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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?)

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In fact, it's hard to find an example of a PDO which has a right inverse that is not smooth tame. It's certainly true for the standard types: elliptic, hyperbolic, and parabolic.

On the other hand, why do you need this for hyperbolic operators? Local solvability of a quasilinear hyperbolic PDE can be proved using the standard implicit function theorem on a Banach space (or, equivalently, the contraction mapping argument). Moreover, even a fully nonlinear hyperbolic PDE can be turned into a quasilinear hyperbolic PDE by "prolongation" (add new unknown functions that are set equal to the partial derivatives of the original unknown functon). In practice, Nash-Moser is rarely needed for most useful or interesting PDE's.

But if you really want to see the smooth tame estimates, you can look at the appendix of:

Klainerman, Sergiu Global existence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101.

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  • $\begingroup$ Well Deane, if a (say) quasilinear hyperbolic PDE has a source term on the right hand side $$P(\phi)=g(\phi,\partial\phi)\partial^2\phi+H(\phi,\partial\phi)=f\ ,$$ I don't see any obvious way of avoiding using Nash-Moser and solving it with a contraction mapping argument, and that is where you really have to deal with local invertibility - typically, $f$ has the form $f=P(\phi_0)+\tilde{f}$ with $\tilde{f}$ "small". You had to face such a problem in that DMJ paper of yours with Robert Bryant and Phillip Griffiths on low-dimensional isometric embeddings. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Jun 20, 2016 at 13:29
  • $\begingroup$ Pedro, I suggest consulting the papers of Klainerman written after the one I cited. The point is that you can use the energy estimates, the Gagliardo-Nirenberg inequalities, and the "right" functional norms (which distinguish the time coordinate from the space coordinates), to get the estimates needed for the Banach space implicit function theorem. $\endgroup$
    – Deane Yang
    Commented Jun 21, 2016 at 10:24
  • $\begingroup$ I did that, he deals with the homogeneous Cauchy problem only (i.e. without sources). The point is that the loss of derivatives is not that severe for the fixed-point iteration scheme aiming at solving the homogeneous Cauchy problem to work, that's why choosing appropriate norms and interpolating with Gagliardo-Nirenberg saves the day. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Jun 21, 2016 at 18:30
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An addendum to Deanne Yang's answer. Actually, if a linear hyperbolic equation is degenerate, meaning that the characteristic roots are real but not distinct, then the inverse operator presents a loss of derivatives, meaning that it takes $H^s$ to $H^{s-m}$ for some $m>0$. This is an actual phenomenon and not a defect of the estimates. Indeed, the solution of the 1D equation $u_{tt} − t^2 u_{xx} = (4k+1) u_x $ can be written explicitly for integer $k$ and loses exactly $k$ derivatives (this was proved by Qi Min-you in 1958).

Thus the inverse of a degenerate hyperbolic equation is a perfect example of a smooth tame map, and the Nash-Moser theorem can be used to handle the nonlinear equations. I used myself this idea in a couple of papers.

EDIT: I lost track of the hardcopy of QMY's paper, which I had at some point, anyway from my notes the solution with initial data 0 and $\phi(x)$ should be $$u(t,x)=\sum_{j=1}^{k} \frac{\sqrt{\pi}t^{2j}}{j!(k-j)!\Gamma(j+1/2)} \phi^{(j)}(x+t^{2}/2).$$

EDIT 2: Wait: in section 6 of this paper the example is extended to a more general class of equations.

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