Restricted testing for positive operators

TL;DR

This paper establishes a characterization of two weight inequalities for certain positive operators, like the Hardy-Littlewood maximal function, using restricted testing conditions that depend only on a dimension-related constant.

Contribution

It introduces a restricted testing condition framework that simplifies the verification of two weight inequalities for positive operators, extending previous characterizations.

Findings

Two weight inequality holds iff fractional A2 condition and restricted testing conditions are satisfied.

Restricted testing conditions only need to be checked on specific cubes with controlled measure ratios.

Results apply to operators like the Hardy-Littlewood maximal function and fractional integrals.

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 $\left\ | 2Q\right\ |_{\omega }\leq D\left\ | Q\right\ |_{\omega }$.