Sharp weighted fractional Hardy inequalities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908741976195072 |
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| author | Dyda, Bartłomiej Kijaczko, Michał |
| author_facet | Dyda, Bartłomiej Kijaczko, Michał |
| contents | We investigate the weighted fractional order Hardy inequality $$ \int_Ω\int_Ω\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partialΩ)^{-α}\text{dist}(y,\partialΩ)^{-β}\,dy\,dx\geq C\int_Ω\frac{|f(x)|^{p}}{\text{dist}(x,\partialΩ)^{sp+α+β}}\,dx, $$ for $Ω=\mathbb{R}^{d-1}\times(0,\infty)$, $Ω$ being a convex domain or $Ω=\mathbb{R}^d\setminus\{0\}$. Our work focuses on finding the best (i.e. sharp) constant $C=C(d,s,p,α,β)$ in all cases. We also obtain weighted version of the fractional Hardy-Sobolev-Maz'ya inequality. The proofs are based on general Hardy inequalities and the non-linear ground state representation, established by Frank and Seiringer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_06760 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp weighted fractional Hardy inequalities Dyda, Bartłomiej Kijaczko, Michał Analysis of PDEs Functional Analysis 46E35, 39B72, 26D15 We investigate the weighted fractional order Hardy inequality $$ \int_Ω\int_Ω\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partialΩ)^{-α}\text{dist}(y,\partialΩ)^{-β}\,dy\,dx\geq C\int_Ω\frac{|f(x)|^{p}}{\text{dist}(x,\partialΩ)^{sp+α+β}}\,dx, $$ for $Ω=\mathbb{R}^{d-1}\times(0,\infty)$, $Ω$ being a convex domain or $Ω=\mathbb{R}^d\setminus\{0\}$. Our work focuses on finding the best (i.e. sharp) constant $C=C(d,s,p,α,β)$ in all cases. We also obtain weighted version of the fractional Hardy-Sobolev-Maz'ya inequality. The proofs are based on general Hardy inequalities and the non-linear ground state representation, established by Frank and Seiringer. |
| title | Sharp weighted fractional Hardy inequalities |
| topic | Analysis of PDEs Functional Analysis 46E35, 39B72, 26D15 |
| url | https://arxiv.org/abs/2210.06760 |