495 problems

Let $\mathcal{R}f=(R_1f,\ldots,R_nf)$ be the vector Riesz transform on $\mathbb{R}^n$, where $R_j$ is the Fourier multiplier with symbol $-i\xi_j/|\xi|$. Does there exist a univers…

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…

Orthonormal wavelet implies wavelet-set conjecture. For each pair $(B,\Gamma^*)$ such that there exists a $(B,\Gamma^*)$ orthonormal wavelet, there exists an $(A,\Gamma)$ wavelet s…

Cone-and-cylinder Fourier-dimension conjecture. Then

Setting. For a Navier-Stokes velocity field with vorticity $\omega$, direction $\xi = \omega/|\omega|$, and peak set $P(t)$ (the high-vorticity region), define the scale-invariant…

For every initial datum $u_0\in\dot H^{1/2}(\mathbb R^3)$ and every time interval $I$ containing $0$, if $u,v\in C(I;\dot H^{1/2}(\mathbb R^3))$ are solutions of the cubic nonlinea…

For every integer $n\ge 0$ and every nonzero polynomial $p$ of degree $n$ whose zeros all lie on the unit circle $\mathbb{T}$, define $Q_n(z)=1+z^n$. For all $0\le s\le t\le\infty$…

For every fixed integer $d\ge 1$, let $C_{\rm LY}(\beta,d)$ denote the optimal constant such that every positive solution $u$ of $\partial_tu+(-\Delta)^{\beta/2}u=0$ on…

For all locally compact groups $G$ and $H$, prove that $\Lambda_{\mathrm{WH}}(G\times H)=\Lambda_{\mathrm{WH}}(G)\Lambda_{\mathrm{WH}}(H)$.

The Schiffer conjecture asserts that if a bounded connected domain $\Omega\subset\mathbb{R}^n$ with sufficiently smooth boundary admits some $\lambda>0$ and a nonconstant function…

For each dimension $N\ge 2$ and higher order $m\ge 2$, determine the sharp constants and all extremal divergence-free vector fields in the corresponding higher-order Heisenberg unc…

Let $n\ge 2$, let $\mathcal P^{n-1}=\{(\xi,|\xi|^2):\xi\in\mathbb R^{n-1}\}\subset\mathbb R^n$, and let $f$ satisfy…

Given an analytic family $\mathcal{F}$ of curves in $\mathbb{R}^{2}$, characterize when there exists a Borel set $E\subset\mathbb{R}^{2}$ with Lebesgue measure $|E|=0$ such that, f…

For every compact group $G$, the central Fourier algebra $\operatorname{ZA}(G)$ is amenable if and only if $G$ is virtually abelian, i.e. $G$ has an abelian subgroup of finite inde…

Let $G$ be a compact connected abelian group. Is there a sufficiently small-measure regime in which every compact set $A\subseteq G$ satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained…

Determine the set of thresholds $\delta>0$ for which there exists a constant $C_M(\delta)<\infty$, independent of the locally compact abelian group $G$, such that every measure…

For every exponent $1<p<\infty$, the Beurling--Ahlfors transform $B$ satisfies…

For $d\ge 3$, consider the mass-critical Hartree equation $i\partial_t u+\Delta u=\mu\bigl(|x|^{-2}*|u|^2\bigr)u$ on $\mathbb{R}^d$, with $u(0)=u_0\in L^2(\mathbb{R}^d)$ and…

Let $\varphi$ be a highest-wave profile for a Whitham-type equation, with crest at $x=0$ and crest regularity $\varphi\in C^{s}$ for some $s\in(0,1)$. The conjecture asserts that,…

For every $n\ge 2$ and complexity bound $E$, there is a constant $C_{n,E}$ such that the following holds. Let $0<\delta,\lambda\le 1$, let $\mathbb{T}$ be a family of unit-length…

Let $(M,g)$ be a smooth closed Riemannian manifold of dimension at least $3$, let $\Sigma\subset M$ be a fixed smooth submanifold of codimension $2$, and let $\phi_\lambda$ be an…

Let $(X,\mu)$ be a probability space and let $(\varphi_n)_{n\ge 1}$ be a complete uniformly bounded orthonormal system in $L^2(X,\mu)$, meaning that…

Let $n\geq 2$, let $u$ solve $\partial_t^2u-\Delta_xu=0$ on $\mathbb{R}^{n+1}$ with $u(\cdot,0)=u_0\in H^s(\mathbb{R}^n)$ and $\partial_tu(\cdot,0)=u_1\in H^{s-1}(\mathbb{R}^n)$, w…

Let $G$ be a Kunze--Stein locally compact group, let $\lambda_G$ denote its left regular representation, and let $\pi$ be a unitary representation of $G$ such that…

Let $\mu_{\gamma,\mathrm{GMC}}$ be the one-dimensional Gaussian multiplicative chaos measure on the circle associated with a log-correlated Gaussian field. The Garban--Vargas conje…