Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that $P$ is elliptic on $M \setminus \{q\}$. Can we say that $(Pu, u) \cong \Vert u\Vert^2\_{H^1}$, or at least $(P(\varphi u), \varphi u) \cong \Vert \varphi u\Vert\_{H^1}^2$ for all $\varphi \in C\_c^\infty (M \setminus \{q\})$? If $P$ were elliptic on $M$, this is standard material. But here the lack of ellipticity at a point is throwing me off.



Elliptic regularity of second order pseudos
At least the answer to your second question is yes.


Let $\varphi\in C^\infty\_c(M\backslash\{q\})$ be given. Choose a function $\tau\in C^\infty\_c(M,[0,1])$ which is equal to $1$ in a neighbourhood of $q$ and which has a support disjoint from the support of $\varphi$. Put a Riemannian metric on $M$, so that we have the negative Laplacian $-\Delta$, a positive elliptic second order differential operator. Let $\widetilde{P}:=(1-\tau)P+\tau (-\Delta)$. Then $\widetilde P$ is a positive second order pseudodifferential operator that is elliptic everywhere on $M$ and one has$$\widetilde P\varphi u=(1-\tau)P\varphi u+\tau (-\Delta)\varphi u=(1-\tau)P\varphi u=P\varphi u -\tau P\varphi u\qquad \forall\;u\in H^2(M),$$
because $\Delta$ is a local operator. Now, as the supports of $\varphi$ and $\tau$ are disjoint, one gets
\begin{align}(P(\varphi u),\varphi u)\_{L^2(M)}&= (\widetilde{P}(\varphi u),\varphi u)\_{L^2(M)}+(\tau P\varphi u,\varphi u)\_{L^2(M)}\\&=
(\widetilde{P}(\varphi u),\varphi u)\_{L^2(M)}+(P\varphi u,\underbrace{\tau\varphi}\_{\equiv0} u)\_{L^2(M)}\\&=(\widetilde{P}(\varphi u),\varphi u)\_{L^2(M)}.\end{align}
Thus, the problem for functions supported away from $q$ reduces to the situation of an elliptic operator, for which the result is standard, as you say.


