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.