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<p>In this paper, we study the Li–Yau’s gradient estimate for the <i>p</i>-Laplacian on weighted graphs. For <i>p</i> ≥ 2, under the condition of <i>CD</i><Stack> <sub><i>p</i></sub> <sup><i>ψ</i></sup> </Stack>(<i>m</i>, <i>K</i>) curvature, we derive a more general type of Li–Yau’s gradient estimate for positive solutions to the <i>p</i>-Laplacian heat equation on finite graphs or locally finite graphs with bounded weighted vertex degree. It is parallel to a result of Kotschwar and Ni on manifolds and an outcome of Wang on smooth metric measure spaces. In particular, when <i>p</i> = 2, our estimate deduces to Lü and Wang’s result on graphs with the <i>CDψ</i>(<i>m</i>, <i>K</i>) curvature. As an application of our main result, we derive the corresponding Harnack inequality.</p>

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Gradient Estimates and CD p ψ (m, K) Curvature for the p-Laplacian on Weighted Graphs

  • 期刊论文

  • 发表日期：

    2025年1月5日

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  • Yongtao Liu

摘要

In this paper, we study the Li–Yau’s gradient estimate for the p-Laplacian on weighted graphs. For p ≥ 2, under the condition of CD p ψ (m, K) curvature, we derive a more general type of Li–Yau’s gradient estimate for positive solutions to the p-Laplacian heat equation on finite graphs or locally finite graphs with bounded weighted vertex degree. It is parallel to a result of Kotschwar and Ni on manifolds and an outcome of Wang on smooth metric measure spaces. In particular, when p = 2, our estimate deduces to Lü and Wang’s result on graphs with the CDψ(m, K) curvature. As an application of our main result, we derive the corresponding Harnack inequality.