Maximal inequalities for continuous martingales and their differential subordinates

Abstract:

Let $X=(X_t)_{t\geq 0}, Y=(Y_t)_{t\geq 0}$ be continuous-path martingales such that $Y$ is differentially subordinate to $X$. The paper contains the proofs of the sharp inequalities \[ \sup _{t\geq 0}||Y_t||_p \leq \sqrt {\frac {2}{p}} ||\sup _{t\geq 0}|X_t| ||_p, \quad 1\leq p< 2\] and \[ \sup _{t\geq 0}||Y_t||_p \leq (p-1) ||\sup _{t\geq 0}|X_t| ||_p, \quad 2\leq p<\infty .\]