Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an estimate of the form $$ ||f(x)||_k \leq C ||x||_{k+r}$$ for some $C$ and $r$ (and all $k$), where $|| \cdot ||_k$ denotes the $k$-th seminorm. Hamilton (1982) shows that every partial differential operator is a tame map and, moreover, he claims that many inverses of partial differential operators are also tame. A few pages later, he shows that the Green's operator of an elliptic differential operator is tame.

What are examples of other differential operators with tame inverses? (In particular, is there any hope that inverses to hyperbolic operators are tame?)