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

Randrianantoanina, Narcisse

Canadian journal of mathematics, Tome 74 (2022) no. 6, pp. 1700-1744

Résumé

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

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.

Détail

DOI : 10.4153/S0008414X21000419

Mots-clés : Noncommutative martingales, martingale Hardy spaces, interpolation spaces