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 \begin{eqnarray*}\sup_{t\geq0}\big\|X_{t}\big\|_{p}&\leq&C_{p}\big\|-\inf_{t\geq0}X_{t}\big\|_{p},\\\phantom{\Vert }\Vert \sup_{t\geq0}X_{t}\Vert _{p}&\leq&c_{p}\|-\inf_{t\geq0}X_{t}\|_{p}\end{eqnarray*} and \[\Vert \sup_{t\geq0}\vert X_{t}\vert \Vert _{p}\leq\mathfrak{c}_{p}\|-\inf_{t\geq0}X_{t}\|_{p}.\] 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.