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

Abstract

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 weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\epsilon$-range. Building on these results, we demonstrate: The $L_{\varphi }^1$-Liouville theorem for $\varphi$-subharmonic functions, $L_{\varphi }^1$-uniqueness property for solutions of the $\varphi$-heat equation and lower bounds for eigenvalues of the weighted Laplacian ${\Delta }_{\varphi }$. Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted $L^p(\mu )$-norm constraint on $|\nabla \varphi {|}^2$.