The class of Bounded Oscillation ( \(\textrm{BO}\) ) operators were introduced and studied by the author in two recent papers. It was shown that many operators in harmonic analysis (Calderón–Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are \(\textrm{BO}\) operators. \(\textrm{BO}\) 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 the various properties of BO operators established in those papers, it was also proved that \(\textrm{BO}\) operators admit pointwise sparse domination, establishing the \(A_2\) -conjecture for those operators. In the present paper we study boundedness properties of \(\textrm{BO}\) operators on \(\textrm{BMO }\) spaces. In particular, we prove that general \(\textrm{BO}\) operators boundedly map \(L^\infty \) into \(\textrm{BMO }\) , and under a logarithmic localization condition those map \(\textrm{BMO }\) 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 \(\textrm{BMO }\) estimations of Calderón–Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces \(H^1\) .
