Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\epsilon$-range and it's application

Wen-Qi Li, Zhikai Zhang

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\varphi$-heat equation and prove Gaussian upper and lower bounds for the $\varphi$-heat kernel on weig…