Two Weight Inequalities for Positive Operators: Doubling Cubes.

For the maximal operator $ M $ on $ \mathbb R ^{d}$, and $ 1< p , \rho < \infty $, there is a finite constant $ D = D _{p, \rho }$ so that this holds. For all weights $ w, \sigma $ on $ \mathbb R ^{d}$, the operator $ M (\sigma \cdot )$ is bounded from $ L ^{p} (\sigma ) \to L ^{p} (w)$ if and only the pair of weights $ (w, \sigma )$ satisfy the two weight $ A _{p}$ condition, and this testing inequality holds: \begin{equation*} \int _{Q} M (\sigma \mathbf 1_{Q} ) ^{p} \; d w \lesssim \sigma ( Q), \end{equation*} for all cubes $ Q$ for which there is a cube $ P \supset Q$ satisfying $ \sigma (P) < D \sigma (Q)$, and $ \ell P = \rho \ell Q$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.