In terms of the equivalence of Poincare inequality and the existence of spectral gap, the super-Poincare inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric di

By using logarithmic transformations, an explicit lower bound estimate of heat kernels is obtained for diffusion processes on Riemannian manifolds. This estimate is sharp for both short and long times, especially for heat kernels on a compact manifold, and is extended to manifolds with unbounded curvature.