A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales

Abstract:

If $(d_{n})_{n\geq 0}$ is a martingale difference sequence, $(\varepsilon _{n})_{n\geq 0}$ a sequence of numbers in $\{ 1,-1\}$, and $n$ a positive integer, then \[ P(\max _{0\leq m\leq n}\vert \sum _{k=0}^{m} \varepsilon _{k}d_{k}\vert \geq 1) \leq \alpha _{p}\Vert \sum _{k=0}^{n} d_{k}\Vert _{p}^{p}. \] Here $\alpha _{p}$ denotes the best constant. If $1\leq p\leq 2$, then $\alpha _{p}= 2/\Gamma (p+1)$ as was shown by Burkholder. We show here that $\alpha _p=p^{p-1}/2$ for the case $p > 2$, and that $p^{p-1}/2$ is also the best constant in the analogous inequality for two martingales $M$ and $N$ indexed by $[0,\infty )$, right continuous with limits from the left, adapted to the same filtration, and such that $[M,M]_t-[N,N]_t$ is nonnegative and nondecreasing in $t$. In Section 7, we prove a similar inequality for harmonic functions.