A heat kernel Gaussian estimate, L1 Liouville theorem, uniqueness, eigenvalue bounds, and a Li-Yau gradient estimate are derived on weighted manifolds with lower N-Ricci curvature in the epsilon-range.
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Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $\epsilon$-range and it's application
A heat kernel Gaussian estimate, L1 Liouville theorem, uniqueness, eigenvalue bounds, and a Li-Yau gradient estimate are derived on weighted manifolds with lower N-Ricci curvature in the epsilon-range.