Two weight inequalities for positive operators: doubling cubes

Abstract

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:

for all cubes $Q$ for which there is a cube $P\supset Q$ satisfying $\sigma (P)\le D\sigma (Q)$, and $\ell (P)\ge \rho \ell (Q)$. 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.