The derivative of the fractional discrete Laplacian is an exotic Riesz potential

Abstract

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 (Comm. PDEs. 2019) to the discrete setting.

13 pages; to appear in C.R. Math. Acad. Sci. Paris