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

Let $\{d_k{\}}_{k\ge 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\mathrm{\infty },$ and $\{{\text{e}}_k{\}}_{k\ge 0}$ a sequence in $\{\pm 1\}\mathrm{.}$ 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$((p^-1)^2 + \tau^2)^{\frac 12}$issharpand$p^-1 = \max{p-1, \frac 1{p-1}}.$For$2\leq p<\infty$ the result is also true with sharp constant for $\tau \in \mathbb{R}\mathrm{.}$