Restricted testing for positive operators

Abstract

We prove that for certain positive operators $T$, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant $D>1$, depending only on the dimension $n$, such that the two weight norm inequality \begin{equation*} \int_{\mathbb{R}^{n}}T\left( fσ\right) ^{2}dω\leq C\int_{\mathbb{ R}^{n}}f^{2}dσ\end{equation*} holds for all $f\ge 0$ if and only if the (fractional) $A_2$ condition holds, and the restricted testing condition \begin{equation*} \int_{Q}T\left( 1_{Q}σ\right) ^{2}dω\leq C\left\ | Q\right\ |_{σ} \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{σ}\leq D\left\ | Q\right\ |_{σ}$. If $T$ is linear, we require as well that the dual restricted testing condition \begin{equation*} \int_{Q}T^{\ast }\left( 1_{Q}ω\right) ^{2}dσ\leq C\left\ | Q\right\ |_{ω} \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |_{ω}\leq D\left\ | Q\right\ |_{ω}$.

This version also updates arXiv:1811.11032, 18 pages