42 problems

Separated bump conjecture. One should have

Let $T$ be a Calderón–Zygmund operator, let $p>1$, and let $(u,v)$ be weights. Let $A$ and $B$ be Young functions, with $\bar A\in B_{p'}$ and $\bar B\in B_p$, and use the normaliz…

Separated bumps conjecture. For all Calderón-Zygmund operators $T$, there holds

Let $w$ be a weight and define … Set $\sigma=w^{1-p'}$, where $1<p<\infty$, and let $T$ be a Calderón–Zygmund operator. Lacey's mixed $A_p$–$A_\infty$ conjecture. The estimate … sh…

Sharp $A_2$ exponent conjecture. The best possible power of $Q_2(w)$ in such a weighted operator bound is $1$.

Let $1<p<\infty$, let $(\mu,\nu)$ be a pair of weights, and let $\Phi$ and $\Psi$ be Young functions with complementary functions $\bar{\Phi}$ and $\bar{\Psi}$. For a cube $Q$, wri…

Two-weight T1 conjecture. The two-weight $T1$ theorem should hold for general measure pairs and general Calderón–Zygmund operators.

Let $T$ be a Calderón–Zygmund operator with a standard kernel, and let $\{\varepsilon_j\}_j$ be a fixed sequence with $\varepsilon_j\to0$ such that, for every…

Bilinear sparse square-function domination conjecture. There exist $\eta=\eta(d,T)\in(0,1)$ and an $\eta$-sparse family $\mathcal{S}$ of cubes in $\mathbb{R}^d$, depending on…

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 $T$ be a Calderón–Zygmund operator and suppose that, for every $w\in A_{1}$, … Here $\varphi$ is the function governing the dependence of the weighted weak-type bound on the…

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…

Weighted bilinear Calderón–Zygmund conjecture. If the maximal operator satisfies the associated unweighted bilinear inequality, then

Let $T$ be a multilinear Calderón–Zygmund operator, let $p_1,\ldots,p_m\in(1,\infty)$, and define $p$ by $1/p=1/p_1+\cdots+1/p_m$. For weights…

Let $0\leq \alpha <1$, $k\in\mathbb{N}$, and $0<\delta <1$. Let $T^{\alpha }$ be a $(k+\delta)$-smooth $\alpha$-fractional Calderón–Zygmund operator on the real line. Suppose…

Let $w$ be a weight, and let $[w]_{A_\infty}$ denote its $A_\infty$ characteristic. Consider the weighted estimate for the maximal truncated Calderón–Zygmund operator established i…

Let $W$ be a matrix weight, and consider matrix-weighted matrix martingale transforms and paraproducts on the associated matrix-weighted space. Their $A_2$-characteristic is denote…

The $A_2$ conjecture. The operator norm should satisfy a bound of the form

Let $T$ be a Calderón–Zygmund operator, let $1<p<\infty$, and let $w$ be an $A_p$ weight, with … Here $\|T\|_{L^p(w)}$ denotes the operator norm on the weighted space $L^p(w)$. The…

Mateu–Prat–Verdera conjecture. The condition $$ should hold for $s1$ with $snotinmathbb{Z}$.

Let $w$ be an $A_2$ weight, meaning that … and let $T$ be a Calderón–Zygmund operator. The $A_2$ conjecture. The sharp dependence of the $L^2(w)$ norm of $T$ on the $A_2$ constant…

Let $1<p<\infty$ and let $X$ be a space of homogeneous type. Let $T$ be a Calderón–Zygmund operator, let $w$ and $\sigma$ be weights, and let $\Phi$ and $\Psi$ be Young functions.…

Christ's conjecture. The class of operators that are $p$-weakly localized relative to the continuous Haar wavelet frame and the weight $p(a,b)=b^{1/2-\delta}$ coincides with the cl…

A2 conjecture. The constant in the corresponding weighted norm inequality for $T$ should depend linearly on the $A_2$ constant; equivalently, one should have