Characterizations of weighted BMO space and its application

In this paper, we prove that the weighted BMO space as follows $${\mathrm{B}\mathrm{M}\mathrm{O}}^p(\omega )=\{f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1:\underset{Q}{\sup }\mathrm{\parallel }{\chi }_Q{\mathrm{\parallel }}_{L^p(\omega )}^{-1}\parallel (f-f_Q){\omega }^{-1}{\chi }_Q{\parallel }_{L^p(\omega )}<\mathrm{\infty }\}$$ is independent of the scale $p\in (0,\mathrm{\infty })$ in sense of norm when $\omega\in A_{1}$\mathrm{.}Moreover,wecanreplace$L^{p}(\omega)$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 $1<p_1,p_2<\mathrm{\infty }$, $1\mathrm{/}p=1\mathrm{/}{p}_1+1\mathrm{/}{p}_2$ and $\omega \in A_1$. Thus we answer the open problem proposed in \cite{C} affirmatively.