The $p$-Hardy-Rellich-Birman inequalities on the half-line
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908875196727296 |
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| author | Štampach, František Waclawek, Jakub |
| author_facet | Štampach, František Waclawek, Jakub |
| contents | The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08864 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $p$-Hardy-Rellich-Birman inequalities on the half-line Štampach, František Waclawek, Jakub Classical Analysis and ODEs 26D15, 39A12, 47J05 The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal. |
| title | The $p$-Hardy-Rellich-Birman inequalities on the half-line |
| topic | Classical Analysis and ODEs 26D15, 39A12, 47J05 |
| url | https://arxiv.org/abs/2603.08864 |