Weighted Laplace-Beltrami Operator And Related Problems

This paper focuses on the weighted Laplace-Beltrami operator,weighted heat equation and related problems on a complete Riemannian manifold.Using weighted Laplacian comparison theorem,we discuss about weighted volume growth,in particular,on the following condition:moreover,we prove a result about Calderon-Zygmund decomposition.We consider the gradient estimate,the Harnack inequality of weighted heat kernel.On the condition of (2) we get integral estimate of weighted heat kernel's covariant derivative with the result of weighted volume growth and prove that weighted Riesz transformation is almost bounded.In this paper,we give the definition of entropy in the following two cases: weighted linear heat equation and Ricci curvature tensor bounded from be-low,then we get two monotone formulas and the differential inequality on the latter circumstance,with these results,we not only estimate the lower bound of heat kernel and point out the relation between the entropy and the geometric properties of the manifold,but also prove that weighted volume has pointwise Euclidean volume growth on the condition that the entropy is bounded from below.Finally,we estimate the lower bound of weighted Laplace-Beltrami operator's first nonzero eigenvalue on a closed Riemannian manifold,discuss the relation between the spectral gap and the m-dimension Bakry-Emery curvature tensor and prove that the heat kernel weighted Laplace-Beltrami operator has compact resolvent on the Hyperbolic space.