P. Jones' interpolation theorem for noncommutative martingale Hardy spaces II

TL;DR

This paper establishes a new real interpolation identity for noncommutative martingale Hardy spaces, extending known results to more general filtrations and including Orlicz-Hardy spaces, thus advancing the theoretical understanding of noncommutative martingale spaces.

Contribution

It proves a novel interpolation identity for noncommutative Hardy spaces without the regular filtration assumption, also extending results to Orlicz-Hardy spaces.

Findings

Interpolation identity for noncommutative Hardy spaces established

Results hold without regular filtration assumption

Extension to noncommutative Orlicz-Hardy spaces

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $1\leq p \leq\infty$, let $\mathcal{H}_p^c(\mathcal{M})$ denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove the following real interpolation identity: if $0<\theta<1$ and $1/p=1-\theta$, then \[ \big(\mathcal{H}_1^c(\mathcal{M}), \mathcal{H}_\infty^c(\mathcal{M})\big)_{\theta,p}=\mathcal{H}_p^c(\mathcal{M}). \] This is new even for classical martingale Hardy spaces as it is previously known only under the assumption that the filtration is regular. We also obtain analogous result for noncommutative column martingale Orlicz-Hardy spaces.