ψ-Li-Yau inequality for the p-Laplacian on weighted graphs with the [formula omitted] curvature

Abstract

In this article, we investigate the Li-Yau inequality for the p-Laplacian on weighted graphs. Under the condition of CDpψ(m,0) curvature for p≥2, we derive a more general type of ψ-Li-Yau inequality for positive solutions to the p-Laplace heat equation on finite graphs or locally finite graphs with bounded weighted vertex degree. This is a generalization of Münch's results for the CDψ(m,0) curvature to the CDpψ(m,0) curvature. To do this, we apply the maximum principle and the semigroup method, which are two different strategies, respectively. The unified version of Münch, which is connected to the log-Li-Yau inequality on manifolds and the ⋅-Li-Yau inequality on graphs, is hence enlarged to the broader p-Laplacian setting. As an application of our main conclusions, we demonstrate that the relevant Harnack inequality for the p-Laplacian.