Fractional boundary Hardy inequality for the critical cases
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915789213270016 |
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| author | Adimurthi Roy, Prosenjit Sahu, Vivek |
| author_facet | Adimurthi Roy, Prosenjit Sahu, Vivek |
| contents | We establish generalised fractional boundary Hardy-type inequality, in the spirit of Caffarelli-Kohn-Nirenberg inequality for different values of $s$ and $p$ on various domains in $\mathbb{R}^d, ~ d \geq 1$. In particular, for Lipschitz bounded domains any values of $s$ and $p$ are admissible, settling all the cases in subcritical, supercritical and critical regime. In this paper we have solved the open problems posed by Dyda for the critical case $sp =1$. Moreover we have proved the embeddings of $W^{s,p}_{0}(Ω)$ in subcritical, critical and supercritical uniformly without using Dyda's decomposition. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11956 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractional boundary Hardy inequality for the critical cases Adimurthi Roy, Prosenjit Sahu, Vivek Analysis of PDEs 46E35 (Primary), 26D15 (Secondary) We establish generalised fractional boundary Hardy-type inequality, in the spirit of Caffarelli-Kohn-Nirenberg inequality for different values of $s$ and $p$ on various domains in $\mathbb{R}^d, ~ d \geq 1$. In particular, for Lipschitz bounded domains any values of $s$ and $p$ are admissible, settling all the cases in subcritical, supercritical and critical regime. In this paper we have solved the open problems posed by Dyda for the critical case $sp =1$. Moreover we have proved the embeddings of $W^{s,p}_{0}(Ω)$ in subcritical, critical and supercritical uniformly without using Dyda's decomposition. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case. |
| title | Fractional boundary Hardy inequality for the critical cases |
| topic | Analysis of PDEs 46E35 (Primary), 26D15 (Secondary) |
| url | https://arxiv.org/abs/2308.11956 |