Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Abstract

AbstractLet$\mathcal {M}$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.