Josephy's theorem, revisited

TL;DR

This work extends Josephy's theorem from real-valued BV functions to normed-space‑valued maps by characterizing when a composition operator $C_f$ maps $BV_p([a,b],E)$ into $BV_q([a,b],E)$ for $1\le p\le q<\infty$. The authors prove a necessary-and-sufficient condition: $f$ must be Hölder continuous on precompact subsets of $E$ with exponent $p/q$, and they show that in the vector-valued setting this does not automatically imply local boundedness of $C_f$. A counterexample demonstrates the separation between domain variation properties and operator boundedness. They then establish a precise local boundedness criterion: $C_f$ is locally bounded iff $f$ is Hölder on bounded subsets with exponent $p/q$, and they remark that any bounded $C_f$ must be constant. Overall, the paper provides a comprehensive, sharp extension of Josephy’s result to BV$_p$-valued maps, clarifying the roles of precompactness, Hölder continuity, and local boundedness in this abstract setting.

Abstract

The main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient conditions for local boundedness of such operators are also discussed.

Theorems & Definitions (10)

• Definition 1
• Proposition 2
• proof
• Example 3
• Theorem 4
• proof
• Example 5
• Theorem 6
• proof
• Remark 7