Doob L2 maximal inequality ($\mathbb{E}{\sup }_{s\le t}|M_s{|}^2\le 4\mathbb{E}|M_t{|}^2$)

For a square-integrable martingale starting at zero,For a continuous local martingale stopped so that its quadratic variation is integrable, the Itô isometry makes the right-hand side $4\mathbb{E}[M{]}_t$.

This is the $p=2$ specialization of Doob's martingale inequality.