BMO from dyadic BMO

Then clearly BMO c BMO, with \\φ\ £\\φ\\, but BMO and BMO, are not the same space; the function log9 {ttlj>0] is in BMO, but not in BMO. In analysis BMO is more important than BMO, because BMO is translation invariant, but BMO, is not. On the other hand, BMO, is very much the easier space to work with because dyadic cubes are nested (if two open daydic cubes intersect then one of them is contained in the other). For example, for BMO the original proofs [1], [6], [8], [11] of the four theorems stated below were rather technical, while for BMO, the analogous results are comparatively trivial. In this paper we derive the four theorems from their dyadic counterparts. Here is the idea. Let Taφ(x) = φ(x — a). Then