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

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 [12]. 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.