Characterizations of weighted BMO space and its application

Abstract

In this paper, we prove that the weighted BMO space as follows $${\rm BMO}^{p}(\omega)=\Big\{f\in L^{1}_{\rm loc}:\sup_{Q}\|\chi_{Q}\|^{-1}_{L^{p}(\omega)}\big\|(f-f_{Q})\omega^{-1}\chi_{Q}\big\|_{L^{p}(\omega)}<\infty\Big\}$$ is independent of the scale $p\in (0,\infty)$ in sense of norm when $\omega\in A_{1}$. Moreover, we can replace $L^{p}(\omega)$ by $L^{p,\infty}(\omega)$. As an application, we characterize this space by the boundedness of the bilinear commutators $[b,T]_{j} (j=1,2)$, generated by the bilinear convolution type Calder\'{o}n-Zygmund operators and the symbol $b$, from $L^{p_{1}}(\omega)\times L^{p_{2}}(\omega)$ to $L^{p}(\omega^{1-p})$ with $1<p_{1},p_{2}<\infty$, $1/p=1/p_{1}+1/p_{2}$ and $\omega\in A_{1}$. Thus we answer the open problem proposed in \cite{C} affirmatively.