BMO from dyadic BMO
Copyright policy )BMO from dyadic BMO
Then clearly BMO c BMO, with \\φ\\d £\\φ\\, but BMO and BMO, are not the same space; the function log\x9 \X{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
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Harmonic analysis in several variables, bounded mean oscillation, Carleson's decomposition theorem, 42B30, dyadic BMO, Muckenhoupt class, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Harmonic analysis in several variables, bounded mean oscillation, Carleson's decomposition theorem, 42B30, dyadic BMO, Muckenhoupt class, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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