BMO Functions Generated by AXℝn Weights on Ball Banach Function Spaces

Let X be a ball Banach function space on ℝn. We introduce the class of weights AXℝn. Assuming that the Hardy-Littlewood maximal function M is bounded on X and X′, we obtain that BMOℝn=αlnω:α≥0,ω∈AXℝn. As a consequence, we have BMOℝn=αlnω:α≥0,ω∈ALp·ℝnℝn, where Lp·ℝn is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space Xω for any ω∈AXℝn, then the commutator b,T is bounded on X with b∈BMOℝn.