Sharpness of Lenglart's domination inequality and a sharp monotone version

TL;DR

This paper establishes the sharpness of the best known constant in Lenglart's domination inequality and provides the optimal constant under a monotonicity assumption, solving an open problem and enhancing maximal inequality bounds.

Contribution

The paper proves the sharpness of the constant in Lenglart's domination inequality and determines the optimal constant under monotonicity, addressing open questions and improving related inequalities.

Findings

The constant c_p = p^{-p}/(1-p) is proven to be sharp.

The sharp constant under monotonicity is identified.

The results solve an open problem posed by Revuz and Yor.

Abstract

We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\, p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor. Motivated by the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy inequality, we also study the domination inequality under an additional monotonicity assumption. In this special case, a constant which stays bounded for $p$ near $1$ was proven by Pratelli and Lenglart. We provide the sharp constant for this case.