12 problems

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…

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

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…

Multiplier conjecture. For $1\leq p\leq\infty$, the estimate

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…

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

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 $\chi$ 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…

Bañuelos's conjecture. For every $1<p<\infty$, $T_{m_r}$ is bounded on $L^p(\mathbb{R}^n)$ and

Let $\mathbb{S}$ be the unit sphere in $\mathbb{R}^n$, let $\varphi:\mathbb{S}\to\mathbb{C}$ satisfy $\|\varphi\|_{L^{\infty}}\leq 1$, let $0<r<\infty$ and $n\geq 2$, and let…