Inequality for Burkholder's martingale transform

Abstract

We find the sharp constant $C=C(\tau,p, \mathbb{E}G, \mathbb{E}F)$ of the following inequality $\|(G^{2}+ \tau^{2} F^{2})^{1/2} \|_{p} \leq C \|F\|_{p},$ where $G$ is the transform of a martingale $F$ under a predictable sequence $\varepsilon$ with absolute value 1, $1<p< 2$, and $\tau$ is any real number.