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Interpolation between noncommutative martingale Hardy and BMO spaces: the case $\textbf {0<p<1}$

Part of: Stochastic processes Selfadjoint operator algebras Topological linear spaces and related structures Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  25 August 2021

Narcisse Randrianantoanina
Narcisse Randrianantoanina*
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA
*
e-mail: randrin@miamioh.edu
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Abstract

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ .

We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ ,

$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$
with equivalent quasi norms.

For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ ,

$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$
with equivalent quasi norms.

These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces.

We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Keywords

Noncommutative martingalesmartingale Hardy spacesinterpolation spaces

MSC classification

Primary: 46L53: Noncommutative probability and statistics 46B70: Interpolation between normed linear spaces 46L52: Noncommutative function spaces
Secondary: 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 60G50: Sums of independent random variables; random walks
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1700 - 1744
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Copyright
© Canadian Mathematical Society 2021

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