I encountered a following formulation of the Doob's $L^p$ inequality.
Let $T\neq\emptyset$ be a totally ordered set and suppose that $X=(X_t)_{t\in T}$ is a positive submartingale. Define $X^*=\mathrm{ess}\,\mathrm{sup}_{t\in T}X_t$ and $X_u^*= \mathrm{ess}\,\mathrm{sup}_{t\in T, t\le u}X_t$ for every $u\in T$. Then, for every $p>1$, $$ \mathbb{E}[(X_u^*)^p]\le \left(\frac{p}{p-1}\right)^p\mathbb{E}[(X_u)^p], \quad u\in T $$ and $$ \mathbb{E}[(X^*)^p]\le \left(\frac{p}{p-1}\right)^p \sup_{u \in T} \mathbb{E}[(X_u)^p].$$
In the textbook it is mentioned as a side note, that the factor $p/(p-1)$ is optimal. I am missing an explanation to this. I tried to show it by taking another smaller factor and (hopefully) show that the inequality is not valid anymore, but I can‘t seem to get to a solution. I hope someone can give me some input whether my approach is right and if not, how can I show that this factor is optimal. Thanks.
For simplicity, I'm going to restrict myself to intervals with finite time horizon; i.e. I'm going to write $[0,T]$ for $T$.
To show that this factor is optimal, it suffices to show that there is a continuous martingale for which equality in Doob's $L^p$ inequality can be attained. We may do so by constructing a martingale such that $\alpha S_T = S_T^*$ (i.e. the supremum is a constant multiple of the running maximum).
Let $B = (B)_{t \in [0,T]}$ be a Brownian motion with $B_0 = 1$. Define the stopping time $$\tau_\alpha = \inf \{ t > 0 \, : \, B_t \leq B_t^* / \alpha \}$$ Observe that $B^{\tau_\alpha}$ is a uniformly integrable martingale. Finally, the process $$S_t = B_{\frac{t}{T-t} \land \tau^\alpha}$$ is a non-negative martingale which, by definition of the stopping time $\tau_\alpha$, satisfies $\alpha S_T = S_T^*$. Setting $\alpha = \frac{p-1}{p}$, we get equality in Doob's $L^p$-maximal inequality.
Reference
Acciaio, Beatrice, Mathias Beiglböck, Friedrich Penkner, Walter Schachermayer, and Johannes Temme. "A trajectorial interpretation of Doob’s martingale inequalities." The Annals of Applied Probability 23, no. 4 (2013): 1494-1505.