## Sharp Norm Inequalities for Martingales and their Differential Subordinates

### Volume 55 / 2007

Bulletin Polish Acad. Sci. Math. 55 (2007), 373-385
MSC: Partially supported by MEiN Grant 1 PO3A 012 29.
DOI: 10.4064/ba55-4-9

#### Abstract

Suppose $f=(f_n)$, $g=(g_n)$ are martingales with respect to the same filtration, satisfying $$ |f_n-f_{n-1}| \leq |g_n-g_{n-1}|,\quad \ n=1,2,\ldots, $$ with probability $1$. Under some assumptions on $f_0$, $g_0$ and an additional condition that one of the processes is nonnegative, some sharp inequalities between the $p$th norms of $f$ and $g$, $0< p< \infty$, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.