Generalized α-variation and Lebesgue equivalence to differentiable functions

2005/12/18 by Jakub Duda, Duda, Jakub
Engineering · #Reservoir Engineering and Simulation Methods #Advanced Numerical Analysis Techniques #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.math/0512428

Abstract

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 CBVG_1/n and SBVG_1/n functions, which play analogous roles for the n-th order differentiability as 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 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^(n-1)(⋅) 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.

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