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

Abstract

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.

Theorems & Definitions (11)

• Proposition 2.1
• proof
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• proof : Proof of Theorem \ref{['thm1']}
• Proposition 3.1
• proof
• Theorem 3.2
• Remark 3.3