BMO with respect to Banach function spaces

Abstract

For every cube Q ⊂ ℝⁿ 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.