Inequality for Burkholder’s martingale transform

Ivanisvili, Paata

doi:10.2140/apde.2015.8.765

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Inequality for Burkholder's martingale transform

Paata Ivanisvili

Vol. 8 (2015), No. 4, 765–806

DOI: 10.2140/apde.2015.8.765

Abstract

We find the sharp constant $C = C (τ, p, E G, E F)$ of the inequality $∥ {(G^{2} + τ^{2} F^{2})}^{1 ∕ 2} ∥_{p} \leq C ∥ F ∥_{p}$ , where $G$ is the transform of a martingale $F$ under a predictable sequence $ε$ with absolute value 1, $1 < p < 2$ , and $τ$ 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

Vol. 8, No. 4, 2015