15 problems

$(n,k)$ maximal-function conjecture. For all $\epsilon>0$ and $0<\delta\ll1$,

Let $\Phi:\mathbb{Z}^{d-1}\to\mathbb{Z}$ be monotone increasing in each argument, and let $\mathcal{Z}_d$ be the collection of dyadic rectangles in $\mathbb{R}^d$ whose sidelengths…

Let $p_{cr}\ge 2$ and let $\Gamma$ be a compact analytic hypersurface. Let $M_\Gamma$ denote its associated maximal operator, let $\mu$ be the measure on $\Gamma$, and let…

Let $\rho>0$ and let $\rho$-Fourier decay mean that a measure $u$ satisfies the decay inequality … Suppose $\Gamma$ is a smooth hypersurface and a measure $mu$ defined on $\Gamma$ s…

Stein's conjecture. For every $p\in(1,\infty]$ there is a constant $C_p>0$ such that

Strong spherical maximal function conjecture. For all $n\geq 2$ and $p>\frac{n+1}{n-1}$, there exists a constant $C_{n,p}>0$ such that

For $1\leq d<n$, let $\mathcal N_\delta$ denote the $(d,n)$-Nikodym maximal function. For $1\leq p\leq n-d+1$, set $q=p'(n-d)$, where $p'$ is the Hölder conjugate of $p$. Off-diago…

Nikodym maximal-function conjecture. For $1<p<n-d+1$,

Let $d\in\mathbb{N}$ and let $B^q(d)\subset\mathbb{R}^d$ be a $q$-ball. The discrete restricted maximal function is denoted by $\mathcal{M}_{\ast,>D}^{G}$, and $\mathcal{C}(G,p)$ d…

Magyar–Stein–Wainger conjecture. If $p>2$, then

Let $d\geq5$, let $\mathcal{R}$ be a lacunary subsequence of $\mathcal{R}_{full}$, and let $A_{\mathcal{R}}^*$ be the associated discrete spherical maximal operator. Weak-type…

Let $A(1)=\{\xi:|\xi|\sim 1\}$. For $R\geq 1$, let $B(0,R)$ denote the ball of radius $R$ centered at the origin, and let $e^{it\Delta}f$ be the free Schrödinger evolution of a fun…

Let $\Phi(x,y)=\rho(x,y)$, where $\rho$ is a Riemannian metric sufficiently close to the Euclidean metric. The bounded conic intersection conjecture. Then $\Phi$ satisfies the boun…

Fixed-width Lipschitz Kakeya conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$, all Lipschitz vector fields $v$, and…

Lipschitz Kakeya maximal-function conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$ and all Lipschitz vector fields $v$, the operator…