Hardy Inequality and Heat Semigroup Estimates for Riemannian Manifolds with Singular Data

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on ∂D, and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L 2 (D) satisfies a strong Hardy inequality with weight δ 2 , (ii) the initial temperature distribution, and the specific heat of D are given by δ −α and δ −β respectively, where δ is the distance to ∂D, and 1