Restricted Testing for Positive Operators

We prove that for certain positive operators T, such as the Hardy–Littlewood maximal function and fractional integrals, there is a constant D>1$$D>1$$, depending only on the dimension n, such that the two weight norm inequality ∫RnTfσ2dω≤C∫Rnf2dσ$$\begin{aligned} \int _{{\mathbb {R}}^{n}}T\left( f\sigma \right) ^{2}\mathrm{{d}}\omega \le C\int _{{\mathbb {R}}^{n}}f^{2}\mathrm{{d}}\sigma \end{aligned}$$holds for all f≥0$$f\ge 0$$if and only if the (fractional) A2$$A_{2}$$ condition holds, and the restricted testing condition ∫QT1Qσ2dω≤CQσ$$\begin{aligned} \int _{Q}T\left( 1_{Q}\sigma \right) ^{2}\mathrm{{d}}\omega \le C\left| Q\right| _{\sigma } \end{aligned}$$holds for all cubes Q satisfying 2Qσ≤DQσ$$\left| 2Q\right| _{\sigma }\le D\left| Q\right| _{\sigma }$$. If T is linear, we require as well that the dual restricted testing condition ∫QT∗1Qω2dσ≤CQω$$\begin{aligned} \int _{Q}T^{*}\left( 1_{Q}\omega \right) ^{2}\mathrm{{d}}\sigma \le C\left| Q\right| _{\omega } \end{aligned}$$holds for all cubes Q satisfying 2Qω≤DQω$$\left| 2Q\right| _{\omega }\le D\left| Q\right| _{\omega }$$.