141 problems

Let $w$ be a weight or measure, let $f$ be a suitable integrable function on bR}^n, and let $\textup{H}$ denote the Hilbert transform and $\textup{M}$ the Hardy–Littlewood maxima…

Let $u$ and $v$ be weights in the Muckenhoupt class $A_1$, and let $H$ denote the Hilbert transform. Sawyer proved that the operator $S_v f=M(fv)v^{-1}$ satisfies a mixed weak-type…

Sarason's Poisson $A_2$ conjecture. The Hilbert transform maps boundedly on $L^2$ in the two-weight setting if and only if the weights have a joint Poisson $A_2$ characteristic.

Let ${\mathcal{A}}_p({\omega},{\sigma})$ denote the two-weight Muckenhoupt condition, and let the testing conditions be … for every cube $I$, where $\mathfrak{T}$ and…

Pérez–Rela conjecture. The inequality remains valid with

Weighted-minimizer profile conjecture. The minimizers are conjectured to have the form

Let $\Omega$ be a collection of pairwise disjoint intervals, let $T^{\Omega}_{\operatorname{RF}}$ be the Rubio de Francia square function associated with the Fourier restrictions t…

Let $T$ be a Calderón–Zygmund operator, let $1<p<\infty$, and let $(\mu,\nu)$ be a pair of weights. For Young functions $\Phi$ and $\Psi$, write $\|\cdot\|_{\Phi,Q}$ and…

Muckenhoupt's sufficiency conjecture. The $C_p$ condition should be sufficient for the Coifman–Fefferman inequality

Hytönen–Lacey one supremum conjecture. There is a dimensional constant $c_n$ such that

Let $0<\alpha<1/2$, and let $M_\alpha$ be the random discrete maximal operator introduced in the paper. Let $f$ and $g$ be finitely supported functions on the integers. A sparse op…

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

Bennett–Grosse-Erdmann conjecture. The displayed inequality, and respectively its reverse inequality, remain valid with the same best possible constant $(\alpha p+1)^p$ when…

Let $w\in A_1$ be a weight, with … and let $T$ be a Calderón–Zygmund operator. Weak Muckenhoupt–Wheeden conjecture. The weak-type estimate … should hold with linear dependence on…

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…

Matrix $A_2$ conjecture. The norm of the vector Hilbert transform on $L^2(W)$ should be bounded by a constant times the linear power of the matrix $A_2$ characteristic:

Let $w\in A_\infty$, and denote its $A_\infty$ characteristic by $[w]_{A_\infty}$. For dimensional constants $C_n>0$, consider the embedding of $A_\infty$ into an $A_p$ class, wher…

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

Let $N>p$ and let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$. For $p^*:=Np/(N-p)$, let $u\in C_0^\infty(\mathbb{R}^N)$ and let $\gamma>p-1$. Weighted Hardy-Sobolev-Maz'ya conjectu…

Let $alpha>-1$ and let $w\buttonin B_{1,alpha}$ be a weight on the unit disc. Write $[w]_{B_{1,alpha}}$ for its $B_{1,alpha}$-constant, and let $P_{alpha}$ denote the Bergman proje…

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 $M_a(q)$ denote the Maz'ya constant associated with the purely cylindrical weighted minimization problem, and let $a$ and $q$ be as in that problem. A minimizer is radially dec…

Let $\mathbb{D}$ be the unit disk, let $\omega$ be a weight on $\mathbb{D}$, and define the Hardy-kernel operator … For an interval $I\subset\mathbb{T}$, let … where $|I|$ is norma…

Cruz–Isralowitz–Moen conjecture. In the scalar case, the conclusion $I_\alpha:L^p(V^{p/q})\to L^q(U)$ remains valid if the hypothesis $\bar{\Psi}\in B_{q'}$ in the stated fractiona…