In this paper, motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Finally, as an application, we obtain sharp Harnack inequalities on hyperbolic spaces.

中文翻译：

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双曲空间上的Li-Yau乘子集和最优Li-Yau梯度估计

在本文中，通过在具有负Ricci曲率下界的完整黎曼流形上找到热方程的正解的尖锐的Li-Yau型梯度估计，我们首先介绍Li-Yau乘子集的概念，并证明可以进行计算通过歧管的热核。然后，通过利用双曲空间上的热核的递推关系，获得双曲空间上的最优的Li-Yau型梯度估计。最后，作为一种应用，我们在双曲空间上获得了尖锐的Harnack不等式。