Perturbation of Burkholder's martingale transform and Monge--Amp\`ere equation

TL;DR

This paper generalizes Burkholder's martingale transform inequality by introducing a perturbation parameter, providing sharp bounds related to the Monge--Ampère equation, with implications for complex martingales in $L^p$ spaces.

Contribution

It extends Burkholder's classical result to include a perturbation parameter, establishing sharp bounds and connecting to the Monge--Ampère equation.

Findings

Derived sharp bounds for perturbed martingale transforms.

Extended results to complex $L^p$ martingales with perturbation parameter.

Established connections to the Monge--Ampère equation.

Abstract

Let $\{d_k\}_{k \geq 0}$ 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 $\tau \in [-\frac 12, \frac 12]$ and $n \in \Z_+$ then $$|\sum_{k=0}^n{(\{c} \e_k \tau) d_k}|_{L^p([0,1], \C^2)} \leq ((p^*-1)^2 + \tau^2)^{\frac 12}|\sum_{k=0}^n{d_k}|_{L^p([0,1], \C)},$$ where $((p^*-1)^2 + \tau^2)^{\frac 12}$ is sharp and $p^*-1 = \max\{p-1, \frac 1{p-1}\}.$ For $2\leq p<\infty$ the result is also true with sharp constant for $\tau \in \R.$