15 problems

Let $T$ satisfy the same conditions as in Theorem 1.2 of the source, let $U$ and $V$ be matrix $A_p$ weights, let $b$ be a scalar function, and let $m\in\mathbb{N}$. Write…

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…

Treil's matrix $A_2$ conjecture. The matrix $A_2$ condition is necessary and sufficient for the boundedness of the Hilbert transform on the corresponding weighted vector-valued…

In what follows, let $T$ be a Dini-continuous Calderón–Zygmund operator, let $W$ be a matrix weight, and let $1<p<\infty$. Define … and … The weak-type estimate … also holds with…

For $d\in\mathbb{N}$, let $T$ be a Calderón–Zygmund singular integral operator acting coordinatewise on vector-valued functions in $(\mathbb{R}^d)^t$, and let…

The paper considers matrix-weighted Besov-type and Triebel--Lizorkin-type sequence spaces $\dot a^{s,\tau}_{p,q}(W)\in\{\dot b^{s,\tau}_{p,q}(W),\dot f^{s,\tau}_{p,q}(W)\}$ and alm…

Let $0<\alpha<n$, let $1\leq p<n/\alpha$, and define $q$ by … Let $W\in\mathcal{A}_{p,q}$ be a matrix weight, and let $I_\alpha$ denote the fractional integral operator. The scalar…

Let $W$ be a matrix weight and let $T$ be a Calderón-Zygmund singular integral operator. The matrix $A_2$ characteristic is denoted by $[W]_{\mathcal{A}_2}$. Matrix $A_2$ conjectur…

Let $W$ be a matrix weight, let $1<p<\infty$, and let $T$ be a Calderón–Zygmund operator acting componentwise on vector-valued functions. Write $L^p(W)$ for the space with norm ind…

Let $W$ be a matrix weight, and let $[W]_{A_2}$ denote its matrix $A_2$ characteristic. Write $H$ for the Hilbert transform acting on vector-valued functions. Matrix $A_2$ conjectu…

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…

Rarity conjecture. Type 2 matrix power functions are rarely $A_2$ matrix weights.

Let $1<p<\infty$ and let $W=(w_{ij})\colon \mathbb{R}^n\times \mathbb{R}^m\to\mathbb{C}^{N\times N}$ be a matrix weight in the set of Muckenhoupt weights…

Let $W$ be a $d\times d$ matrix weight in $A_2$, and let $\{\sigma_I\}_{I\in\mathcal{D}}$ be a sequence of matrices satisfying $\|\sigma\|_\infty<\infty$. Let $H$ be the Hilbert tr…

Let $W$ be a $d\times d$ matrix weight in $A_2$, and let $S_W$ denote the associated square function on $L^2(W)$. Matrix weighted square function conjecture. … The scalar analogue…