Abstract. Given a domain Ω of a complete Riemannian manifold M and define A to be the Laplacian with Neumann boundary condition on Ω. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper boundHere d is the geodesic distance on M, V Ω (x, r) is the Riemannian volume of B(x, r) ∩ Ω, where B(x, r) is the geodesic ball of center x and radius r, and δ is a constant related to the doubling property of Ω. As a consequence we obtain analyticity of the semigroup e −tA on L p (Ω) for all p ∈ [1, ∞) as well as a spectral multiplier result. Introduction and main resultsThis short note is devoted to the Gaussian upper bound for the heat kernel of the Neumann Laplacian. Let us start with the Euclidean setting in which Ω is a bounded Lipschitz domain of R n . Let ∆ N be the Neumann Laplacian. It is well known that the corresponding heat kernel h(t, x, y) satisfiesOne can replace the extra term e t by (1 + t) n/2 but the decay h(t, x, y) ≤ Ct −n/2 cannot hold for large t since e t∆N 1 = 1. We refer to the monographs [5] or [17] for more details.In applications, for example when applying the Gaussian bound to obtain spectral multiplier results one can apply (1) to −∆ N + I (or ǫI for any ǫ > 0) and not to −∆ N . It is annoying to add the identity operator especially it is not clear how the functional calculus for −∆ N can be related to that of −∆ N + I. The same problem occurs for analyticity of the semigroup e t∆N on L p (Ω) for p ∈ [1, ∞). One obtains from (1) analyticity of the semigroup but not a bounded analytic semigroup. This boundedness (on sectors of the right half plane) is important in order to obtain appropriate estimates for the resolvent or for the time derivatives of the solution to the corresponding evolution equation on L p . In this note we will show in an elementary way how one can resolve this question. The idea is that (1) can be improved into a Gaussian upper bound of the type

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish a Gaussian lower bound for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for Gaussian two-sided bounds for the fundamental solution.

Let Ω be a C ∞ -smooth bounded domain of R n , n 1, and let the matrix a ∈ C ∞ (Ω; R n 2 ) be symmetric and uniformly elliptic. We consider the L 2 (Ω)-realization A of the operator − div(a∇·) with Dirichlet boundary conditions. We perturb A by some real valued potential V ∈ C ∞ 0 (Ω) and note A V = A + V . We compute the asymptotic expansion of tr(e −tA V − e −tA ) as t ↓ 0 for any matrix a with constant coefficients. In the particular case where A is the Dirichlet Laplacian in Ω, that is when a is the identity of R n 2 , we make the four main terms appearing in the asymptotic expansion formula explicit and prove that L ∞ -bounded sets of isospectral potentials of A are bounded in H 2 (Ω).