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

Nikoloz Devdariani

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.

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

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 for which the corresponding variable-exponent Lebesgue space shares the property with such that the space of continuous functions forms a closed linear subspace in . In particular, we derive the necessary and sufficient conditions on the decreasing rearrangement of the exponent for which there exists an equimeasurable exponent of such that the corresponding variable-exponent Lebesgue space possesses the aforementioned property.
Paper Structure (3 sections, 3 theorems, 31 equations)

This paper contains 3 sections, 3 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. definitions and auxiliary results
  3. Proof of Theorem \ref{['thm_necess_suff_on_decreasing_rearrangement']}

Key Result

Theorem 1

For the existence of $\bar{p}(\cdot) \in W(p)$ for which $C(\Omega)$ is a closed subspace in $L^{\bar{p}(\cdot)}(\Omega)$, it is necessary and sufficient that

Theorems & Definitions (4)

  • Theorem 1
  • Definition 2
  • Proposition 3: Edmunds, Lang, Nekvinda
  • Theorem 4