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

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.