Sharp weak type estimates for weights in the class $A_{p_1, p_2}$

TL;DR

This paper derives sharp distribution function estimates for weights satisfying the $A_{p_1, p_2}$ condition, generalizing known conditions and providing maximizers, with applications to self-improvement properties of weights.

Contribution

It introduces sharp estimates for $A_{p_1, p_2}$ weights, including maximizers, and demonstrates their use in deriving self-improvement results for weights in $A_2$.

Findings

Sharp estimates for $A_{p_1, p_2}$ weights' distribution functions.

Identification of maximizers for these estimates.

Application to $A_2$ weights showing self-improvement to Reverse H"older condition.

Abstract

We get sharp estimates for the distribution function of nonnegative weights, which satisfy so called $A_{p_1, p_2}$ condition. For particular choices of parameters $p_1$, $p_2$ this condition becomes an $A_p$-condition or Reverse H\"{o}lder condition. We also get maximizers for these sharp estimates. We use the Bellman technique and try to carefully present and motivate our tactics. As an illustration of how these results can be used, we deduce the following result: if a weight $w$ is in $A_2$ then it self-improves to a weight, which satisfies a Reverse H\"{o}lder condition.