Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J\_\epsilon$ on a compact manifold $M$, defined by the following three properties:


1) $J\_\epsilon \in \Psi^{-\infty}(M)$ for each $\epsilon \in (0,1]$


2) $\{J\_\epsilon: 0 < \epsilon \leq 1\}$ is a bounded subset of $\Psi^0(M)$.


3) $J\_\epsilon \to u$ in $L^2(M)$ as $\epsilon \to 0$ for each $u \in L^2(M)$.


Here $\Psi^m$ is the space of pseudodifferential operators of order $m$. Then for $A \in \Psi^m$, the commutator $[A,J\_\epsilon]$ is bounded in $\Psi^{m-1}$. Here the topology on $\Psi^m$ is that induced by the operator norms $H^{s} \to H^{s-m}$.


This is said to be a simple consequence of arguments in section 5, with which it seemingly bears little relation. There's a (somewhat tricky) proof of this in Treves' book for standard mollifiers, but it relies heavily on the fact that they are convolution operators. Any ideas on why this generalization should hold?



Generalized Friedrichs Lemma
The topology on $\Psi^m$ is a ''bornology''. A sequence of symbols is converging if it is bounded in the Fréchet space $\Psi^m$ and if it converges in $C^\infty$. That bornology implies the strong convergence of operators. Then $J\_\epsilon$ is bounded in $\Psi^0$ implies that the commutator $[A,J\_\epsilon]$ is bounded in $\Psi^{m-1}$.


