Sharp Inequalities for Linear Combinations of Orthogonal Martingales

For any two real-valued continuous-path martingales X = {X_t}_t≥0 and Y = {Y_t}_t≥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 [Studia Math., 2003, 157(3): 237–278]. We also give applications of our martingale inequalities to Riesz transforms and some discrete operators.