BMO with respect to Banach function spaces

For every cube $$Q \subset \mathbb {R}^n$$ we let $$X_Q$$ be a quasi-Banach function space over Q such that $$\Vert \chi _Q\Vert _{X_Q} \simeq 1$$ , and for $$X= \{X_Q\}$$ define $$\begin{aligned} \Vert f\Vert _{{{\,\textrm{BMO}\,}}_X}&:=\sup _Q \,\Vert f-{\textstyle \frac{1}{|Q|}\int _Qf} _{X_Q},\\ _{{{\,\textrm{BMO}\,}}_X^*}&:=\sup \,\inf _c\, f-c\Vert _{X_Q}. \end{aligned}$$ We study necessary sufficient conditions on X {{\,\textrm{BMO}\,}}= {{\,\textrm{BMO}\,}}_X = {{\,\textrm{BMO}\,}}_{X}^*. In particular, give full characterization of the embedding $${{\,\textrm{BMO}\,}}\hookrightarrow {{\,\textrm{BMO}\,}}_X$$ in terms so-called sparse collections cubes easily checkable rather weak $${{\,\textrm{BMO}\,}}_X^* \hookrightarrow {{\,\textrm{BMO}\,}}$$ . Our main theorems recover improve all previously known results this area.