AI 中文总结
本文针对满足CDψ(n,-K)条件的局部有限加权图，建立Li-Yau型梯度估计，结合热核Harnack不等式得到曲率依赖的体积加倍估计，K=0时还推出无限图上-Δ谱底为零的结论。
AI 中文摘要
我们研究满足CDψ(n,-K)条件（K≥0）的局部有限加权图上的梯度估计与体积增长问题。我们建立了热半群的变分不等式，并由此推导出一族Li-Yau型梯度估计。此外，在ψ满足适当假设时，我们建立了度量球的一致热保留估计。结合从所得梯度估计得到的热核Harnack不等式，这给出了依赖曲率的指数体积加倍估计：V(x,2r)≤Ce^{cKr²}V(x,r)。当K=0时，该结果退化为一致体积加倍，进一步意味着无限图上-Δ的谱底为零。
英文摘要
We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CDψ(n,-K)$ condition with $K\geq0$. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on $ψ$, we establish a curvature dependent heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate \begin{equation*} V(x,2r)\leq C e^{c\sqrt{K}r}V(x,r). \end{equation*} When $K=0$, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of $-Δ$ vanishes on infinite graphs.
