BMO with respect to Banach function spaces

For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that ‖ χ Q ‖ X Q ≃ 1 , and for X = { X Q } define ‖ f ‖ BMO X : = sup Q ‖ f - 1 | Q | ∫ Q f ‖ X Q , ‖ f ‖ BMO X ∗ : = sup Q inf c ‖ f - c ‖ X Q . We study necessary and sufficient conditions on X such that BMO = BMO X = BMO X ∗ . In particular, we give a full characterization of the embedding BMO ↪ BMO X in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMO X ∗ ↪ BMO . Our main theorems recover and improve all previously known results in this area.