Interpolation of noncommutative quasimartingale spaces

Abstract

Let ${\hat{L}}_p\left(\mathrm{M}\right)$ be the space of bounded $L_p\left(\mathrm{M}\right)$-quasimartingales. We prove that, with equivalent norms, $\left({\hat{L}}_{p_0}\right(\mathrm{M}),{\hat{L}}_{p_1}(\mathrm{M}){)}_{\theta ,p}={\hat{L}}_p(\mathrm{M})$, where $1<p_0,p_1\le \infty$, $1<\theta <1$, and $\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}$. We also prove that, for $1<p<q<\infty$, $\left({\widehat{BMO}}^c\right(\mathrm{M}),{\hat{\mathrm{H}}}_p^c(\mathrm{M}){)}_{\frac{p}{q},q}={\hat{\mathrm{H}}}_q^c(\mathrm{M})$ and $\left({\widehat{BMO}}^r\right(\mathrm{M}),{\hat{\mathrm{H}}}_p^r(\mathrm{M}){)}_{\frac{p}{q},q}={\hat{\mathrm{H}}}_q^r(\mathrm{M})$, where ${\hat{\mathrm{H}}}_p\left(\mathrm{M}\right)$ and $\widehat{BMO}\left(\mathrm{M}\right)$ are, respectively, the Hardy space and the bounded mean oscillation space of noncommutative quasimartingales.