Maps of several variables of finite total variation. I: Mixed differences and the total variation
The following three technical results are proved for functions \(f\) defined on an \(n\)-dimensional rectangle with values in a metric semigroup (i.e., with a metric which is invariant under addition): 1. An additivity formula for a variation of Vitaly-Hardy-Kraus type with respect to the domain. 2. An estimate of the distance of \(f(x)\) and \(f(y)\) in terms of the total variation of \(f\) in the sense of Hildebrandt-Leonov on the rectangle with corners \(x\) and \(y\). 3. A Fatou type lemma for the total variation: If a sequence of functions with bounded total variation converges pointwise, its limit is of bounded total variation (with the bound being the corresponding limes inferior of bounds). The technicalities arise of course from the general setting, since for scalar functions on intervals these properties are well-known and easy to see. All results are preparations for a Helly type selection theorem shown in part II [ibid. 369, No. 1, 82--93 (2010; Zbl 1200.26017)].
functions of several variables with values in a metric semigroup
scientific article; zbMATH DE number 5757988
scientific article