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

Abstract

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