A critical Hardy-Rellich inequality
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909044749369344 |
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| author | Castro, Hernán |
| author_facet | Castro, Hernán |
| contents | In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\geq 1$ there exists a constant $C_N>0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N\,\mathrm{d}x\leq C_N\int_{\mathbb R^N}\left|Δu(x)\right|^N\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A critical Hardy-Rellich inequality Castro, Hernán Analysis of PDEs Functional Analysis 26D10, 35A23, 46E35 In this work, we prove a critical version of a Hardy-Rellich type inequality. We show that for $N\geq 1$ there exists a constant $C_N>0$ such that \[ \int_{\mathbb R^N}\left|\nabla\left(\frac{u(x)}{|x|}\right)\right|^N\,\mathrm{d}x\leq C_N\int_{\mathbb R^N}\left|Δu(x)\right|^N\,\mathrm{d}x, \] for any $u\in C^\infty_c(\mathbb R^N\setminus\left\{0\right\})$. |
| title | A critical Hardy-Rellich inequality |
| topic | Analysis of PDEs Functional Analysis 26D10, 35A23, 46E35 |
| url | https://arxiv.org/abs/2511.16537 |