Sharp inequalities for linear combinations of orthogonal martingales

TL;DR

This paper establishes sharp $L^p$ inequalities for linear combinations of orthogonal martingales that are differentially subordinate, linking the best constants to the norms of specific operators related to the Hilbert transform.

Contribution

It provides the exact $L^p$ bounds for linear combinations of orthogonal martingales, connecting them to well-known operator norms, which were previously computed by Hollenbeck, Kalton, and Verbitsky.

Findings

Derived sharp $L^p$ inequalities for $aX+bY$

Connected martingale inequalities to Hilbert transform operator norms

Identified the best constants as operator norms from $L^p$ to $L^p"

Abstract

For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of the form $aX+bY$ with $a, b$ real numbers. The best $L^p$ constant is equal to the norm of the operator $aI+bH$ from $L^p$ to $L^p$, where $H$ is the Hilbert transform on the circle or real line. The values of these norms were found by Hollenbeck, Kalton and Verbitsky \cite{HKV}.