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
$\int_{Q}T\left( 1_{Q}\sigma
\right) ^{2}d\omega \leq C\left\ | Q\right\ |_{\sigma } $
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 $\left\ | 2Q\right\ |_{\omega }\leq D\left\ | Q\right\ |_{\omega }$.