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

TL;DR

This paper establishes curvature-dimension conditions for discrete long-range jump operators, including the fractional discrete Laplacian, leading to Li-Yau inequalities, Harnack inequalities, and heat kernel bounds.

Contribution

It introduces new curvature-dimension conditions for a class of discrete operators and derives key inequalities and bounds from these conditions.

Findings

Curvature-dimension conditions are satisfied for various discrete jump operators.

Li-Yau inequalities are derived for solutions of the heat equation.

Heat kernel bounds are established based on the curvature-dimension framework.

Abstract

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 certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_\Upsilon (0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_\Upsilon (0,F)$ implies a Li-Yau inequality for positive solutions of the heat equation associated with the operator $L$. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian.