63 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…

For $p\in(1,\infty)$, let $T_{BA}$ be the Beurling–Ahlfors transform acting on $L^p(\mathbb C)$, and let $p^*=\max(p',p)$, where $p'$ is the Hölder conjugate of $p$. Its norm satis…

Stein's conjecture. The operator $H_{\Gamma}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1<p<\infty$.

Let $\mu$ be an Ahlfors regular measure, and consider its Riesz transform. David–Semmes conjecture. The $L^{2}$-boundedness of the Riesz transform should be sufficient to imply rec…

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…

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

Let $E\subset\mathbb{R}^{n+1}$ be an Ahlfors–David regular set. The Riesz transform conjecture. Boundedness in $L^2(E)$ of the Riesz transform should imply that $E$ is uniformly re…

Let $\bold{H}_{\mathbb{C}}$ denote the complex Riesz transform on $4\mathbb{R}^24$, let $k\in\mathbb{N}$, and let $1<p<\infty$. Write $q=\frac{p}{p-1}$. Carlen–Dragicevic–Kovač con…

Let $\mu$ be a codimension-one Ahlfors regular measure on parabolic space $\mathbb R^{n+1}$. Suppose that the parabolic Riesz transform is bounded on $L^2(\mu)$, and that $\mu$ sat…

Let $v(cdot)$ be a Lipschitz vector field on $\mathbb{R}^2$, and let $M_{v(cdot)}$ denote the associated maximal function; consider its truncated version. Zygmund's conjecture. The…

For $m\in\mathbb{N}$, let $\Lambda_m$ be the operator … and let $w(\Lambda_m)$ denote its weak-type $(1,1)$ constant. Gill's conjecture. For every $m\in\mathbb{N}$, … The paper sta…

Let $\mathcal{B}$ denote the Beurling–Ahlfors transform on $L^1(\mathbb{C})$, and let $\Lambda_m$ be the operators defined by … The weak-type constant of an operator $T$ is denoted…

Positive boundedness conjecture. Every such datum $\mathbf{H}_\mathbf{M}$ is $\mathbf{p}$-bounded.

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 $\gamma(t)=(a_1t,a_2t,a_3t)$, where $a_1,a_2,a_3$ are nonzero and pairwise distinct, so that $\gamma$ is a nondegenerate line in $\mathbb{R}^3$. For $\vec f=(f_1,f_2,f_3)$ and…

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

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…

Critical-exponent conjecture. The same conclusion holds for non-disjoint $\Gamma$ under the assumption of the open set condition.

The two-weight Hilbert transform conjecture. The Hilbert transform $H$ is bounded from $L^p(\sigma)$ to $L^p(\omega)$ if and only if both local testing conditions hold together wit…

For $d\geq2$, let $K_{\mathbb{H}^{(k)}}$ and $K_{R_\mathrm{dis}^{(k)}}$ be the kernels of the probabilistic and Calderón–Zygmund discrete Riesz transforms, respectively, with…

For $d>1$, let $\mathbb{K}_{\mathbb{H}^{(k)}}$ denote the probabilistic continuous Riesz transform in the $k$th coordinate, acting on $L^p(\mathbb{R}^d)$, where $1<p<\infty$ and…

Let $R^{(k)}_\mathrm{dis}$ be the Calderón–Zygmund discrete Riesz transform on $\mathbb{Z}^d$, for $k=1,\dots,d$, and let $p^*=\max\{p,p/(p-1)\}$. The sharp norm conjecture. For al…

Let $\Omega$ be homogeneous of degree zero on $\mathbb{R}^d$, satisfy the vanishing moment condition, and let $A$ be a function on $\mathbb{R}^d$ whose derivatives of order one bel…

Under the immersed boundary integral method formulation, apply the two-dimensional quadrature rule on each plane through a target point and sum the resulting errors. Additional can…