32 problems

Let $X$ be a space of homogeneous type, let $M$ be the uncentered Hardy–Littlewood maximal operator, and let $p(39)\in\mathrm{LH}$ satisfy $p_+<\infty$. Let $w$ be a weight, and le…

Separated bump conjecture. Given $0<\alpha<n$ and $1<p\leq q<\infty$, the strong-type inequality for the Riesz potential holds for every pair of weights $(u,\sigma)$ satisfying the…

The $\textup{A}_\infty$ weighted Sobolev conjecture. One has

Reverse Hölder self-improvement conjecture. If $w\in\operatorname{RH}_p$ and $1<p<\infty$, then for

Let $S_{\operatorname{int}}$ be the intrinsic square function and let $w\in A_2$ be a Muckenhoupt weight, with characteristics $[w]_{A_2}$ and $[w]_{A_{\infty}}$. The weighted weak…

Let $\alpha\geq -1/2$, $p\in(1,\infty)$, and let $w\colon\mathbb{R}\to[0,\infty)$ satisfy … where the supremum is over all intervals $B\subset\mathbb{R}$ and $1/p+1/p'=1$. The weig…

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…

Weak-type weighted Sobolev conjecture. For such $w$ and $f$,

The NTV conjecture. If $\mathcal{A}_{2}(\sigma,\omega)$ is finite and both testing characteristics $\mathfrak{T}_{H}(\sigma,\omega)$ and $\mathfrak{T}_{H}(\omega,\sigma)$ are finit…

Let $p_1,\ldots,p_m\in(1,\infty]$ with $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}>0$, and let $w=\prod_{j=1}^m w_j$. Suppose $T$ is well-defined on all $m$-tuples…

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. For a rectangle $R=I_1\times I_2$, let $\pi_R(f)$ denote the projection used in the Poincaré inequality, and let…

Let $1\leq p_1,\dots,p_m<\infty$ and define $p$ by … Let $w_1,\dots,w_m$ and $\nu$ be weights such that $(w_1,\dots,w_m,\nu)\in A_{\vec P}^{\mathcal R}$, and let $v$ be a weight sa…

Sparse form domination conjecture. The same estimate should also hold after replacing $q_0$ by any $q\in(0,q_0]$.

Weaker optimality conjecture. There exist $c_1,c_2,m_1,m_2>0$ such that

Optimality conjecture. There exist $c_1,c_2>0$ such that

Sparse square-function criterion. Suppose that for every $0<\eta<1$ and every $\eta$-sparse family $\mathcal{S}$ of cubes in $\mathbb{R}^d$,

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 $T$ be a one-sided Calderón–Zygmund operator: an operator bounded on $L^2$ with a kernel $K(x,y)$ satisfying … … for fixed $0<\alpha\leq 1$, and $K(x,y)=0$ when $x<y$. For…

Unified weighted-condition conjecture. There is a constant $C$ such that

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

Bump conjecture. There is a constant $C>0$, independent of $w$ and $\sigma$, such that

Hytönen–Lacey one-supremum conjecture. The estimate

Power-improved two-weight conjecture. If $(\mu_1^r,\mu_2^r)\in A_p^+(\mathbb{R}_{+}^2)$ for some $r>1$, then the displayed inequality holds for some $C>0$. This is presented as a v…

Let $1<p<\infty$, let $p'=p/(p-1)$, let $u$ and $v$ be weights, and let $T$ be a Calderón–Zygmund operator. Write $M:L^p(u)\to L^p(v)$ for boundedness of the maximal operator betwe…