Nonlinear composition operators in bv_p spaces: continuity and compactness
Daria Bugajewska, Piotr Kasprzak
TL;DR
The paper advances the theory of nonlinear composition operators $C_f$ on sequence spaces, notably $bv_p(E)$, by characterizing a full spectrum of continuity properties (pointwise, local uniform, uniform, Lipschitz, and Hölder) and establishing precise compactness criteria. It systematically relates the regularity of the generator $f:E o E$ to the corresponding regularity of $C_f$ across a range of source–target spaces, using a detailed hierarchy of Hölder classes and new growth conditions. A central finding is that, except in some special one-dimensional cases, compactness and local compactness force the generator to be constant, mirroring classical results for function spaces; several sharp thresholds delineate when nontrivial $f$ can yield Hölder or Lipschitz operator behavior. The study connects these sequence-space results to the parallel theory of Wiener variation, offering a coherent view of when nonlinear superposition operators preserve, enhance, or destroy regularity, with implications for nonlinear analysis on sequence spaces and related applications.
Abstract
Continuing the study initiated in our earlier article [7], this paper aims to characterize various continuity properties of nonlinear composition operators acting on some sequence spaces, giving special attention to the space of sequences of bounded variation. In addition to pointwise, uniform, and locally uniform continuity, we investigate Lipschitz continuity as well as several types of Hölder continuity. Furthermore, we provide a characterization of the compactness properties of these operators.