The derivative of the fractional discrete Laplacian is an exotic Riesz potential
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911475997605888 |
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| author | Li, Bo Lin, Qingze Wu, Huoxiong |
| author_facet | Li, Bo Lin, Qingze Wu, Huoxiong |
| contents | Let $Δ_{N}$ be the multidimensional discrete Laplacian on $\mathbb{Z}^N$ ($N\ge1$). In this note, we prove that, when $N=1$, the right hand derivative of $(-Δ_1)^s$ at $0$ is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math. 2000), and when $N\ge 2$, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Comm. PDEs. 2019) to the discrete setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The derivative of the fractional discrete Laplacian is an exotic Riesz potential Li, Bo Lin, Qingze Wu, Huoxiong Analysis of PDEs 39A12, 35R11, 39A12 Let $Δ_{N}$ be the multidimensional discrete Laplacian on $\mathbb{Z}^N$ ($N\ge1$). In this note, we prove that, when $N=1$, the right hand derivative of $(-Δ_1)^s$ at $0$ is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math. 2000), and when $N\ge 2$, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Comm. PDEs. 2019) to the discrete setting. |
| title | The derivative of the fractional discrete Laplacian is an exotic Riesz potential |
| topic | Analysis of PDEs 39A12, 35R11, 39A12 |
| url | https://arxiv.org/abs/2603.01039 |