Precise Gaussian Lower Bounds on Heat Kernels

Abstract

Let g be a Riemannian metric on a Euclidean space. The Levi-Civita Laplace-Beltrami operator Δ generates a diffusion semi-group. We denote the heat kernel by p(t,x, y). The following estimate is called a Gaussian bound on the heat kernel: there exist 0 < δ₁ < 1,δ₂ > 0 and C₁, C2 > 0 such that for any t > 0, here d(x, y) denotes the Riemannian distance between x and y. In this article, we will study more precise estimates on the lower bound for a fixed x as follows: there exists a positive constant C such that for any 0t<1, ost bounds (1.1) in the literature (cf. [3]) are not precise in the sense that δ₁ > 0, except for Li-Yau’s result in [7] which asserts that (1.2) holds with under the assumption that Ric > 0. To my knowledge, there seems to be no other criteria.