Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and \(p\)-moments, \(1\leqslant p \leqslant 2\), of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on \(\mathrm{BMO}([0,1])\). In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.