Generalized Α-variation and Lebesgue Equivalence to Differentiable Functions

We find an equivalent condition for a real function f : [a, b] → R to be Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of CBV G1/n and SBV G1/n functions, which play analogous rôles for the n-th order differentiability as the classical notion of a V BG∗ function for the first order differentiability, and the classes CBV1/n and SBV1/n (introduced by Preiss and Laczkovich) for C 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 (·) 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.