Atomic blocks for noncommutative martingales
Abstract
Given a probability space , the Hardy space 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 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.
Cite
@article{arxiv.1407.5451,
title = {Atomic blocks for noncommutative martingales},
author = {Jose M. Conde-Alonso and Javier Parcet},
journal= {arXiv preprint arXiv:1407.5451},
year = {2014}
}
Comments
A proof of our main result for classical martingales follows from a weak notion of atom which was known to experts long ago (sketched in the paper for completeness). It however does not apply to the noncommutative setting, which remains the main novelty of the paper