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

Let Δ N be the multidimensional discrete Laplacian on ℤ N ( N ≥ 1 ). 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 ≥ 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.