Perturbation of Burkholder's martingale transform and Monge--Ampère equation

Abstract

Let $\{d_k{\}}_{k\ge 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\infty ,$ and $\{\e_k\}_{k \geq 0}$ a sequence in $\{\pm 1\}.$ We obtain the following generalization of Burkholder's famous result. If $\tau \in [-\frac{1}{2},\frac{1}{2}]$ and $n\in {\mathbb{Z}}_{+}$ then $$|\sum_{k=0}^n{(\{c} \e_k τ) d_k}|_{L^p([0,1], \C^2)} \leq ((p^*-1)^2 + τ^2)^{\frac 12}|\sum_{k=0}^n{d_k}|_{L^p([0,1], \C)},$$ where $((p^{\ast }-1{)}^2+{\tau }^2{)}^{\frac{1}{2}}$ is sharp and $p^{\ast }-1=\max \{p-1,\frac{1}{p-1}\}.$ For $2\le p<\infty$ the result is also true with sharp constant for $\tau \in \mathbb{R}.$

45 pages, 13 figures