Transport Equations via the Weyl–Hörmander Calculus

In this chapter we construct concretely heat Green operators of the initial boundary value problem on a Riemannian manifold under the Dirichlet, Neumann and Robin conditions. More precisely, by making use of a family of Weyl symbol functions as in Tables 17.1 , 17.2 , 17.3 , 17.4 , and 17.5 introduced in Chap. 17 we solve the five transport equations for the Dirichlet (D), Neumann (N), regular Robin (R), generalized Robin (G) and hypoelliptic Robin (S) problems on the half space \({\mathbf {R}}^{n}_{+}\) in terms of the Weyl–Hörmander calculus as in Table 19.1 (see Lemmas 19.1–19.5).