We find conditions on a real function f:[a,b] → ℝ equivalent to being 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 and functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes and (introduced by Preiss and Laczkovich) for Cⁿ smoothness. As a consequence, we deduce that Lebesgue equivalence to an n-times differentiable function is the same as Lebesgue equivalence to a function f which is (n-1)-times differentiable with pointwise Lipschitz. We also characterize functions that are Lebesgue equivalent to n-times differentiable functions with a.e. nonzero derivatives. As a corollary, we establish a generalization of Zahorski’s Lemma for higher order differentiability.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-3-1,
author = {Jakub Duda},
title = {Generalized $\alpha$-variation and Lebesgue equivalence to differentiable functions},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {191-217},
zbl = {1191.26003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-3-1}
}
Jakub Duda. Generalized α-variation and Lebesgue equivalence to differentiable functions. Fundamenta Mathematicae, Tome 205 (2009) pp. 191-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-3-1/