Precompactness in bivariate metric semigroup-valued bounded variation spaces

TL;DR

This paper establishes conditions under which sets in various bivariate metric semigroup-valued bounded variation spaces are precompact, introducing the concept of equimetric sets to facilitate this analysis.

Contribution

It introduces the concept of equimetric sets and proves precompactness criteria for sets in a broad class of bivariate variation spaces.

Findings

Pointwise total boundedness and joint equivariatedness imply precompactness.

Applicable to multiple variation spaces including Jordan, Wiener, Waterman, Riesz, and Korenblum.

Provides a unified framework for precompactness in these spaces.

Abstract

In this paper, we show that if a set in bivariate metric semigroups-valued bounded variation spaces is pointwise totally bounded and joint equivariated then it is precompact. These spaces include bounded Jordan variation spaces, bounded Wiener variation spaces, bounded Waterman variation spaces, bounded Riesz variation spaces and bounded Korenblum variation spaces. To do so, we introduce the concept of equimetric set.