Maximal Inequalities for Martingales and Their Differential Subordinates

We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if $X=(X_t)_{tge 0},, Y=(Y_t)_{tge 0}$ are local martingales such that $Y$ is differentially subordinate to $X$ , then $$begin{aligned} ||Y||_1le beta ||sup _{tge 0}|X_t|;||_1, end{aligned}$$ where $beta =2.585ldots $ is the best possible.