28 problems

Let $\mathbb T$ be the circle, let $\varphi$ be a continuous self-mapping of $\mathbb T$, and let $A(\mathbb T)$ denote the Wiener algebra. The Beurling–Helson condition is that ……

Radial Fourier multiplier conjecture. The inequality should hold for

Restricted lacunary-sided polygon conjecture. For $2\le p<\infty$, the maximal operator $f\mapsto\sup_{k\in\mathbb N}|T_kf|$ maps $L^p(\mathbb R^2)$ into itself. The source calls t…

Let $V_2$ be the class of multipliers whose frequency components have uniformly bounded $2$-variation. The spaces $H^1$ and $L\log^{1/2}L$ are understood locally, and…

Let $1\leq p<\infty$, let $m$ be a bounded function, and write $\mathcal M_p$ for the Fourier multiplier class on $L^p$. The implication … holds. Converse-growth conjecture. The co…

Hörmander's $C^1$ conjecture. If $\varphi\in C^1$ satisfies this uniform boundedness condition, then $\varphi$ is linear. The condition is associated with unimodular Fourier multip…

Let $n\geq 3$, write $\xi=(\xi',\xi_n)\in\mathbb{R}^n$ with $\xi'\in\mathbb{R}^{n-1}$, and let $\lambda>0$ and $R>0$. For a non-negative function $\varphi\in C_c^{\infty}([1/2,2])$…

Technicality conjecture. The restriction $\beta\in\{0\}\cup(\frac{1}{2},+\infty)$ in the improved $L^1$ estimate

Cone maximal function conjecture. The operator $\widetilde M_\varphi$ is bounded on $L^2(\mathbb{R}^n)$.

Fourier multiplier conjecture. The function $\mathcal{F}(\lambda)$ is a Fourier multiplier from $H^1(\mathbb{R}_+^2\times\mathbb{R}_+^2)$ into $L^1$ if and only if there exists a c…

Free-group Fourier multiplier conjecture. There exists a completely bounded Fourier multiplier

Inner amenability–$(\kappa_\infty)$ conjecture. Inner amenability of $G$ is equivalent to property $(\kappa_\infty)$.

Inner amenability and decomposable multipliers conjecture. The group $G$ is inner amenable if and only if

The nonlinear Schrödinger equation under consideration is posed on the torus $\mathbb T^d$ with a bounded Fourier multiplier in the linear part; the relevant multiplier is expected…

Let $\mathrm{G}$ be an $n$-dimensional stratified Lie group with homogeneous dilation weights $(\ell_1,\ell_2,\ldots,\ell_n)$, rescaled so that $\min(\ell_j)=1$, and let…

Let $\mathrm{T}\colon S^{d-1}\to\mathbb{R}^{\ell}$ be sufficiently smooth, at least Hölder continuous, and weakly cancelling: … For $\alpha\in(0,d)$, define … let…

Characteristic tame cuts conjecture. Cocompact lattices in higher rank semisimple Lie groups have characteristic tame cuts.

Amenability characterization. $\Gamma$ is amenable if and only if

Let $\mathbb{G}$ be a locally compact quantum group and let $1<p{\leqslant}\infty$. Write $\mathfrak{M}^{1,p}(\mathbb{G})$ for the space of the relevant $(1,p)$-multipliers and…

Optimal logarithmic power conjecture. If $K'<\infty$, then there is a constant $C=C(s_1,\dots,s_n,p,n,d,\psi)$ such that every $f\in\mathscr C_0^\infty(\mathbb R^n)$ satisfies

Let $m$ be a compactly supported radial multiplier on $\mathbb{R}^N$, let $T_m$ be the associated Fourier multiplier operator, and let $\check{m}$ denote its convolution kernel. Ra…

Assume $d\geq 2$, $\alpha\in(0,1)$, and define … For the operator $\widetilde T_\alpha h=\mathcal F_d^{-1}\left(1_B(\cdot)(1-|\cdot|^2)^{-\alpha}\mathcal F_dh\right)$, the set…

Let $\psi$ be a smooth function on $I^d$, let $\chi_\circ$ be smooth and supported near the origin, and define … where $\xi=(\zeta,\tau)\in\mathbb R^{d-1}\times\mathbb R$. Assume t…

Let $\chi$ be a Schwartz function satisfying … and let $S_\tau$ be the Fourier multiplier with symbol $\chi((|\xi|-1)/\tau)$, where $0<\tau<1$. Thin-annuli formulation of the Bochn…

Let the disc multiplier be the Fourier multiplier on $\mathbb{R}^d$ whose symbol is the indicator function of a Euclidean ball. A weighted inequality of the general form … should h…