Atomic blocks for noncommutative martingales

Abstract

Given a probability space (Ω, Σ, µ), the Hardy space H 1 (Ω) which is associated to the martingale square function does not admit a classical atomic decomposition when the underlying filtration is not regular. In this paper we construct a decomposition of H 1 (Ω) into 'atomic blocks' in the spirit of Tolsa, which we will introduce for martingales. We provide three proofs of this result. Only the first one also applies to noncommutative martingales, the main target of this paper. The other proofs emphasize alternative approaches for commutative martingales. One might be well-known to experts, using a weaker notion of atom and approximation by atomic filtrations. The last one adapts Tolsa's argument replacing medians by conditional medians.

Nondoubling Calderón–Zygmund theory: a dyadic approach

“…In addition, we could also hope for an alternative form of regularity more adapted to the necessities in harmonic analysis. This is made precise in theorem A below, which confirms and makes rigorous the belief -somehow reflected in our recent work [3,4]-on a strong relationship between nondoubling measures and irregular filtrations satisfying the conditions above. Recall that a ball B(x, r) is called (α, β)-doubling when µ(B(x, αr)) ≤ βµ(B(x, r)).…”

“…• Fefferman-Stein duality: the predual of RBMO Σ (A) is, as in the commutative case, the Hardy space H 1 associated to the noncommutative dyadic square function. However, as shown in [4], it also admits an atomic block decomposition very similar to that explained in section 2.…”

Large BMO spaces vs interpolation

“…Our results above give some insight on the relation between nondoubling and martingale BMO theories, see [6,18] for other results along this line. In [6], we adapt Tolsa's ideas to give an atomic block description of martingale H 1 .…”

“…Our results above give some insight on the relation between nondoubling and martingale BMO theories, see [6,18] for other results along this line. In [6], we adapt Tolsa's ideas to give an atomic block description of martingale H 1 . Semigroup BMO spaces are used in [18] to construct a Calderón-Zygmund theory which incorporates noncommutative measure spaces (von Neumann algebras) to the picture.…”

“…The term 'atom' unfortunately appears here in several settings -σ-algebras, measures and Hardy spaces with different meanings-but it will be clear which one is used from the context. Atomic descriptions are not possible for arbitrary H 1 -see [6] for an 'atomic block' description both in the commutative/noncommutative settings-but there are such results for h 1 (defined above). A Σ-measurable function a ∈ L 2 (Ω) is called an atom when there exists k ≥ 1 and…”