The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914650383187968 |
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| author | Devdariani, Nikoloz |
| author_facet | Devdariani, Nikoloz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_01817 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables Devdariani, Nikoloz Functional Analysis Classical Analysis and ODEs 46E30, 46E15 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. |
| title | The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables |
| topic | Functional Analysis Classical Analysis and ODEs 46E30, 46E15 |
| url | https://arxiv.org/abs/2310.01817 |