The distance in Morrey spaces to $C^{\infty}_{\mathrm{comp}}$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910804507361280 |
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| author | Yamaguchi, Satoshi |
| author_facet | Yamaguchi, Satoshi |
| contents | In this paper we characterize the distance between the function $f$ and the set $C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)$ in generalized Morrey spaces $L_{p,ϕ}(\mathbb{R}^d)$ with variable growth condition. We also prove that the bi-dual of $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,ϕ}(\mathbb{R}^d)}$ is $L_{p,ϕ}(\mathbb{R}^d)$. As an application of the characterization of the distance we show the boundedness of Calderón-Zygmund operators on $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,ϕ}(\mathbb{R}^d)}$. By the duality we also see that these operators are bounded on its dual and bi-dual spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The distance in Morrey spaces to $C^{\infty}_{\mathrm{comp}}$ Yamaguchi, Satoshi Functional Analysis 42B35 (Primary) 46E30, 46B10, 42B20 (Secondary) In this paper we characterize the distance between the function $f$ and the set $C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)$ in generalized Morrey spaces $L_{p,ϕ}(\mathbb{R}^d)$ with variable growth condition. We also prove that the bi-dual of $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,ϕ}(\mathbb{R}^d)}$ is $L_{p,ϕ}(\mathbb{R}^d)$. As an application of the characterization of the distance we show the boundedness of Calderón-Zygmund operators on $\overline{C^{\infty}_{\mathrm{comp}}(\mathbb{R}^d)}^{L_{p,ϕ}(\mathbb{R}^d)}$. By the duality we also see that these operators are bounded on its dual and bi-dual spaces. |
| title | The distance in Morrey spaces to $C^{\infty}_{\mathrm{comp}}$ |
| topic | Functional Analysis 42B35 (Primary) 46E30, 46B10, 42B20 (Secondary) |
| url | https://arxiv.org/abs/2501.17620 |