Open Access
Spring 2014 Sharp maximal $L^{p}$-estimates for martingales
Rodrigo Bañuelos, Adam Osȩkowski
Illinois J. Math. 58(1): 149-165 (Spring 2014). DOI: 10.1215/ijm/1427897172

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.

Citation

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Rodrigo Bañuelos. Adam Osȩkowski. "Sharp maximal $L^{p}$-estimates for martingales." Illinois J. Math. 58 (1) 149 - 165, Spring 2014. https://doi.org/10.1215/ijm/1427897172

Information

Published: Spring 2014
First available in Project Euclid: 1 April 2015

zbMATH: 1316.60059
MathSciNet: MR3331845
Digital Object Identifier: 10.1215/ijm/1427897172

Subjects:
Primary: 31B05 , 60G40 , 60G44

Rights: Copyright © 2014 University of Illinois at Urbana-Champaign

Vol.58 • No. 1 • Spring 2014
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