Boundary Value Problems and Sharp Inequalities for Martingale Transforms

Let $p^\ast$ be the maximum of $p$ and $q$ where $1 < p < \infty$ and $1/p + 1/q = 1$. If $d = (d_1, d_2, \cdots)$ is a martingale difference sequence in real $L^p(0, 1), \varepsilon = (\varepsilon_1, \varepsilon_2, \cdots)$ is a sequence of numbers in $\{-1, 1\}$, and $n$ is a positive integer, then $\|\sum^n_{k=1} \varepsilon_kd_k\|_p \leq (p^\ast - 1) \|\sum^n_{k=1} d_k\|_p$ and the constant $p^\ast - 1$ is best possible. Furthermore, strict inequality holds if and only if $p \neq 2$ and $\|\sum^n_{k=1} d_k\|_p > 0$. This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if $\varepsilon$ is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case $p = 2$. The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of $L^p(0, 1)$ is $p^\ast - 1$. The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.