Inequality for Burkholder's martingale transform

Abstract

We find the sharp constant $C=C\left(\tau ,p,\mathbb{E}G,\mathbb{E}F\right)$ of the inequality $\parallel {\left(G^2+{\tau }^2{F}^2\right)}^{1∕2}{\parallel }_p\le C\parallel F{\parallel }_p$, where $G$ is the transform of a martingale $F$ under a predictable sequence $\epsilon$ with absolute value 1, $1<p<2$, and $\tau$ is any real number.