21 problems

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 $J_H(y)=\boldsymbol{1}_{[-1,1]}(y)y^{-1}$ be the Hilbert transform kernel with a sharp cutoff, and let $\operatorname{Osc}_n(J_H;f)$ denote the corresponding oscillation operat…

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 $d>0$ and let $\beta_d$ be the kernel of the periodic Hilbert transform on the strip of depth $d$, defined for $s\in\mathbb{R}\setminus 2\pi\mathbb{Z}$ by the kernel formulas i…

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…

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…

The two-weight T1 conjecture. The Hilbert transform $H$ is bounded from $L^{p}(\sigma)$ to $L^{p}(\omega)$ if and only if, for every interval $I$, the local testing conditions

Let $H$ be the Hilbert transform, let $I$ be a unit interval, let $w\in A_1$, and let $M$ denote the maximal operator. For functions $f$ and $\lambda>0$, write…

Quadratic two-weight characterization. The inequality above holds if and only if $H^\sigma$ and $H^\omega$ satisfy the global quadratic testing conditions in $L^p$ and $L^{p'}$, re…

Let $H$ be the Hilbert transform, let $w\in A_1$, and denote the $A_1$ characteristic of $w$ by $[w]_{A_1}$. For a measurable function $g$, write $\|g\|_{L^{1,\infty}(w)}$ for its…

Alaifari–Pierce–Steinerberger conjecture. In this lower bound, $\left\|f'\right\|_{L^2(I)}$ may be replaced by $\left\|f'\right\|_{L^1(I)}$.

Let $\tau$! be a pair of nonnegative locally finite measures $\tau=(\sigma,\omega)$ on $\mathbb{R}^{n}$, let $H$ be the Hilbert transform, and let…

Let $\theta$ be a bounded solution on $[0,T]$ of … where $H$ is the Hilbert transform. The $C^{1/2}$ Hölder seminorm measures the quotient $|\theta(t,x)-\theta(t,y)|/|x-y|^{1/2}$.…

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 … mathbb{R}$. For each interval$I$, define the Poisson integral by … Let$b_I$be the unique, if it exists, point such that … Let$\widetilde\sigma_I:=\sigma-\sigma{b_I}\del…

Let $I,J\subset\mathbb{R}$ be intervals in the configuration of Case 3 or Case 4, and let $f\in BV(I)$ be supported on $I$. Let $h:[0,\infty)\to\mathbb{R}_{+}$ be the positive func…

Let $w$ and $\sigma$ be weights on the real line. The Muckenhoupt $A_2$ condition is the usual two-weight condition controlling the averages of $w$ and $\sigma$ on intervals, while…

Let $v$ be a vector field with $1+\eta>1$ derivatives, assume the fixed-width Lipschitz Kakeya maximal-function conjecture for $v$, and let $\operatorname H_{v,\epsilon}$ be the tr…

Lipschitz Hilbert-transform conjecture. There is a universal constant $K$ such that, with $\epsilon=\|v\|_{\operatorname{Lip}}/K$,

Stein's conjecture. There is an absolute constant $K>0$ such that, if $\epsilon^{-1}=K\|v\|_{\operatorname{Lip}}$, then