Bounded oscillation operators on BMO spaces

Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calder\'on-Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are $BO$ operators. $BO$ operators are defined on abstract measure spaces equipped with a basis of abstract balls. The abstract balls in their definition owe four basic properties of classical balls in $\mathbb{R}^n$, which are crucial in the study of singular operators on $\mathbb{R}^n$. Among various properties studied in these papers it was proved that $BO$ operators allow pointwise sparse domination, establishing the $A_2$-conjecture for those operators. In the present paper we study boundedness properties of $BO$ operators on $BMO$ spaces. In particular, we prove that general $BO$ operators boundedly map $L^\infty$ into $BMO$, and under a logarithmic localization condition those map $BMO$ into itself. We obtain these properties as corollaries of new local type bounds, involving oscillations of functions over the balls. We apply the results in the $BMO$ estimations of Calder\'on-Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces $H^1$.