#### Vol. 8, No. 4, 2015

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
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
##### Milestones
Received: 8 March 2014
Revised: 1 February 2015
Accepted: 25 March 2015
Published: 21 June 2015
##### Authors
 Paata Ivanisvili Department of Mathematics Michigan State University 619 Red Cedar Road East Lansing, MI 48824 United States