  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.
