A sharp maximal inequality for continuous martingales and their differential subordinates

Assume that X, Y are continuous-path martingales taking values in ℝν, ν ⩾ 1, such that Y is differentially subordinate to X. The paper contains the proof of the maximal inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {Y_t } \right|} \right\|_1 \leqslant 2\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {X_t } \right|} \right\|_1 .$$\end{document} 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.