Let $X$ be a continuous local martingale such that $\vert X_t\vert^p$ is uniformly integrable for each $t\ge 0$ and some $p\ge 1$. Does it follow that $X$ is uniformly integrable?
I am not sure how to use the $p$ here. Since $X$ is a continuous local martingale, we have a sequence $(\tau_n)$ of stopping times (increasing and tending to infinity) for which $X^{\tau_n}$ is uniformly integrable. Therefore, for each $n$, $X_t ^{n\wedge \tau_n}=X_{n\wedge t}^{\tau_n}=\mathbb{E}(X_n^{\tau_n}\vert \cal{F}_{n\wedge t})$.
Is it possible to show that $X$ is uniformly integrable?
As the following example shows, $(X_t)_{t \geq 0}$ is, in general, not uniformly integrable:
Let $(X_t)_{t \geq 0}$ be a one-dimensional Brownian motion. Then
$$\mathbb{E}(|X_t| 1_{\{|X_t| \geq R\}}) = 2\int_{R}^{\infty} x \frac{1}{\sqrt{2 \pi t}} \exp \left(- \frac{x^2}{2t} \right) \, dx= \sqrt{\frac{2t}{\pi}} e^{-R^2/2t} \tag{1}$$
which shows that $|X_t|$ is uniformly integrable for each $t>0$; for $t=0$ this is trivial. In fact, $|X_t|^p$ is uniformly integrable for any $p \geq 1$ and fixed $t \geq 0$. However, $(1)$ also shows that $(X_t)_{t \geq 0}$ is not uniformly integrable since the right-hand side of $(1)$ converges to $\infty$ as $t \to \infty$.