  Let $X$ be a continuous-path martingale and let $Y$ be a stochastic integral,
with respect to $X$, of some predictable process with values in $[-1,1]$. We
provide an explicit formula for Burkholder's function associated with the
weighted $L^2$ bound $$ \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}.$$
