Merryfield's inequality for multiparameter martingales

TL;DR

This paper extends Merryfield's inequality to discrete multiparameter martingales, establishing an $L^p$ comparison between the maximal and square functions for regular filtrations, enhancing understanding of their behavior.

Contribution

It introduces a discrete version of Merryfield's inequality for multiparameter martingales, providing new $L^p$ bounds for maximal and square functions.

Findings

Established $L^p$ comparison between maximal and square functions

Extended Merryfield's inequality to discrete multiparameter setting

Applicable for regular multiparameter filtrations

Abstract

We extend an inequality of Merryfield, valid in the continuous setting, to discrete multiparameter martingales. As a consequence, we obtain the $L^p$ comparison of the maximal function with the square function: \begin{align*} E[(Sf)^p] \lesssim E[(f^*)^p] \end{align*} for regular multiparameter filtrations and $0 < p < \infty$.