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arXiv · 1907.11633

Vector-valued $q$-variational inequalities for averaging operators and Hilbert transform

Guixiang Hong·Wei Liu·Tao Ma
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Abstract

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.

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Guixiang Hong, Wei Liu, Tao Ma. 2019-07-26. Vector-valued $q$-variational inequalities for averaging operators and Hilbert transform. https://arxiv.org/abs/1907.11633

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