Summary: For \(w: \mathbb{R}^ n\times \mathbb{R}_ +\to \mathbb{R}_ +\) and \(1\leq p<\infty\), let \(\text{bmo}_{w,p} (\mathbb{R}^ n)\) be the set of locally integrable functions \(f\) on \(\mathbb{R}^ n\) for which \[ \sup_ I \Biggl( {1\over {w(I)}} \int_ I | f(x)- f_ I|^ p dx \Biggr)^{1/p} <\infty, \] where \(I=I(a,r)\) is the cube with center \(a\) whose edges have length \(r\) and are parallel to the coordinate axes, \(w(I)= w(a,r)\) and \(f_ I\) is the average of \(f\) over \(I\). If \(w\) satisfies appropriate conditions, then the following are equivalent: \[ fg\in \text{bmo}_{w,p} (\mathbb{R}^ n) \quad \text{ whenever } \quad f\in \text{bmo}_{w,p} (\mathbb{R}^ n), \tag{1} \] \[ g\in L^ \infty (\mathbb{R}^ n) \quad \text{and } \sup_ I \Biggl( {1\over {w^*(I)}} \int_ I | g(x)- g_ I|^ p dx\Biggr)^{1/p} <\infty, \tag{2} \] where \(w^*= w/\Psi\), \(\Psi= \Psi_ 1+ \Psi_ 2\) and \[ \Psi_ 1(a,r)= \Biggl( \int_ 1^{\max(2,| a|,r)} {{w(0,t)^{1/p}} \over {t^{n/p+1}}} dt\Biggr)^ p, \qquad \Psi_ 2(a,r)= \Biggl( \int_ r^{\max(2,| a|,r)} {{w(a,t)^{1/p}} \over {t^{n/p+1}}} dt\Biggr)^ p. \]