Analysis of the heat kernel of the Dirichlet-to-Neumann operator

We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the right half-plane and for all its derivatives.