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.

设 $\mathcal {M}$ 是一个半有限 von Nemann 代数，它配备了$\mathcal {M}的（半有限）von Neumann 子代数的递增过滤 $(\mathcal {M}_n)_{n\geq 1} $ $ 。对于 $0<p <\infty $ ，让 $\mathsf {h}_p^c(\mathcal {M})$ 表示非交换列条件鞅哈代空间和 $\mathsf {bmo}^c(\mathcal {M} )$ 表示与过滤 $(\mathcal {M}_n)_{n\geq 1}$ 关联的列“小”鞅 BMO 空间。

我们证明了以下实插值恒等式：如果 $0<p <\infty $ 且 $0<\theta <1$ ，则对于 $1/r=(1-\theta )/p$ ， $$ \begin{align*} \ big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{ r}^c(\mathcal{M}), \end{align*} $$ 具有等效的准范数。

对于复杂插值的情况，我们得到如果 $0<p<q<\infty $ 和 $0<\theta <1$ ，那么对于 $1/r =(1-\theta )/p +\theta /q$ ， $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta} =\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ 具有等效的准范数。

这些将先前已知的结果从 $p\geq 1$ 扩展到整个范围 $0<p<\infty $ 。其他相关空间，如自适应序列空间和 Junge 的非交换条件 $L_p$ 空间，也显示在使用实数方法或复数方法时形成整个范围 $0<p<\infty$ 的 插值尺度。我们的证明方法基于 Junge 的非交换条件 $L_p$ - 空间的 Orlicz 空间版本的新代数原子分解。

我们应用这些结果来推导非交换对称拟 Banach 空间中鞅的各种不等式。