Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application (2505.19113v1)
Abstract: 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 $L1_\phi$-Liouville theorem for $\phi$-subharmonic functions, $L1_\phi$-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 $Lp(\mu)$-norm constraint on $|\nabla\phi|2$.