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( f\sigma \right) ^{2}d\omega \leq C\int_{\mathbb{ R}^{n}}f^{2}d\sigma \end{equation*} holds for all $f\geq 0$ if and only if the (fractional) $A_{2}$ condition holds, and the restricted testing condition \begin{equation*} \int_{Q}T\left( 1_{Q}\sigma \right) ^{2}d\omega \leq C\left\ | Q\right\ |{\sigma } \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |{\sigma }\leq D\left\ | Q\right\ |{\sigma }$. If $T$ is linear, we require as well that the dual restricted testing condition \begin{equation*} \int{Q}T^{\ast }\left( 1_{Q}\omega \right) ^{2}d\sigma \leq C\left\ | Q\right\ |{\omega } \end{equation*} holds for all cubes $Q$ satisfying $\left\ | 2Q\right\ |{\omega }\leq D\left\ | Q\right\ |_{\omega }$.