Compactness in spaces of functions of bounded variation from ideal perspective
Jacek Gulgowski, Adam Kwela, Jacek Tryba
TL;DR
The paper addresses when embeddings between spaces of bounded variation are compact for Waterman spaces and Chanturia classes. It adopts a unified ideal-theoretic framework, linking ABV, BV(\phi), and V[g(n)] to summable and simple density ideals via Exh$(\phi)$ and Katětov order, to derive intrinsic sequence- and ideal-based criteria. The main contributions are complete characterizations of compact embeddings ${ABV}\subseteq{BBV}$ and $V[g(n)]\subseteq V[h(n)]$ in terms of asymptotics of partial sums and ideal containments (e.g., $\sum_{k=1}^n b_k = o(\sum_{k=1}^n a_k)$ and $\mathcal I_A \le_K \mathcal I_{B^M}$ for some $M$). These results unify and extend prior compactness analyses for BV-type spaces by translating analytic conditions into combinatorial ideal-theoretic terms, enabling broad applicability to different variation notions.
Abstract
Recently we have presented a unified approach to two classes of Banach spaces defined by means of variations (Waterman spaces and Chanturia classes), utilizing the concepts from the theory of ideals on the set of natural numbers. We defined correspondence between an ideal on the set of natural numbers, a certain sequence space and related space of functions of bounded variation. In this paper, following these ideas, we give characterizations of compact embeddings between different Waterman spaces and between different Chanturia classes: both in terms of sequences defining these function spaces and in terms of properties of ideals corresponding to these function spaces.