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 $(Pu, u) \cong \Vert u\Vert^2_{H^1}$? If $P$ were elliptic on $M$, this is standard material. But here the lack of ellipticity at a point is throwing me off.

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.