| Popis: |
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 $1Comment: Recently, Jarod Hart and Rodolfo H. Torres(arXiv:1707.01141) obtained some characterizations of weighted BMO space, Theorem 1.2 in my paper can be seem as a corollary of Theorem 5.9 in their paper. It should point out that our paper has submitted to Forum Mathematicum on 18-Apr-2017 |
