BMO with respect to Banach function spaces

Abstract

For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, and forWe 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.