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 $bmo_{ϕ,p}(X)$ the set of all functions $f ∈ L^{p}_{loc}(X)$ such that $sup_{a ∈ X, r>0} 1/ϕ(a,r) (1/μ(B(a,r)) ʃ_{B(a,r)} |f(x) -f_{B(a,r)}|^p dμ)^{1/p} < ∞$, 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 $bmo_{ϕ}(X) = bmo_{ϕ,1}(X)$ and $bmo(X) = bmo_{1,1}(X)$. In this paper, we characterize $PWM(bmo_{ϕ1,p_1}(X), bmo_{ϕ2,p_2}(X))$. The following are examples of our results. $PWM(bmo_{(log(1/r))^{-α}}(𝕋^n),bmo_{(log(1/r))^{-β}}(𝕋^n)) = bmo_{(log(1/r))^{α-β-1}}(𝕋^n)$, 0≤β < α < 1, $PWM (bmo_{(log(1/r))^{-1}}(𝕋^n),bmo(𝕋^n)) = bmo_{(log log(1/r))^{-1}}(𝕋^n),$ $PWM (bmo(ℝ^n),bmo_{log(|a|+r+1/r),p}(ℝ^n)) = bmo(ℝ^n)$, 1 < p < ∞, etc.
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