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

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with an increasing filtration $({\mathcal{M}}_n{)}_{n\ge 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $1\le p\le \infty$, let ${\mathcal{H}}_p^c(\mathcal{M})$ denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration $({\mathcal{M}}_n{)}_{n\ge 1}$ and the index $p$. We prove the following real interpolation identity: if $0<\theta <1$ and $1/p=1-\theta$, then \[ \big(\mathcal{H}_1^c(\mathcal{M}), \mathcal{H}_\infty^c(\mathcal{M})\big)_{θ,p}=\mathcal{H}_p^c(\mathcal{M}). \] 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.