A counterexample to Ljamin’s theorem

Maćkowiak, Piotr

doi:10.1090/S0002-9939-2014-11912-5

A counterexample to Ljamin’s theorem
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by Piotr Maćkowiak PDF

Proc. Amer. Math. Soc. 142 (2014), 1773-1776 Request permission

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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