Characterizations of weighted BMO space and its application

TL;DR

This paper characterizes weighted BMO spaces, showing their independence from scale p for A_1 weights, and links these spaces to boundedness of bilinear commutators of Calderón-Zygmund operators.

Contribution

It proves the invariance of weighted BMO spaces across different p and characterizes these spaces via bilinear commutator boundedness, solving an open problem.

Findings

Weighted BMO space is independent of p for A_1 weights.

L^{p}( ext{or } L^{p, ext{infinity}}) norms characterize the space.

Bilinear commutators are bounded if and only if the symbol is in the weighted BMO space.

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.