Burkholder's function and a weighted $L^2$ bound for stochastic integrals

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 $$\mathrm{\parallel }Y{\mathrm{\parallel }}_{L^2(W)}\lesssim [w{]}_{A_2}\mathrm{\parallel }X{\mathrm{\parallel }}_{L^2(W)}\mathrm{.}$$