Construction of function spaces close to $L^\infty$ with associate space close to $L^1$

TL;DR

This paper constructs a new variable exponent function space close to $L^ Infty$ with an associate space near $L^1$, featuring properties related to Fourier series divergence and containing continuous functions.

Contribution

It introduces a novel variable exponent space with specific structural properties and a rich associate space, expanding the understanding of function spaces near $L^ Infty$ and $L^1$.

Findings

The space $X$ contains $C([0,1])$ as a closed subspace.

The associate space of $X$ includes functions with divergent Fourier series.

The space bridges properties of $L^ Infty$ and $L^1$ with specific divergence features.

Abstract

The paper introduces a variable exponent space $X$ which has in common with $L^{\infty}([0,1])$ the property that the space $C([0,1])$ of continuous functions on $[0,1]$ is a closed linear subspace in it. The associate space of $X$ contains both the Kolmogorov and the Marcinkiewicz examples of functions in $L^{1}$ with a.e. divergent Fourier series.