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Precompactness in bivariate metric semigroup-valued bounded variation spaces

Jingshi Xu, Yinglian Niu

TL;DR

The paper addresses precompactness of sets in bivariable metric semigroup-valued bounded variation spaces by developing a unifying framework based on equimetric sets and joint equivariation. It defines appropriate total and joint variation notions for five BV settings—Wiener, Riesz, Waterman, and Korenblum—proves completeness of the associated metric spaces, and establishes a general precompactness criterion: a set is precompact if it is $\text{joint equivariated}$ and its pointwise slices are precompact in the codomain. The key contribution is the introduction of the equimetric concept and the extension of univariate precompactness results to the bivariate, metric semigroup-valued case, showing that precompactness can be reduced to coordinate-wise precompactness plus a structural equivariance condition. This provides a robust, widely applicable toolkit for compactness in BV-type function spaces valued in complete metric semigroups, with potential implications for Fourier analysis and related areas.

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.

Precompactness in bivariate metric semigroup-valued bounded variation spaces

TL;DR

The paper addresses precompactness of sets in bivariable metric semigroup-valued bounded variation spaces by developing a unifying framework based on equimetric sets and joint equivariation. It defines appropriate total and joint variation notions for five BV settings—Wiener, Riesz, Waterman, and Korenblum—proves completeness of the associated metric spaces, and establishes a general precompactness criterion: a set is precompact if it is and its pointwise slices are precompact in the codomain. The key contribution is the introduction of the equimetric concept and the extension of univariate precompactness results to the bivariate, metric semigroup-valued case, showing that precompactness can be reduced to coordinate-wise precompactness plus a structural equivariance condition. This provides a robust, widely applicable toolkit for compactness in BV-type function spaces valued in complete metric semigroups, with potential implications for Fourier analysis and related areas.

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.
Paper Structure (5 sections, 16 theorems, 111 equations)

This paper contains 5 sections, 16 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Bivariate bounded Wiener $p$ variation spaces as $p\in(0,\infty)$
  3. Bivariate bounded Riesz variation spaces
  4. Bivariate bounded Waterman variation spaces
  5. Bivariate bounded Korenblum variation spaces

Key Result

Lemma 1

Suppose $\{\rho_\tau: \tau\in T\}$ is a family of pseudo-metrics on $X$ and $\rho=\sup_{\tau\in T}\rho_\tau$. Let $A\subset X$. If $A$ is $T$-equimetric and totally bounded for $\rho_\tau$ for each $\tau \in T$, then $A$ is totally bounded for $\rho$.

Theorems & Definitions (38)

  • Lemma 1
  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • ...and 28 more