Fractional Hardy inequalities and capacity density
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916197553930240 |
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| author | Ihnatsyeva, Lizaveta Mohanta, Kaushik Vähäkangas, Antti V. |
| author_facet | Ihnatsyeva, Lizaveta Mohanta, Kaushik Vähäkangas, Antti V. |
| contents | We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional Hardy inequalities and capacity density Ihnatsyeva, Lizaveta Mohanta, Kaushik Vähäkangas, Antti V. Classical Analysis and ODEs Analysis of PDEs 31C15 (26D15, 46E35, 31E05) We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting. |
| title | Fractional Hardy inequalities and capacity density |
| topic | Classical Analysis and ODEs Analysis of PDEs 31C15 (26D15, 46E35, 31E05) |
| url | https://arxiv.org/abs/2404.05222 |