Moderate maximal inequalities for the Ornstein-Uhlenbeck process

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The maximal inequalities for diffusion processes have drawn increasing attention in recent years. Here we prove the moderate maximal inequality for the Ornstein-Uhlenbeck process, which includes the $L^p$ maximal inequality as a special case and generalizes the $L^1$ maximal inequality obtained by Graversen and Peskir [Proc. Amer. 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 an extension of the classical Burkholder-Davis-Gundy inequality.