Generalized α-variation and Lebesgue equivalence to differentiable functions

• Autorzy: Jakub Duda
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26A45: Functions of bounded variation, generalizations
• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 205
• Numer: 3
• Strony: 191-217

Daty

wydano

2009

Twórcy

autor

Jakub Duda

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

Identyfikatory

DOI

10.4064/fm205-3-1