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Uniformly continuous superposition operators in the space of bounded variation functions

✍ Scribed by Janusz Matkowski

Publisher: John Wiley and Sons
Year: 2010
Tongue: English
Weight: 85 KB
Volume: 283
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200710126

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✦ Synopsis

Abstract

Let I, J ⊂ ℝ be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ℝ,

H (φ)(x) ≔ h (x, φ (x)),

maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ℝ) and is uniformly continuous with respect to the BV ‐norm, then

h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J,

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