Embedding $BMO$ into weighted $BMO$

Abstract

A classical result of harmonic analysis asserts that if a weight $w$ satisfies Muckenhoupt's condition $A_\infty$, then the unweighted class $\mathit{BMO}$ is contained in the weighted space $\mathit{BMO}(w)$. The paper identifies the norm of this embedding in the one-dimensional setting. Specifically, for any function $f\in \mathit{BMO}(\mathbb{R})$ and any weight $w\in A_\infty(\mathbb{R})$ of characteristic $[w]_{A_\infty}$, we have the estimate $$ \|f\|_{\mathit{BMO}(w)}\leq e\sqrt{2}[w]_{A_\infty}\|f\|_{\mathit{BMO}}. $$ The constant $e\sqrt{2}=3.8442\dots$ is the best possible. We also prove a sharp version of this result in which the characteristic $[w]_{A_\infty}$ of the weight is fixed. Further extensions to the theory of martingales are obtained.