Vol. 8, No. 4, 2015

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Inequality for Burkholder's martingale transform

Paata Ivanisvili

Vol. 8 (2015), No. 4, 765–806
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, and $\tau$ is any real number.

Keywords
martingale transform, martingale inequalities, Monge–Ampère equation, torsion, least concave function, concave envelopes, Bellman function, developable surface
Mathematical Subject Classification 2010
Primary: 42B20, 42B35, 47A30