Generalized $\alpha$-variation and Lebesgue equivalence to differentiable functions

Tom 205 / 2009

Jakub Duda Fundamenta Mathematicae 205 (2009), 191-217 MSC: Primary 26A24; Secondary 26A45. DOI: 10.4064/fm205-3-1


We find conditions on a real function $f:[a,b]\to\mathbb R$ equivalent to being 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 rôles for the $n$th order differentiability to 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, 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 $f^{(n-1)}(\cdot)$ 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.


  • Jakub DudaDepartment of Mathematics
    Weizmann Institute of Science
    Rehovot 76100, Israel
    PIRA Energy Group
    3 Park Ave FL 26
    New York, NY 10016, U.S.A.

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