A criterion for absolute continuity (Q2435173)

The authors, using methods based uniquely on evaluation of integrals, give a necessary and sufficient condition for a summable function to be equivalent to an absolutely continuous function. The main result says that a summable function \(f: [ a,b] \rightarrow \overline{\mathbb{R}}\) is equivalent to a function \(g :[a,b] \rightarrow \overline{\mathbb{R}}\) which is a \(C^{n-1}\)-function and \(g^{n-1}\) is absolutely continuous iff there exists a nonnegative summable function \(h:[a,b] \rightarrow \overline{\mathbb{R}}\) such that for all \(C^{\infty}\)-functions \(\varphi : [a,b] \rightarrow \overline{\mathbb{R}}\) with \(s\left( \varphi \right) \subset ( a,b) \), \[ \left| \int_{a}^{b}f \varphi ^{\left( n\right) }dx\right| \leq \left\| \varphi \right\| \int_{s\left( \varphi \right) }^{{}}hdx, \] where \(s\left( \varphi \right) \) denotes the support of \(\varphi \) and \(\left\| \varphi \right\| \) stands for the sup-norm of \(\varphi \). Moreover, the case of several variables is investigated.