448 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…

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 connected complex reductive algebraic group $G$ and every involutive automorphism $\theta:G\to G$, let $H=G^\theta$ and let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{s}$…

With $\mu_a$ the biased-coin product probability measure on the product group $G$ defined in Section 3, Conjecture 6 states: Given $a>0$, there exist $C(a)>0$ such that for…

Naive bilinear Kakeya conjecture. Let $\varepsilon>0$. For every $\delta\in(0,1)$ and every pair of transversal families of Heisenberg $\delta$-tubes…

Let $X$ be a space of homogeneous type, let $M$ be the uncentered Hardy–Littlewood maximal operator, and let $p(39)\in\mathrm{LH}$ satisfy $p_+<\infty$. Let $w$ be a weight, and le…

Let $n$ be a positive integer, let $g\in L^{2}(\mathbb{R}^{n})$ be nonzero, and for $x,\omega\in\mathbb{R}^{n}$ let $T_x$ and $M_\omega$ denote the translation and modulation opera…

Consider the upper half-plane $\mathbb{R}_{+}^2=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$, two weights $\mu_1$ and $\mu_2$ on $\mathbb{R}_{+}^2$, and the two-weight inequality … Her…

Separated bump conjecture. Given $0<\alpha<n$ and $1<p\leq q<\infty$, the strong-type inequality for the Riesz potential holds for every pair of weights $(u,\sigma)$ satisfying the…

Scattering convergence and continuity conjecture. Condition $$ holds for all $fin {bf L}^1({{mathbb{R}}}^d)$. Moreover, if $fin {bf L}^1({{mathbb{R}}}^d)$, then $overline S f(q)…

Let $I^{d}$ denote the $d$-dimensional unit cube, and let $L\subset\mathbb{R}^{d}$. The cube spectral-tiling conjecture. $(I^{d},L)$ is a spectral pair if and only if $(I^{d},L)$ i…

Let $L\subset\mathbb{R}^{d}$. A spectrum is a set $L$ for which there exists a set $\Omega$ such that $(\Omega,L)$ is a spectral pair, and a tiling set is a set $L$ for which there…

Bilinear Kakeya range conjecture. $K^*(q'\times q'\to p'/2)$ holds if and only if

Bilinear restriction range conjecture. If $n\geq2$, then $R^*(p\times p\to q)$ holds whenever

The sharp restriction conjecture. $R^*(p\to q)$ holds whenever

The smoothness conjecture. If $\operatorname{codim}\mathcal{F}>0$ and the leaves are dense, then

The subunit-$L^p$ discrepancy conjecture. For every dimension $d\geq2$ and every $0<p<1$, there is a constant depending only on $d$ and $p$ such that

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

The short square-set energy conjecture. One should have

For a finite set $E$ of squares in the short interval … let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The local energy conjecture. There exists…

The short interval conjecture. For every $\alpha<1$, every such trigonometric polynomial satisfies

For a set $E\subset\{1^2,\ldots,N^2\}$ of squares, let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The logarithmic energy conjecture. There exists a…

Logarithmic Fourier-norm conjecture. If, for every finite subgroup $V\leq\widehat G$ with $|V|\leq M$, one has

Modified Newton distance conjecture. If $S\in\mathfrak S^m\mathbb R^{n_X+n_Z}$, then

Let $J_H(y)=\boldsymbol{1}_{[-1,1]}(y)y^{-1}$ be the Hilbert transform kernel with a sharp cutoff, and let $\operatorname{Osc}_n(J_H;f)$ denote the corresponding oscillation operat…