29 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…

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…

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

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 $S$ be a smooth hypersurface in ${\mathbb R}^d$ given by the graph of a smooth function, let $\sigma_1$ be the associated measure, and define … where…

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

Tao's conjecture. For any $1\leq p<2$, the operator $T^\lambda_\ast$ extends boundedly on $L^p(\mathbb R^n)$ whenever

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…

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

Stokolos's reformulation of Zygmund's conjecture. There exists an integer $k$ with $1\leq k\leq n$ such that $M_{\mathcal{B}}$ is sharply bounded from…

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 $L^{1,\infty}$.

Nonconstant fixed-point conjecture. Under these assumptions, there is no non-constant fixed point $f\in L^{1}_{\operatorname{loc}}(\nu)$ of $M_\nu$.

$\Omega$-Kakeya maximal conjecture. Fix any probability measure $\mu$ on $\mathbb{S}^{n-1}$ satisfying, for some $d\in[0,n-1]$,

Let $p e(1/2,1]$ and let $f:\mathbb{Z}tomathbb{R}$ be a function in $ell^{p}(\mathbb{Z})$. Write ${rm Var}p f$ for the $p$-variation of $f$, and let $Mf$ denote the centered discrete…

Density-basis differentiation conjecture. If $M_{\mathcal{B}}f(x)<\infty$ almost everywhere, then

Strong differentiation conjecture. If $\widetilde{M}_{\mathcal{B}}f(x)$ is finite almost everywhere, then $f$ is strongly differentiable.

Let $1<p<q$, let $w\in C_q$, let $T$ be a Calderón--Zygmund operator, and let $f\in C_c^\infty(\mathbb{R}^n)$. Linear quantitative CFI conjecture. The estimate … should hold. The p…

Let $n\ge 2$ and $1< p < \infty$. Let $M^{\beta}_*$ denote the generalized spherical maximal operator on $\mathbb{R}^n$, and let condition (cnd18) be … with the last inequality wea…

Let $M$ be the Hardy–Littlewood maximal operator on $\mathbb{R}^n$, and let $u,v\in A_1$. Sawyer's mixed weak-type conjecture. There exists a dimensional constant $c_n$ such that ……

Sawyer's extrapolation conjecture. There is a finite constant $c$, depending on the $A_1$ constant of $u$ and on $c_v$, such that