Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda. "Generalized α-variation and Lebesgue equivalence to differentiable functions." Fundamenta Mathematicae 205.3 (2009): 191-217. <http://eudml.org/doc/283012>.

@article{JakubDuda2009,
abstract = {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 $CBVG_\{1/n\}$ and $SBVG_\{1/n\}$ 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 $CBV_\{1/n\}$ and $SBV_\{1/n\}$ (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 $f^\{(n-1)\}(·)$ 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.},
author = {Jakub Duda},
journal = {Fundamenta Mathematicae},
keywords = {differentiability; Lebesgue equivalence; differentiability via homeomorphism; Zahorski Lemma; pointwise Lipschitz function; fractional variations},
language = {eng},
number = {3},
pages = {191-217},
title = {Generalized α-variation and Lebesgue equivalence to differentiable functions},
url = {http://eudml.org/doc/283012},
volume = {205},
year = {2009},
}

TY - JOUR
AU - Jakub Duda
TI - Generalized α-variation and Lebesgue equivalence to differentiable functions
JO - Fundamenta Mathematicae
PY - 2009
VL - 205
IS - 3
SP - 191
EP - 217
AB - 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 $CBVG_{1/n}$ and $SBVG_{1/n}$ 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 $CBV_{1/n}$ and $SBV_{1/n}$ (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 $f^{(n-1)}(·)$ 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.
LA - eng
KW - differentiability; Lebesgue equivalence; differentiability via homeomorphism; Zahorski Lemma; pointwise Lipschitz function; fractional variations
UR - http://eudml.org/doc/283012
ER -