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

Abstract: Let ${d_k}<em>{k \geq 0}$ be a complex martingale difference in $L<sup>p[0,1],$ where $1&lt;p&lt;\infty,$ and ${\e_k}</em>{k \geq 0}$ 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 $$|\sum_{k=0}^n{({c} \e_k \tau) d_k}|<em>{L^p([0,1], \C^2)} \leq ((p^*-1)² + \tau^{2)^{\frac} 12}|\sum</em>{k=0}^{n{d_k}|_{L^p([0,1],} \C)},$$ where $$((p^*-1)² + \tau^{2)^{\frac} 12}$$ is sharp and $p^*-1 = \max{p-1, \frac 1{p-1}}.$ For $2\leq p&lt;\infty$ the result is also true with sharp constant for $\tau \in \R.$