An equivalent quasinorm for the Lipschitz space of noncommutative martingales

Abstract

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 h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0 < p < 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) and h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) for p = 1 p=1 or 2 < p < ∞ 2\lt p\lt \infty .