I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it only received one, rather unsatisfactory to me, answer there. I'm wondering if people here can have a look. Some related questions have been posed here and here (and I noted in particular the literature link provided in the answer to the latter question). However, neither of those discussions cover the possibility of boundary conditions being present, or possible lack of compactness.

# Repost of the original question

Let me illustrate my question by starting with the simplest possible example: Let us consider $P := - \mathrm{d}^2/\mathrm{d}x^2$, an elliptic partial differential operator on $\mathbb{R}$; let us also consider the following boundary-value problem on the interval $\overline{\Omega} = [0,1]$: \begin{equation} P u = f, \qquad u(0)=u(1)=0. \end{equation} As is (I think) well-known, when seen as an operator $L^2(\Omega) \to L^2(\Omega)$, $P$ is unbounded. However, it is closed on the dense domain $D(P) := H^2(\Omega) \cap H_0^1(\Omega)$ where $H_0^1(\Omega)$ is the closure of $C_{\mathrm{c}}^\infty(\Omega)$ in the $H^1$ norm (so that any element of this space has vanishing trace on $\partial \Omega = \{0,1\}$, i.e. it satisfies the Dirichlet boundary condition above in a weak sense). Furthermore, $0$ is in the resolvent of $(P,D(P))$, i.e. there exists a bounded inverse $P^{-1} : L^2(\Omega) \to L^2(\Omega)$. In fact, in this example the inverse is easily computed: it is the integral operator defined by the (continuous, as it happens) kernel \begin{equation} G(x,y) = \begin{cases} x(1-y) & x \leq y \\ y(1-x) &x > y \end{cases}, \quad (x,y) \in \Omega \times \Omega. \end{equation} Of course, when viewed as a distribution in $\mathscr{D}'(\Omega \times \Omega)$, $G$ is the Schwartz kernel of $P^{-1}$ which we know on abstract grounds must exist since $P^{-1} : C_{\mathrm{c}}^{\infty}(\Omega) \to \mathscr{D}'(\Omega)$ is continuous.

My question is the following: in this example and in more general examples where $P$ is a second-order elliptic differential operator on, say, an open (and not necessarily compact) region $\Omega$ with smooth boundary in $\mathbb{R}^n$, and assuming that we can find a suitable dense domain $D(P)$ for $P$ as above so that $(P,D(P))$ has a bounded inverse $P^{-1} : L^2(\Omega) \to L^2(\Omega)$, **does the Schwartz kernel $G$ of $P^{-1}$ always define a pseudodifferential operator on $\Omega$?**

# Addendum for MO

User mcd on math.SE points out that the Boutet de Monvel calculus ought to be relevant here. Aside from wishing to see exactly how this is, I wonder whether the possible lack of compactness (of $\Omega$) might cause problems in such an approach.

# UPDATE

I have reduced my question to the following subproblem: Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold, as explained in a comment there.