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We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y)\big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric,integrable kernel $k$. We prove several results stating that under certainconditions on the kernel the operator $L$ satisfies the curvature-dimensioncondition $CD_\Upsilon (0,F)$ (recently introduced by two of the authors) withsome $CD$-function $F$, where attention is also paid to the asymptoticproperties of $F$ (exponential growth at infinity and power-type behaviour nearzero). We show that $CD_\Upsilon (0,F)$ implies a Li-Yau inequality forpositive solutions of the heat equation associated with the operator $L$. TheLi-Yau estimate in turn leads to a Harnack inequality, from which we alsoderive heat kernel bounds. Our results apply to a wide class of operatorsincluding the fractional discrete Laplacian.

Li-Yau and Harnack inequalities via curvature-dimension conditions for  discrete long-range jump operators including the fractional discrete  Laplacian