Compactness in spaces of functions of bounded variation from ideal perspective

TL;DR

This paper explores the compactness properties of embeddings between Waterman and Chanturia function spaces, using ideal theory to characterize these relationships in terms of sequences and ideals.

Contribution

It provides new characterizations of compact embeddings between these function spaces via ideal and sequence-based methods.

Findings

Characterizations of compact embeddings between Waterman spaces.

Characterizations of compact embeddings between Chanturia classes.

Connections between ideal properties and function space compactness.

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.