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

2023/10/03 by Devdariani, Nikoloz
#46E15 #46E30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2310.01817

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(⋅) for which the corresponding variable-exponent Lebesgue space L^p(⋅)([0;1]ⁿ) shares the property with L^∞([0;1]ⁿ) such that the space of continuous functions C([0;1]ⁿ) forms a closed linear subspace in L^p(⋅)([0;1]ⁿ) . In particular, we derive the necessary and sufficient conditions on the decreasing rearrangement of the exponent p(⋅) for which there exists an equimeasurable exponent of p(⋅) such that the corresponding variable-exponent Lebesgue space possesses the aforementioned property.

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