6 problems

Let $T$ be a sparse operator, let $1<p<q<\infty$, and let $\mathcal{D}_{p,p}$ and $\mathcal{D}^{\ast}_{p,p}$ denote the corresponding direct comparison bump quantities for the case…

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 $T$ be a vector of standard singular integrals in higher dimensions, and suppose that $T$ is both strongly elliptic and strongly gradient elliptic. The energy conditions are th…

Let $\sigma$ and $\omega$ be locally finite positive measures, and let $H$ denote the Hilbert transform. Suppose the two-weight Muckenhoupt condition $(A2)$ and the testing conditi…

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 $\sigma$ and $\omega$ be positive Borel measures on $\mathbb{R}^{n}$, let $1<p<\infty$ with conjugate exponent $p'$ defined by $1/p+1/p'=1$, and let $T$ be a standard singular…