In Deane Yang's notes about Gunther's proof of the celebrated isometric embedding theorem, at the end it is stated that $v$ inherits the regularity of $h$ because the operator $I-Q_0(v,\cdot)$ is elliptic.
The relevant formulas are
and $Q_0$ is a smooth linear combination of the $Q_i$'s and $Q_{ij}$'s.
Now: what is the meaning of elliptic in this context? Are we regarding $I-Q_0(v,\cdot)$ as a pseudo-differential operator? (which would bother me a little since usually the symbol of a pseudo-differential operator is assumed to be smooth)
Once the statement is clarified, how can one check the ellipticity condition?