Atomic blocks for noncommutative martingales

Given a probability space $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$, the Hardy space ${\mathrm{H}}_1(\mathrm{\Omega })$ 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 ${\mathrm{H}}_1(\mathrm{\Omega })$ 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.