A sharp maximal inequality for continuous martingales and their differential subordinates

Assume that X, Y are continuous-path martingales taking values in R-nu, nu >= 1, such that Y is differentially subordinate to X. The paper contains the proof of the maximal inequality parallel to sup(t >= 0) vertical bar Y-t vertical bar parallel to(1) <= 2 parallel to sup(t >= 0) vertical bar X-t vertical bar parallel to(1). The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.