Merryfield's inequality for multiparameter martingales

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 $0<p<\infty$.