Alternate characterizations of bounded variation and of general monotonicity for functions (Q2362699)

In this paper, the author develops necessary and sufficient conditions for a function to be equal almost everywhere to a function of bounded variation. With these conditions one can broaden the class of general monotone functions but retaining some of the properties of functions in this class. The key point is to replace the variation of a function by a kind of oscillation. A function \(f\) defined on \((0,\infty)\) is said to be general monotone with majorant function \(\beta(x)\geq 0\) (\(f\in\mathrm{GM}(\beta)\)) if \(f\) is locally of bounded variation on \((0,\infty)\), \(\lim\limits_{x\to\infty}f(x)=0\), and there is a constant \(C>0\) such that \[ \int^{2x}_{x} |df(t)|\leq C\beta (x). \] Let now denote by \(Af[a,b]\) the average of \(f\in L^{1} ([a,b])\) on~\([a,b]\). For \(I\) a subinterval of~\(\mathbb{R}\), \(f\) locally integrable on~\(I\) and \(a,b\in I\) define \(\lambda'_{f,a}(b)=\lambda''_{f,a}(b)=0\) when \(b\leq a\) and for \(b>a\): \[ \begin{aligned} \lambda'_{f,a}(b)&=\sup \sum^{n}_{k=1} \frac{1}{d_{k}-d_{k-1}} \int^{d_{k}}_{d_{k-1}} |f(t)-Af[d_{k-1},d_{k}]|\,dt,\\ \lambda''_{f,a}(b)&=\sup \sum^{n-1}_{k=1} |Af [d_{k},d_{k-1}] -Af [d_{k-1},d_{k}]|, \end{aligned} \] where the suprema are taken over all partitions~\(\{d_{k}\}\) of~\([a,b]\). The class \(\mathrm{GM}'(\beta)\) (resp.~\(\mathrm{GM}''(\beta)\)) consists of functions~\(f\) locally integrable on~\((0,\infty)\) with \(\lim\limits_{x\to\infty}f(x)=0\), such that there is a constant~\(C\) with \[ \lambda'_{f,x}(2x)\leq C\beta(x)\quad (\text{resp.\;}\lambda''_{f,x}(2x)\leq C\beta(x)). \] It turns out that \(\lambda'_{f,x}(2x)\) and \(\lambda''_{f,x}(2x)\) are of the same order and as a consequence it is \(\mathrm{GM}'(\beta)=\mathrm{GM}''(\beta)\). The main results are the following: -- Let \(f\) be of bounded variation on~\([a,b]\). Then \[ \lambda''_{f,a}(b)\leq \int^{b}_{a} |df(t)|. \] -- Let \(f\) be locally integrable on \([a,b]\), \(\lambda''_{f,a}(b)<\infty\). Then there exists \(g\) of bounded variation on~\([a,b]\) such that \(f=g\) a.e., and \[ \int^{b}_{a}|dg(t)\leq \lambda''_{f,a}(b). \] From this, it follows that any general monotone function with majorant function~\(\beta(x)\) is in the class~\(\mathrm{GM}''(\beta)\) and for every function~\(f\in \mathrm{GM}''(\beta)\) there is a general monotone function~\(g\) such that \(f=g\) a.e. As well, functions \(f\) equal a.e. to a function of bounded variation are characterized by the finiteness of \(\lambda''_{f,a}(b)\). For the special case of \(\beta(x)=\displaystyle\int^{cx}_{x/c} |f(t)| \dfrac{dt}{t}\), \(c\) some constant, functions in \(\mathrm{GM}(\beta)\) satisfy some essential growth properties. These properties are shared almost everywhere by functions in~\(\mathrm{GM}''(\beta)\).