1,191 problems

A measurable set $E\subset\mathbb{R}^{d}$ of positive Lebesgue measure is spectral if there exists a countable set $\Lambda\subset\mathbb{R}^{d}$ such that the exponential family ……

A Kakeya set in $\mathbf{R}^d$ is a subset containing a unit line segment in every direction. The Hausdorff dimension of a set is its Hausdorff dimension in the ambient Euclidean s…

For an integer $d\geq 2$, let $\mathbb C^d$ be the $d$-dimensional complex vector space. For $x,\omega\in\mathbb Z_d$, define the cyclic translation and modulation operators by … w…

Given $1 \leq p \leq \infty$, define the Bochner–Riesz means … The Bochner–Riesz conjecture. As soon as $\alpha > 0$, there is a range of exponents $p$ for which … for all…

Let $\mathbb{T}^d:=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ be the $d$-dimensional torus, and let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction with eigenvalue parameter…

Stein's square function conjecture. For $p>2$, the estimate

Let a frame be a sequence in a Hilbert space that provides stable expansions, and call it bounded below in norm when its elements have norms uniformly bounded away from zero. Feich…

Let $K$ be a compact connected Lie group with normalized Haar measure $dg$, and let $f$ and $g$ be complex-valued $K$-finite functions. Mathieu's conjecture. If … then … for all su…

Let $u$ solve the wave equation Cauchy problem … with $u(x,0)=f(x)$ and $\partial_tu(x,0)=0$ on $\mathbb{R}^d\times\mathbb{R}$. For $s_p=\frac{d-1}{2}-\frac{d-1}{p}$, Sogge's conje…

Let $w$ be a weight or measure, let $f$ be a suitable integrable function on bR}^n, and let $\textup{H}$ denote the Hilbert transform and $\textup{M}$ the Hardy–Littlewood maxima…

Mizohata–Takeuchi conjecture. If $\Sigma$ is convex, then for all $f\in L^2(\Sigma)$ and measurable weights $w\colon\mathbb{R}^n\to[0,\infty)$,

Let $u$ and $v$ be weights in the Muckenhoupt class $A_1$, and let $H$ denote the Hilbert transform. Sawyer proved that the operator $S_v f=M(fv)v^{-1}$ satisfies a mixed weak-type…

Let $f$ be a square-integrable function, that is, $f\in L^2$ on the relevant domain, and consider its Fourier series. Luzin's conjecture. The Fourier series of every square-integra…

For $p\in(1,\infty)$, let $T_{BA}$ be the Beurling–Ahlfors transform acting on $L^p(\mathbb C)$, and let $p^*=\max(p',p)$, where $p'$ is the Hölder conjugate of $p$. Its norm satis…

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…

Iosevich–Sawyer conjecture. For $p>2$, this condition is necessary and sufficient for the boundedness of $\mathcal{M}$ on $L^p$.

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

Zygmund's conjecture I. The maximal operator $M_{B_{\boldsymbol{\phi}}}$ is bounded from $L^1\left(1+\log^+L^1\right)^{k-1}$ to $L^{1,\infty}$.

Stein's conjecture. The operator $H_{\Gamma}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1<p<\infty$.

For $R>0$, let $S_R^{\alpha}$ be the Bochner–Riesz means on $\mathbb{R}^n$, and let $S_*^{\alpha}f(x)=\sup_{R>0}|S_R^{\alpha}f(x)|$. Tao's maximal Bochner–Riesz conjecture. For…

For $delta\bin(0,1)$, define the Nikodým maximal operator by … where the supremum is over all $delta$-tubes $T$ centered at $x$, and a $delta$-tube is the closed $(delta/2)$-neighb…

Let $d\geq 2$, let $s>0$, and set $1/\tau=s/(d-1)$. The relevant approximation error is measured in the Triebel–Lizorkin space $\mathcal{F}^{0q}_\infty(\mathbb{S}^{d-1})$ using non…

Let $M$ be a complete Riemannian manifold, and let $R_M$ be its Riesz transform … For $1<p<2$, write $\mathrm{L}^p(M)$ for the scalar $L^p$ space and…

Let $1<p<2$. A function in $L^p(\mathbb R)$ is a cyclic vector with respect to translations if the closed linear span of its translates is all of $L^p(\mathbb R)$. N. Wiener charac…

Let $G$ be a reductive group over $F$, let $\rho$ be the representation defining the reductive monoid $X_\rho$, and let $C^(X_\rho(F))$ be the space of locally constant functions o…