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.