Observations on gaussian upper bounds for Neumann heat kernels

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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 bound h(t,x,y)≤ C[V_Ω(x,√t)V_Ω(y,√t)]¹/²(1+d²(x,y)/4t)δe⁻ᵈ²⁽ˣ,ʸ⁾/⁴ᵗ, t0, x,y∈ Ω. Here 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-t A on Lᵖ(Ω) for all p ∈ [1, ∞) as well as a spectral multiplier result.