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} = (2k+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.