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 .

中文翻译：

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非对易鞅的 Lipschitz 空间的等价拟同范数

摘要 本文提出了非对易鞅的Lipschitz空间的等价拟模。作为应用，我们获得了非交换鞅哈代空间 hpc ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. hpr ( ℳ ) {h}_{p }^{r}( {\mathcal M} ) ) 和 Lipschitz 空间 λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) 对于 0 < p < 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1 {p}-1。我们还证明了 hpc ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) 和 hpr ( ℳ ) {h}_{p}^{r}( {\mathcal M } ) 对于 p = 1 p=1 或 2 < p < ∞ 2\lt p\lt \infty 。