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Abstract

For $w : ℝ^{n} × ℝ_{+} → ℝ_{+}$ and 1 ≤ p < ∞, let $bmo_{{}w,p}(ℝ^n)$ be the set of locally integrable functions f on $ℝ^n$ for which $sup_{I}(1/w(I) ʃ_{I} |f(x)-f_{I}|^p dx)^{1/p} < ∞$ 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: (1) $fg ∈ bmo_{w,p}(ℝ^n)$ whenever $f ∈ ℝ bmo_{w,p}(ℝ^n)$, (2) $g ∈ L^∞(ℝ^n)$ and $sup_{I}( 1/w*(I) ʃ_{I} |g(x)-g_{I}|^p dx)^{1/p} < ∞$, where $w* = w/Ψ, Ψ = Ψ_{1} + Ψ_{2}$ and $Ψ_{1}(a,r) = (ʃ_{1}^{max(2,|a|,r)} (w(O,t)^{1/p})/(t^{n/p+1}) dt)^p$, $Ψ_{2}(a,r) = (ʃ_{r}^{max(2,|a|,r)} (w(a,t)^{1/p})/(t^{n/p+1}} dt)^p$.