Generalized \(\alpha\)-variation and Lebesgue equivalence to differentiable functions.

*(English)*Zbl 1191.26003A 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.

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.

Reviewer: Tomasz Natkaniec (Gdańsk)

##### MSC:

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

26A45 | Functions of bounded variation, generalizations |