Optimal discrete Hardy-Rellich-Birman inequalities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916244087635968 |
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| author | Štampach, František Waclawek, Jakub |
| author_facet | Štampach, František Waclawek, Jakub |
| contents | We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\ell=2$) and Birman ($\ell\geq3$) weights. For $\ell=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\ell=2$, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For $\ell\geq3$, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal discrete Hardy-Rellich-Birman inequalities Štampach, František Waclawek, Jakub Classical Analysis and ODEs Spectral Theory 26D15, 47B39, 39A12 We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\ell=2$) and Birman ($\ell\geq3$) weights. For $\ell=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\ell=2$, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For $\ell\geq3$, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal. |
| title | Optimal discrete Hardy-Rellich-Birman inequalities |
| topic | Classical Analysis and ODEs Spectral Theory 26D15, 47B39, 39A12 |
| url | https://arxiv.org/abs/2405.07742 |