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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

A counterexample to Ljamin's theorem


Author: Piotr Maćkowiak
Journal: Proc. Amer. Math. Soc. 142 (2014), 1773-1776
MSC (2010): Primary 47H30, 26A45; Secondary 45G10
Published electronically: February 19, 2014
MathSciNet review: 3168482
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Abstract: One of the well-known results ensuring that a nonautonomous superposition operator maps the set of functions of one variable of bounded variation in the sense of Jordan into itself is the theorem by A. G. Ljamin. According to that theorem it suffices to consider the class of functions which are uniformly Lipschitz w.r.t. the second variable and of uniformly bounded variation w.r.t. the first variable. Unfortunately, Ljamin's result is false. Here we deliver an example contradicting sufficiency of those conditions.


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Additional Information

Piotr Maćkowiak
Affiliation: Department of Mathematical Economics, Poznań University of Economics, Al. Niepodległości 10, 61-875 Poznań, Poland
Email: p.mackowiak@ue.poznan.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11912-5
Keywords: Nonautonomous superposition operator, functions of bounded variation in the sense of Jordan
Received by editor(s): May 2, 2012
Received by editor(s) in revised form: June 26, 2012
Published electronically: February 19, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.