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Let us recall the weighted estimates that we need for our corollaries. A function $ w >0$ is a *Muckenhoupt $ A_p$ weight* if $$[w] _{A_p} = \sup _{Q} \Bigl[\frac { w ^{\frac 1 {1-p}} (Q) } {\lvert  Q\rvert }\Bigr] ^{p-1} \frac {w (Q)} {\lvert  Q\rvert  } < \infty .$$ Above, we are conflating $ w$ as a measure and a density, thus $w ^{\frac 1 {1-p}} (Q)  =\int _{Q} w (x) ^{\frac 1 {1-p}} \;dx $. We have these estimates, which are sharp in the $ A_p$ characteristic. They are an element of the sparse proof of the $ A_2$ conjecture. (See [@MR3085756] for a proof.)

\[t:wts\] These estimates hold for all $ 1  < p < \infty $. $$\lVert \Lambda _{1,1} \;:\; L ^{p} (w) \mapsto L ^{p} (w) \rVert  \lesssim [w] _{A _{p}} ^{\max \{1, \frac 1 {p-1}\}}  .$$

For our applications, we have a second class of operators, a simplified form of those introduced by Benau-Bernicot-Petermichl [@160506401]. For our purposes, we need a much simplified version of their result. Define an additional characteristic of a weight, namely the *reverse Hölder* property. $$\label{e:RH}
[w] _{RH_r} = \sup _{ Q}   \frac {\langle w  \rangle _{Q, r}} {\langle 