Characterizations of weighted BMO space and its application

Abstract

In this paper, we prove that the weighted BMO space as follows $${\mathrm{BMO}}^p(\omega )=\{f\in L_{\mathrm{loc}}^1:\underset{Q}{\sup }\Vert {\chi }_Q{\Vert }_{L^p(\omega )}^{-1}\Vert (f-f_Q){\omega }^{-1}{\chi }_Q{\Vert }_{L^p(\omega )}<\infty \}$$ 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 Calderón-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.

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

References in corpus (2)

• Characterizations of BMO through commutators of bilinear singular integral operators
• Another characterizations of Muckenhoupt $A_p$ class

Cited by in corpus (1)

• A note on commutator in the multilinear setting