Young's functional with Lebesgue-Stieltjes integrals

Abstract

For non-decreasing real functions $f$ and $g$, we consider the functional $ T(f,g ; I,J)=\int_{I} f(x)\di g(x) + \int_J g(x)\di f(x)$, where $I$ and $J$ are intervals with $J\subseteq I$. In particular case with $I=[a,t]$, $J=[a,s]$, $s\leq t$ and $g(x)=x$, this reduces to the expression in classical Young's inequality. We survey some properties of Lebesgue-Stieltjes interals and present a new simple proof for change of variables. Further, we formulate a version of Young's inequality with respect to arbitrary positive finite measure on real line including a purely discrete case, and discuss an application related to medians of probability distributions and a summation formula that involves values of a function and its inverse at integers.