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

In this paper, we establish a parabolic Harnack inequality for positive solutions of the φ-heat equation and prove Gaussian upper and lower bounds for the φ-heat kernel on weighted Riemannian manifolds under lower N-Ricci curvature bound with ε-range. Building on these results, we demonstrate: The L¹_φ-Liouville theorem for φ-subharmonic functions, L¹_φ-uniqueness property for solutions of the φ-heat equation and lower bounds for eigenvalues of the weighted Laplacian Δ_φ. 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(μ)-norm constraint on |∇φ|².