Real Analysis Exchange

BVp-Functions and Change of Variable

N. Merentes and J. L. Sánchez

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Abstract

In this note we discuss some interconnections between the space \(BV_p[a,b]\) (\(1\leq p\lt\infty\)) of functions of bounded \(p\)-variation (in Wiener's sense) and the space \(Lip_\alpha[a,b]\) (\(0\lt\alpha\leq 1\)) of Hölder continuous functions. In particular, we 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.

Article information

Source
Real Anal. Exchange, Volume 37, Number 1 (2011), 177-188.

Dates
First available in Project Euclid: 30 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1335806770

Mathematical Reviews number (MathSciNet)
MR3016858

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

Keywords
bounded \(p\)-variation Hölder continuity homeomorphism, change of variables

Citation

Merentes, N.; Sánchez, J. L. BV p -Functions and Change of Variable. Real Anal. Exchange 37 (2011), no. 1, 177--188. https://projecteuclid.org/euclid.rae/1335806770


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