The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables

TL;DR

This paper extends the characterization of when continuous functions form a closed subspace in variable-exponent Lebesgue spaces from one to multiple variables, using decreasing rearrangement techniques.

Contribution

It generalizes prior one-variable results to multiple variables, providing necessary and sufficient conditions for the closure of continuous functions in these spaces.

Findings

Characterizes exponents p(·) for which C([0,1]^n) is closed in L^{p(·)}.

Derives conditions on decreasing rearrangement of p(·).

Establishes existence of equimeasurable exponents with the same property.

Abstract

In this paper, we generalize a recently obtained result by Kopaliani and Zviadadze from the one-variable case to the several-variable case. Specifically, in terms of decreasing rearrangement, we characterize those exponents $p(\cdot)$ for which the corresponding variable-exponent Lebesgue space $L^{p(\cdot)}([0;1]^n)$ shares the property with $L^\infty([0;1]^n)$ such that the space of continuous functions $C([0;1]^n)$ forms a closed linear subspace in $L^{p(\cdot)}([0;1]^n)$ . In particular, we derive the necessary and sufficient conditions on the decreasing rearrangement of the exponent $p(\cdot)$ for which there exists an equimeasurable exponent of $p(\cdot)$ such that the corresponding variable-exponent Lebesgue space possesses the aforementioned property.