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

## TL;DR

This paper investigates the conditions under which vector-valued $q$-variational inequalities hold for averaging operators and the Hilbert transform, establishing necessity of martingale cotype $q$ and characterizing UMD and cotype properties.

## Contribution

It proves that martingale cotype $q$ is necessary for vector-valued $q$-variational inequalities and characterizes UMD and cotype properties via these inequalities.

## Key findings

- Martingale cotype $q$ is necessary for inequalities.
- Characterization of UMD and cotype properties.
- Extension of previous $L^p$-boundedness results.

## 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.