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Generalized $$\alpha$$-variation and Lebesgue equivalence to differentiable functions. (English) Zbl 1191.26003
A function $$f:[a,b]\to\mathbb R$$ is Lebesgue equivalent to $$g:[a,b]\to\mathbb R$$ provided there exists a homeomorphism $$h: [a,b]\to [a,b]$$ such that $$g=f\circ h$$. The author characterizes functions that are Lebesgue equivalent to $$n$$-times differentiable functions, where $$n\geq 2$$. (A simple solution of this problem for $$n=2$$ was given by the same author in an earlier paper.) For that purpose, he introduces two new classes of functions $$CBVG_{1/n}$$ and $$SBVG_{1/n}$$, analogous to the classes $$CBV_{1/n}$$ and $$SBV_{1/n}$$, introduced by M. Laczkovich and D. Preiss in [Indiana Univ. Math. J. 34, 405–424 (1985; Zbl 0557.26004)] to characterize functions Lebesgue equivalent to $$C^n$$ functions.
The main result of the paper says that the following conditions are equivalent: (1) $$f$$ is Lebesgue equivalent to a function $$g$$ which is $$n$$-times differentiable; (2) $$f$$ is Lebesgue equivalent to a function $$g$$ which is $$n$$-times differentiable and such that $$g^{(i)}(x)=0$$ whenever $$i\in\{ 1,\dots, n\}$$ and $$x\in K_g$$, and $$g'(x)\neq 0$$ whenever $$x\in [a,b]\setminus K_g$$; (3) $$f$$ is Lebesgue equivalent to a function $$g$$ which is $$n-1$$-times differentiable and such that the function $$g^{(n-1)}$$ is pointwise Lipschitz; (4) $$f$$ is $$CBVG_{1/n}$$; (5) $$f$$ is $$SBVG_{1/n}$$. (Here $$K_g$$ denotes the set of all points of varying monotonicity of $$g$$.) Moreover, it is shown that for each $$n\geq 2$$ there exists a continuous function which is $$CBVG_{1/n}$$, but not Lebesgue equivalent to any $$C^n$$ function.
In the next theorem the author characterizes functions that are Lebesgue equivalent to $$n$$-times differentiable functions with a.e. nonzero derivative. As a corollary, he obtains a generalization of Zahorski’s lemma for higher order differentiability. He proves that for a closed set $$M\subset [a,b]$$ there is an $$n$$-times differentiable homeomorphism $$h:[a,b]\to [a,b]$$ with $$M=h(\{ x\in [a,b]: h^{(i)}(x)=0$$ for all $$i=1,\dots, n\})$$ iff there exists a decomposition of $$M$$ such that certain variational conditions closely related to the definition of the class $$CBVG_{1/n}$$ (respectively, $$SBVG_{1/n}$$) are satisfied.

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