IUMJ

Title: Sharp $L^p$-bounds for a small perturbation of Burkholder's martingale transform

Authors: Nicholas Boros, Prabhu Janakiraman and Alexander Volberg

Issue: Volume 61 (2012), Issue 2, 751-773

Abstract:

Let $\{d_k\}_{k\geq 0}$ be a complex martingale difference in $L^p[0,1]$, where $1<p<\infty$, and $\{\epsilon_k\}_{k\geq 0}$ be 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 \[ \bigg\|\sum_{k=0}^n\binom{\epsilon_k}{\tau}d_k\bigg\|_{L^p([0,1),\mathbb{C}^2)} \leq ((p^{*}-1)^2 + \tau^2)^{1/2}\Big\|\sum_{k=0}^n{d_k}\Big\|_{L^p([0,1),\mathbb{C})}, \] where $((p^{*}-1)^2 + \tau^2)^{1/2}$ is sharp and $p^{*}-1=\max\{p-1,1/(p-1)\}$. For $2\leq p<\infty$ the result is also true with sharp constant for $|\tau|\leq 1$.