Subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

TL;DR

This paper investigates how orthogonality in martingales affects $L^p$-norm estimates, showing that the constants in inequalities can be improved when orthogonality is present, especially for $1<p extless 2$.

Contribution

It extends previous results by demonstrating that orthogonality attached to the dominating martingale also reduces the constants in $L^p$ estimates for $1<p extless 2$, broadening the understanding of subordination in martingales.

Findings

Orthogonality reduces the $L^p$-norm estimate constants.

The paper applies the idea to the case where orthogonality is attached to the dominating martingale.

Constants are not proven to be sharp in this new setting.

Abstract

We are given two martingales on the filtration of the two dimensional Brownian motion. One is subordinated to another. We want to give an estimate of $L^p$-norm of a subordinated one via the same norm of a dominating one. In this setting this was done by Burkholder in \cite{Bu1}--\cite{Bu8}. If one of the martingales is orthogonal, the constant should drop. This was demonstrated in \cite{BaJ1}, when the orthogonality is attached to the subordinated martingale and when $2\le p<\infty$. This note contains an (almost obvious) observation that the same idea can be used in the case when the orthogonality is attached to a dominating martingale and $1<p\le 2$. Two other complementary regimes are considered in \cite{BJV_La}. When both martingales are orthogonal, see \cite{BJV_Le}. In these two papers the constants are sharp. We are not sure of the sharpness of the constant in the present note.