Sharp maximal $L^p$-estimates for martingales

Abstract

Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and ${\mathfrak{c}}_p$ such that $$\underset{t\ge 0}{\sup }{\left|\left|X_t\right|\right|}_p\le C_p{\left|\left|-\underset{t\ge 0}{\inf }{X}_t\right|\right|}_p,$$ $$||\underset{t\ge 0}{\sup }{X}_t|{|}_p\le c_p{\left|\left|-\underset{t\ge 0}{\inf }{X}_t\right|\right|}_p$$ and $$||\underset{t\ge 0}{\sup }|X_t||{|}_p\le {\mathfrak{c}}_p{\left|\left|-\underset{t\ge 0}{\inf }{X}_t\right|\right|}_p.$$ The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities are deduced from the existence of special functions, enjoying certain majorization and convexity-type properties. Some applications concerning harmonic functions on Euclidean domains are indicated.