $L^2\to L^{(2n+4)/n}$ extension estimate should hold uniformly in $q$:

Let … where $a_m$ is supported in … For $\lambda>0$, define the superlevel set $U_\lambda(\nu)=\{(x,t):|\nu(x,t)|\geq\lambda\}$. Periodic restriction conjecture. If $|a_m|\leq 1$ f…

Let $P^{n-1} = \{\xi_n = \sum_{j=1}^{n-1} \xi_j^2,\ |\xi_j| \leq 1,\ \forall 1 \leq j \leq n-1\}$ be the truncated unit paraboloid, let $\mathrm{d}\sigma$ be its natural surface me…

Let $n\geq 3$ and let $T^{\lambda}$ be a Hörmander-type operator satisfying the straight condition: for each $\xi$, the associated quantity $G(x,\xi)$ is invariant under changes in…

Let $n\geq 3$, let $\alpha\in(0,1)\cup(1,\infty)$, and define the restriction-type operator … where $e(t):=e^{2\pi it}$, $\psi$ is compactly supported and smooth, and…

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…