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 $ϕ: X×ℝ_{+} → ℝ_{+}$, we denote by $\text{bmo}_{ϕ,p}(X)$ the set of all functions $f ∈ L^{p}_{loc}(X)$ such that $\underset{a ∈ X, r>0}{\text{sup}} 1/{ϕ(a,r)} (1/μ(B(a,r)) \int_{B(a,r)} |f(x) -f_{B(a,r)}|^p dμ)^{1//p} < ∞$, where B(a,r) is the ball centered at a and of radius r, and $f_{B(a,r)}$ is the integral mean of f on B(a,r). Let $\text{bmo}_{ϕ}(X) =\text{bmo}_{ϕ,1}(X)$ and $\text{bmo}(X) = \text{bmo}_{1,1}(X)$. In this paper, we characterize $\text{PWM}(\text{bmo}_{ϕ1,p_1}(X), \text{bmo}_{ϕ2,p_2}(X))$. The following are examples of our results. $\text{PWM} (\text{bmo}_{(log(1//r))^{-α}}(\mathbb{T}^n), \text{bmo}_{(log(1//r))^{-β}}(\mathbb{T}^n)) = \text{bmo}_{(log(1//r))^{α-β-1}}(\mathbb{T}^n)$, 0≤β < α < 1, $\text{PWM} (\text{bmo}_{(log(1//r))^{-1}}(\mathbb{T}^n), \text{bmo} (\mathbb{T}^n)) = bmo_{(log log(1//r))^{-1}}(\mathbb{T}^n),$ $\text{PWM} (\text{bmo}(ℝ^n),\text{bmo}_{log(|a|+r+1//r),p}(ℝ^n)) = \text{bmo}(ℝ^n)$, 1 < p < ∞, etc.