Comptes Rendus
Article de recherche - Analyse harmonique
The derivative of the fractional discrete Laplacian is an exotic Riesz potential
[La dérivée du laplacien discret fractionnaire est un potentiel de Riesz exotique]
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 287-299

Let $\Delta _{N}$ be the multidimensional discrete Laplacian on $\mathbb{Z}^N$ ($N\ge 1$). In this note, we prove that, when $N=1$, the right-hand derivative of $(-\Delta _1)^s$ 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 (Commun. Partial Differ. Equations, 2019) to the discrete setting.

Soit $\Delta _{N}$ le laplacien discret multidimensionnel sur $\mathbb{Z}^N$ ($N\ge 1$). Dans cette note, nous prouvons que, lorsque $N=1$, la dérivée à droite de $(-\Delta _1)^s$ en $0$ est un potentiel de Riesz discret exotique (à savoir, le cas limite : l’ordre est $0$) au sens de Stein–Wainger (J. Anal. Math., 2000), et que lorsque $N\ge 2$, la dérivée correspondante est également un potentiel de Riesz discret exotique avec un correcteur supplémentaire. Une conclusion similaire pour le cas de la dérivée gauche est également envisagée. Tous les résultats obtenus dans cette note étendent le Laplacien logarithmique de Chen–Weth (Commun. Partial Differ. Equations, 2019) au cadre discret.

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DOI : 10.5802/crmath.808
Classification : 39A12, 35R11, 39A12
Keywords: Discrete harmonic analysis, fractional Laplacian, Riesz potential
Mots-clés : Analyse harmonique discrète, Laplacien fractionnaire, potentiel de Riesz
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Bo Li; Qingze Lin; Huoxiong Wu. The derivative of the fractional discrete Laplacian is an exotic Riesz potential. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 287-299. doi: 10.5802/crmath.808
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     language = {en},
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