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

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 $L^{\mathrm{\infty }}([0;1{]}^n)$ 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.