$A_{infty}$ constants between $mathit{BMO}$ and weighted $mathit{BMO}$

摘要

In this short article, we consider estimates of the ratio egin{equation*}|{f}|_{mathit{BMO}(w)} / |{f}|_{mathit{BMO}}end{equation*}from above and below, where $w$ belongs to Muckenhoupt class $A_{infty}$. The upper bound of the ratio was proved by Hytönen and Pérez in~[6] with the optimal power. We establish the lower bound of the ratio and give two other proofs of the upper bound.