  The maximal inequalities for diffusion processes have drawn increasing
attention in recent years. However, the existing proof of the $L^p$ maximum
inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a
rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck
process, which include the $L^p$ maximum inequalities as special cases and
generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and
Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also
obtain a new moderate maximal inequality for continuous local martingales,
which can be viewed as a supplement of the classical Burkholder-Davis-Gundy
inequality.
