Burkholder’s function via Monge–Ampère equation

Abstract

We will show how to get Burkholder’s function from (Ann. Probab. 12 (1984) 647–702) by using Monge–Ampère equation. This method is quite different from those in the series of Burkholder’s papers (Ann. Probab. 12 (1984) 647–702, An extension of classical martingale inequality (1986) Marcel Dekker, Astérisque 157–158 (1988) 75–94, In Harmonic analysis and partial differential equations (1989) 1–23 Springer, In École d’Ete de Probabilités de Saint-Flour XIX (1991) 1–66 Springer, Ann. Probab. 22 (1994) 995–1025, Studia Math. 91 (1988) 79–83).