An Optimal Estimate of the $L^r$-$\delta$-Variation

Abstract

A notion of Lr-δ-variation of a function in Lr which plays an essential role in the theory of the Henstock-Kurzweil integral for functions in Lr (HKr-integral) is studied. We obtain an improved, in fact the best possible, estimate from below for this variation via the classical variation. We show that the class of functions having a finite Lr-δ-variation on an interval coincides with the class of functions of bounded variation on this interval. As a by-product of the results of the paper we obtain a new proof of the uniqueness of the HKr-integral.