58 problems

Iosevich–Sawyer conjecture. For $p>2$, this condition is necessary and sufficient for the boundedness of $\mathcal{M}$ on $L^p$.

Zygmund's conjecture I. The maximal operator $M_{B_{\boldsymbol{\phi}}}$ is bounded from $L^1\left(1+\log^+L^1\right)^{k-1}$ to $L^{1,\infty}$.

For $R>0$, let $S_R^{\alpha}$ be the Bochner–Riesz means on $\mathbb{R}^n$, and let $S_*^{\alpha}f(x)=\sup_{R>0}|S_R^{\alpha}f(x)|$. Tao's maximal Bochner–Riesz conjecture. For…

Let $G$ be a Métivier group with sub-Laplacian $\mathcal{L}$, and let $S_*^{\alpha}(\mathcal{L})$ denote the maximal Bochner–Riesz operator associated with $\mathcal{L}$. Write…

Stein–Iosevich–Sawyer conjecture. If $0<q\leq \tfrac12$ and this estimate holds, then $\mathcal{M}$ is bounded on $L^p$ for every $p>1/q$.

Nieraeth's conjecture. If $M$ is bounded on $X$, then $M$ is bounded on $X'$.

Zygmund's conjecture. The operator $M_{\Gamma}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1<p<\infty$.

Sharp $L^p$ boundedness conjecture. For general $\Lambda$, the operator $M^\Lambda$ is bounded on $L^p$ whenever

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…

Lacunary spherical summability conjecture. The operator $S_{\mathrm{lac}}^*$ maps $L^2(\mathbb R^2)$ into weak $L^2(\mathbb R^2)$. This is a restricted version of the spherical sum…

Spherical summability conjecture. The operator $S_{\mathrm{sph}}^*$ maps $L^2(\mathbb R^2)$ into weak $L^2(\mathbb R^2)$. The question concerns almost-everywhere spherical summabil…

Let $d\ge 3$, let $p_d=d/(d-1)$, and let $E$ be the set defining the spherical maximal operator $\cM_E$. For each integer $k$ and relevant $0<\delta<1$, write $N(E^k,2^k\delta)$ fo…

Sharp $L^p$ boundedness conjecture. The $L^p$ boundedness of $M_{\gamma_n}$ holds if and only if $p>n$.

Let $0<\delta<1$. A $\delta$-tube is a tube of length $1$ and cross-sectional diameter $\delta$, and a family $\mathbb{T}$ of $\delta$-tubes is $\delta$-separated when distinct tub…

Let $S$ be the relevant two-dimensional paraboloid and let $E_Sf$ denote its extension operator. For $p,q\in[1,\infty]$, use the mixed norm $L^q_{x_1}L^\infty_{x_2}$. Maximal exten…

Let $I$ be a compact interval. For $p,q\in[1,\infty]$, write $L^q_xL^\infty_t(\mathbb{R}\times I)$ for the corresponding mixed-norm space. For $\alpha\in\mathbb{R}$, let…

Maximal-operator conjecture. The operator $M_\gamma$ maps $L^p(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedly if and only if $p>n$. This conjecture concerns the $L^p$ theory of geom…

Let $M$ be a Cartan–Hadamard manifold with sectional curvature $K_\sigma$ satisfying … and let $1<a<b<2a$. Write $\mathscr{M}$ for the Hardy–Littlewood maximal operator on $M$. Max…

Let $v(cdot)$ be a Lipschitz vector field on $\mathbb{R}^2$, and let $M_{v(cdot)}$ denote the associated maximal function; consider its truncated version. Zygmund's conjecture. The…

Let $0<\varepsilon<1$, and let $\mathcal{K}^{\ast}(\otimes_1L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon)$ be the statement that for every $0<\delta<1$ and every family…

The planar $L^{3/2}$ conjecture. The estimate

Let $M$ be the centered Hardy--Littlewood maximal operator on $fR$, and let $f\in\operatorname{BV}(fR)$ have limits $f(\infty)$ and $f(-\infty)$ at the two ends. Write…

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

Fix exponents $q3e=1$ and $varsigma>0$ (or $03cvarsigma3c=1$), and write … Let $u$ and $v$ be positive, measurable functions such that $u^{\varsigma}\in A_{\varrho}^{\mathcal R}$ a…

The $n/k$ to $n$ norm conjecture for ${\mathcal N}^k$. There is a constant $C_n$, depending on $n$ and ${\mathbb F}$, such that