Sharp maximal Lp-bounds for continuous martingales and their differential subordinates

Suppose that X, Y are Hilbert-space-valued continuous-path martingales such that Y is differentially subordinate to X. The paper contains the proof of sharp estimates between p-th moments of Y and the maximal function of X for 0 < p < 1. The proof rests on Burkholder's method and exploits a certain special function of three variables, enjoying appropriate size and concavity requirements. The analysis reveals an unexpected phase transition between the cases 0 < p < 1/2 and 1/2 <= p < 1. The latter case is relatively simple: the special function is essentially quadratic and the best constant is equal to root 2/p. The analysis of the former case is much more intricate and involves the study of a non-linear ordinary differential equation.