Let $X$ be a non-compact complete Riemannian manifold and $P$ a first-order elliptic pseudodifferential operator on $X$. Let $Q$ be a parametrix for $P$, so that $PQ - 1 = T$ and $QP - 1 = R$ are lower order operators. 


If $X$ were compact, then the operators $Q$, $T$, $PQ$ and $PT$ would all be bounded on $L^2(X)$, since they have non-positive order. 


My question is: when $X$ is non-compact, is it possible to choose the parametrix $Q$ so that $Q$, $T$, $PQ$, $PT$ are still all bounded on $L^2(X)$?



Choice of parametrix on a non-compact manifold
Yes, at least if $Q(x,y)$ is locally integrable in $y$.


Given two integral kernels $Q(x,y)$ and $Q'(x,y)$ that agree on a neighborhood of the diagonal in $X \times X$, then if one is a parametrix, so is the other one. Hence, you can always restrict a parametrix to have support in an arbitrary neighborhood of the diagonal. Just multiply any given parametrix $Q(x,y)$ by a smooth function $\chi(x,y)$ that has sufficiently small support but is still identically $1$ on a smaller neighborhood of the diagonal.


Since $Q(x,y)$ is locally integrable, you can choose the neighborhood small enough so that $\int\_X |Q(x,y)| dx < C$ and $\int\_X |Q(x,y)| dy < C$. Then, by the [Schur test](https://en.wikipedia.org/wiki/Schur_test), $Q$ is bounded on $L^2(X)$.


Each of the operators $T$ and $PT$ is also represented by a locally integrable integral kernel. Thus, you can make the exact same argument for each of them, possibly restricting the support of $Q$ even closer to the diagonal, so that $T$ and $PT$ both satisfy the Schur test.


Finally, by your definition $PQ = 1+T$, hence it is automatically bounded.


If $Q(x,y)$ is not locally integrable, the above argument may not work, since there might not exist a sufficiently small neighborhood of the diagonal that will make all the $y$-integrals bounded.


**Update:** However, take a look at Thm VI.2.1 of Stein's *Harmonic Analysis* (PUP, 1993). It seems to directly address the $L^2$ boundedness of (scalar) pseudodifferential operators of a certain class, that might cover your operators.


