Doob's inequality gives an estimation of
$$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$
where $X$ is a martingale. Now I wonder how to estimate
$$\mathbb{P}(\sup_{0\leq t,s\leq 1, |t-s|\leq\delta}|X_t-X_s|\geq\varepsilon)\ \ (\ast)$$
If $L^1$ integrability is not enough for the estimation, we can add some assumptions about the integrability of $X$. Does someon have an idea for this estimation of $(\ast)$? Thanks a lot!