Generalized $α$-variation and Lebesgue equivalence to differentiable functions

Abstract

We find an equivalent condition for a real function $f:[a,b]\to\R$ to be Lebesgue equivalent to an $n$-times differentiable function ($n\geq 2$); a simple solution in the case $n=2$ appeared in an earlier paper. For that purpose, we introduce the notions of $CBVG_{1/n}$ and $SBVG_{1/n}$ functions, which play analogous roles for the $n$-th order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability, and the classes $CBV_{1/n}$ and $SBV_{{1}/{n}}$ (introduced by Preiss and Laczkovich) for $C^n$ smoothness. As a consequence of our approach, we obtain that Lebesgue equivalence to $n$-times differentiable function is the same as Lebesgue equivalence to a function $f$ which is $(n-1)$-times differentiable with $f^{(n-1)}(\cdot)$ being pointwise Lipschitz. We also characterize the situation when a given function is Lebesgue equivalent to an $n$-times differentiable function $g$ such that $g'$ is nonzero a.e. As a corollary, we establish a generalization of Zahorski's Lemma for higher order differentiability.