Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-reflexivity of the Banach space $ΛBV^{(p)}$

Batoul S. Mortazavi-Samarin, Mehdi Rostami

TL;DR

The paper addresses whether the Waterman-Shiba space $\Lambda\mathrm{BV}^{(p)}$ is reflexive, extending the classical non-reflexivity results from $\Lambda\mathrm{BV}$ to the $p$-variation setting for $p\in(1,\infty)$. It develops a constructive approach, building a bounded sequence of test functions and a corresponding family of bounded linear functionals $L_{(n_k)}$ to exhibit weak non-convergence of subsequences. By leveraging Hölder's inequality with conjugates $p$ and $q$ (where $1/p+1/q=1$) and carefully arranged supports, it shows the absence of reflexivity in $\Lambda\mathrm{BV}^{(p)}$. This sharpens our understanding of generalized bounded variation spaces and has implications for Fourier analysis and operator theory on such spaces.

Abstract

In this paper, we show that the Waterman-Shiba space is non-reflexive. In fact, Prus-Wiśniowski and Ruckle, in \cite{1}, generalized the well-known fact that states the space of bounded variation functions is non-reflexive. Here, an improvement of that result is provided.

Non-reflexivity of the Banach space $ΛBV^{(p)}$

TL;DR

The paper addresses whether the Waterman-Shiba space is reflexive, extending the classical non-reflexivity results from to the -variation setting for . It develops a constructive approach, building a bounded sequence of test functions and a corresponding family of bounded linear functionals to exhibit weak non-convergence of subsequences. By leveraging Hölder's inequality with conjugates and (where ) and carefully arranged supports, it shows the absence of reflexivity in . This sharpens our understanding of generalized bounded variation spaces and has implications for Fourier analysis and operator theory on such spaces.

Abstract

In this paper, we show that the Waterman-Shiba space is non-reflexive. In fact, Prus-Wiśniowski and Ruckle, in \cite{1}, generalized the well-known fact that states the space of bounded variation functions is non-reflexive. Here, an improvement of that result is provided.
Paper Structure (2 sections, 1 theorem, 22 equations, 1 figure)

This paper contains 2 sections, 1 theorem, 22 equations, 1 figure.

Table of Contents

  1. Introduction and Preliminaries
  2. Main Result

Key Result

Theorem 2.1

Let $p\in (1,\infty)$ and $\Lambda$ be a Waterman sequence. Then the Waterman-Shiba space $\Lambda\rm BV^{(p)}$ is non-reflexive.

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Theorem 2.1
  • proof