Recently, in \cite{GXHTM}, the authors established $L^p$-boundedness of vector-valued $q$-variational inequalities for averaging operators which take values in the Banach space satisfying martingale cotype $q$ property. In this paper, we prove that martingale cotype $q$ property is also necessary for the vector-valued $q$-variational inequalities, which is a question left open. Moreover, we characterize UMD property and martingale cotype $q$ property in terms of vector valued $q$-variational inequalities for Hilbert transform.

中文翻译：

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平均算子和希尔伯特变换的向量值 q 变分不等式

最近，在 \cite{GXHTM} 中，作者建立了向量值 $q$-变分不等式的 $L^p$-有界性，用于取值在 Banach 空间中满足鞅共型 $q$ 属性的平均算子。在本文中，我们证明了鞅cotype $q$ 属性对于向量值$q$-变分不等式也是必要的，这是一个悬而未决的问题。此外，我们根据希尔伯特变换的向量值 $q$-变分不等式来表征 UMD 属性和鞅共型 $q$ 属性。