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 $X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}$ are local martingales such that $Y$ is differentially subordinate to $X$ , then $$\begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|||_1, \end{aligned}$$ where $\beta =2.585\ldots $ is the best possible.