Pointwise multipliers for functions of weighted bounded mean oscillation

For ${\displaystyle w:{ℝ}^n\times {ℝ}_{+}\to {ℝ}_{+}}$ and 1 ≤ p < ∞, let ${\displaystyle bm{o}_{\left\{\right\}w,p}\left({ℝ}^n\right)}$ be the set of locally integrable functions f on ${\displaystyle {ℝ}^n}$ for which ${\displaystyle {\supset }_I{\left(\frac{1}{w}\left(I\right){ʃ}_I{|f\left(x\right)-f_I|}^{p}dx\right)}^{\frac{1}{p}}<\infty }$ 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 ${\displaystyle f_I}$ is the average of f over I. If w satisfies appropriate conditions, then the following are equivalent: (1) ${\displaystyle fg\in bm{o}_{w,p}\left({ℝ}^n\right)}$ whenever ${\displaystyle f\in ℝbm{o}_{w,p}\left({ℝ}^n\right)}$, (2) ${\displaystyle g\in L^{\infty }\left({ℝ}^n\right)}$ and ${\displaystyle {\supset }_I{(\frac{1}{w}\cdot \left(I\right){ʃ}_I{|g\left(x\right)-g_I|}^{p}dx)}^{\frac{1}{p}}<\infty }$, where ${\displaystyle w\cdot =\frac{w}{\Psi },\Psi ={\Psi }_1+{\Psi }_2}$ and ${\displaystyle {\Psi }_1(a,r)={\left({ʃ}_1^{max(2,\left|a\right|,r)}\frac{w{(O,t)}^{\frac{1}{p}}}{t^{\frac{n}{p}+1}}dt\right)}^p}$, ${\displaystyle {\Psi }_2(a,r)={\left({ʃ}_r^{max(2,\left|a\right|,r)}\frac{w{(a,t)}^{\frac{1}{p}}}{t^{\frac{n}{p}+1}}dt\right)}^p}$.