Sharp inequalities for linear combinations of orthogonal martingales

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}.