A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L^ p(0,1)$

Abstract:

Let $1 < p < \infty$. Let $d = ({d_1},{d_2}, \ldots )$ be a real-valued martingale difference sequence, $\theta = ({\theta _1},{\theta _2}, \ldots )$ is a predictable sequence taking values in $[0,1]$. We show that the best constant of the inequality, \[ {\left \| {\sum \limits _{k = 1}^n {{\theta _k}{d_k}} } \right \|_p} \leq {c_p}{\left \| {\sum \limits _{k = 1}^n {{d_k}} } \right \|_p}, \quad n \geq 1,\] satisfies \[ {c_p} = \frac {p}{2} + \frac {1}{2}\;\log \;\left ({\frac {{1 + y}}{2}} \right ) + \frac {{{\alpha _2}}}{p} + \cdots ,\] where $\gamma = {e^{ - 2}}$ and ${\alpha _2} = {\left [ {\frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2}} \right ]^2} + \frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2} - 2{\left ({\frac {\gamma }{{1 + \gamma }}} \right )^2}$. The best constant equals the unconditional basis constant of a monotone basis of ${L^p}(0,1)$.