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

Abstract

In this short article, we consider estimates of the ratio \begin{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.