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 if the pair of weights (w,\sigma) satisfy the two weight A_{p} condition, and this testing inequality holds: \int_{Q} M(\sigma \mathbf 1_{Q})^{p} dw \lesssim \sigma(Q), for all cubes Q for which there is a cube P\supset Q satisfying \sigma(P) \leq D\sigma(Q) , and \ell(P) \geq \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.