Pointwise multipliers on weighted BMO spaces

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for ${\displaystyle \varphi :X\times {ℝ}_{+}\to {ℝ}_{+}}$, we denote by ${\displaystyle {\text{bmo}}_{\varphi ,p}\left(X\right)}$ the set of all functions ${\displaystyle f\in L^p\_\left\{loc\right\}\left(X\right)}$ such that ${\displaystyle \underset{a\in X,r>0}{\text{sup}}\frac{1}{\varphi (a,r)}{(\frac{1}{\mu }\left(B(a,r)\right){\int }_{B(a,r)}{|f\left(x\right)-f_{B(a,r)}|}^{p}d\mu )}^{1/p}<\infty }$, where B(a,r) is the ball centered at a and of radius r, and ${\displaystyle f_{B(a,r)}}$ is the integral mean of f on B(a,r). Let ${\displaystyle {\text{bmo}}_{\varphi }\left(X\right)={\text{bmo}}_{\varphi ,1}\left(X\right)}$ and ${\displaystyle \text{bmo}\left(X\right)={\text{bmo}}_{1,1}\left(X\right)}$. In this paper, we characterize ${\displaystyle \text{PWM}({\text{bmo}}_{\varphi 1,p_1}\left(X\right),{\text{bmo}}_{\varphi 2,p_2}\left(X\right))}$. The following are examples of our results. ${\displaystyle \text{PWM}({\text{bmo}}_{{\left(\log (1/r)\right)}^{-\alpha }}\left(T^n\right),{\text{bmo}}_{{\left(\log (1/r)\right)}^{-\beta }}\left(T^n\right))={\text{bmo}}_{{\left(\log (1/r)\right)}^{\alpha -\beta -1}}\left(T^n\right)}$, 0≤β < α < 1, ${\displaystyle \text{PWM}({\text{bmo}}_{{\left(\log (1/r)\right)}^{-1}}\left(T^n\right),\text{bmo}\left(T^n\right))=bm{o}_{{\left(\log \log (1/r)\right)}^{-1}}\left(T^n\right),}$ ${\displaystyle \text{PWM}(\text{bmo}\left({ℝ}^n\right),{\text{bmo}}_{\log (\left|a\right|+r+1/r),p}\left({ℝ}^n\right))=\text{bmo}\left({ℝ}^n\right)}$, 1 < p < ∞, etc.