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

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 $\bmo^c(\M)$ denote the column \lq\lq little\rq\rq \ 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 $1/r=(1-\theta)/p$, $$\big(\mathsf{h}_p^c(\mathcal{M}), \bmo^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}),$$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty$ and $0<\theta<1$, then for $1/r =(1-\theta)/p +\theta/q$, $$\big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M})$$ 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.