The Borel summable solutions of heat operators on a real analytic manifold

We study heat type equations ∂ t u = Δ ˜ u , where the operator Δ ˜ is given by a sum of squares of commuting, real analytic vector fields acting on a real analytic manifold. We give necessary and sufficient conditions for convergence and Borel summability of formal power series solutions in terms of generalized integral means of the initial data. The results are also valid for affine perturbations of Δ ˜ .