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

In this paper, we establish a parabolic Harnack inequality for positive solutions of the $\phi$-heat equation and prove Gaussian upper and lower bounds for the $\phi$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\varepsilon$-range. Building on these results, we demonstrate: The $L^1_\phi$-Liouville theorem for $\phi$-subharmonic functions, $L^1_\phi$-uniqueness property for solutions of the $\phi$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $\Delta_\phi$. 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 $|\nabla\phi|^2$.