An equivalent quasinorm for the Lipschitz space of noncommutative martingales

Abstract

In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space hpc(ℳ){h}_{p}^{c}( {\mathcal M} ) (resp. hpr(ℳ){h}_{p}^{r}( {\mathcal M} )) and the Lipschitz space λβc(ℳ){\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λβr(ℳ){\lambda }_{\beta }^{r}( {\mathcal M} )) for 0<p<10\lt p\lt 1, β=1p−1\beta =\tfrac{1}{p}-1. We also prove some equivalent quasinorms for hpc(ℳ){h}_{p}^{c}( {\mathcal M} ) and hpr(ℳ){h}_{p}^{r}( {\mathcal M} ) for p=1p=1 or 2<p<∞2\lt p\lt \infty .

Keywords

• noncommutative space
• martingale
• hardy space
• 46l53
• 46l52
• 60g42