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\(BV_{p}\)-functions and change of variable. (English) Zbl 1247.26016

Interconnections between the space \(BV_p[a,b]\) (\(1\leq p <\infty)\) of functions of bounded \(p\)-variation (in Wiener’s sense) and the space \(Lip_{\alpha}[a,b]\) (\(0<\alpha\leq 1\)) of Hölder continuous functions are discussed. In particular, it is shown show that \(f\in BV_p[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being monotone, and that \(f\in BV_p[a,b]\cap C[a,b]\) if and only if \(f=g\circ \tau\), with \(g\in Lip_{1/p}[a,b]\) and \(\tau\) being a homeomorphism.

MSC:

26A45 Functions of bounded variation, generalizations
26A16 Lipschitz (Hölder) classes
26A48 Monotonic functions, generalizations
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References:

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