55 problems

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

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…

Sharp logarithmic norm conjecture. The quantity $(\log \delta^{-1})^{1/p}$ is the sharp $L^p$ operator norm of $\mathcal{N}_{\delta}^{\gamma}$ for $p\in[2,\infty)$; that is, the lo…

Córdoba's maximal Nikodym conjecture. For every $\epsilon>0$, there is a constant $C_\epsilon$ such that

Halo conjecture. The differentiation basis $\mathcal B$ should differentiate every function $f$ such that

DGS conjecture. The number $c_p$ is the operator norm of $M$ on $L^p(\mathbb{R})$, that is,

Finite-field $(n,k)$ maximal-function conjecture. These necessary conditions are also sufficient: $T_{n,k}$ is bounded from $L^p$ to $L^q$ whenever

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

Let $\Phi:\mathbb{Z}^{d-1}\to\mathbb{Z}$ be monotone increasing in each argument, and let $\mathcal{Z}_d$ be the collection of dyadic rectangles in $\mathbb{R}^d$ whose sidelengths…

Let $\mathcal{E}$ be a family of axis-parallel rectangles in $\mathbb{R}^d$, whose sidelengths are monotone increasing functions of $k$ parameters, and let…

Sharp constant conjecture. The inequality holds with $C=1$ for every function $f:\mathbb{R}\to\mathbb{R}$ of bounded variation.

Arithmetic spherical maximal-function conjecture. The arithmetic spherical maximal function $M_\Lambda$ is bounded on $\ell^p(\mathbb{Z}^d)$ for all $p>\eta(\Lambda)$. The precedin…

Global maximal operator conjecture. It is conjectured that $\mathcal{M}_{\mathrm{global}}$ is bounded only for $p>p_c$ for some critical index $p_c>1$, often related to the spectra…

Let $p_{cr}\ge 2$ and let $\Gamma$ be a compact analytic hypersurface. Let $M_\Gamma$ denote its associated maximal operator, let $\mu$ be the measure on $\Gamma$, and let…

Let $M_\Gamma$ be the maximal operator associated with a hypersurface $Gamma$, and let the necessary condition be that … for every hyperplane $H$ not passing through the origin. Io…

Let $\rho>0$ and let $\rho$-Fourier decay mean that a measure $u$ satisfies the decay inequality … Suppose $Gamma$ is a smooth hypersurface and a measure $mu$ defined on $Gamma$ s…

Let $E\subset[1,2]$ have Minkowski dimension $\dim_{\mathrm{M}}E=\beta$ and quasi-Assouad dimension $\dim_{\mathrm{qA}}E=\gamma$. Let $\mathcal{T}_E$ denote the set of exponent pai…

For each integer $n\geq 3$, let $\gamma_n(t)=(t,t^2,\ldots,t^n)$ be the moment curve in $\mathbb{R}^n$, equipped with the associated nonisotropic dilations and Hardy space…

Weak-type $(1,1)$ conjecture. The lacunary maximal function $M_{lac}$ is of weak type $(1,1)$. The corresponding $L^p$ bounds are known for every $p>1$, but the weak-type $(1,1)$ e…

Let $G$ be a symmetric convex body in $\d\mathbb{R}^d$, and let $C(p,\mathbb{R}_+,G)$ denote the best constant in the continuous Hardy--Littlewood maximal inequality … Here…

Best-case equality conjecture. In the best case,

Strong spherical maximal function conjecture. For all $n\geq 2$ and $p>\frac{n+1}{n-1}$, there exists a constant $C_{n,p}>0$ such that

Let $k$ be the number of nonvanishing principal curvatures of the underlying compact smooth hypersurface, and consider the bilinear local maximal operator $\mathrm{M}_\sigma^{loc}$…

Let $m=2$, and let $\mathrm{M}_\sigma^{loc}$ and $\mathrm{M}_\sigma$ denote the multilinear local and global maximal functions associated with a measure whose Fourier transform sat…

Let $E$ be the elliptic maximal function … where the supremum is over all ellipses centered at $(x,y)$ whose semi-major and semi-minor axes have lengths in $[1/2,2]$. Erdogan's con…