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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 $[-1,1]$. Weprovide an explicit formula for Burkholder's function associated with theweighted $L^2$ bound $$ \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}.$$

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