Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
True, this is actually simpler. The proof of the Frobenius theorem Igor refers to rewrites the PDE system as a prescription for an involutive vector sub-bundle of the tangent bundle (involutivity is encoded in the integrability conditions and amounts to saying that the Lie bracket of any two vector fields tangent to this sub-bundle remain so) and finds a local trivialization of this sub-bundle by commuting vector fields, which therefore have commuting flows (solving the ODE's Igor refers to) yielding the adapted coordinate charts of the integral manifold through the given initial condition.
Unfortunately he doesn't, at least not in this reference... this specific fact refers to the Jacobi identity for the cross (vector) product in $\mathbb{R}^3$. A proper outline of this argument can be found e.g. in khudian.net/Etudes/Geometry/jacidentandheights2.pdf
Well... My computer decided to crash today. I've managed to fix it just now, but then Stefan and Tobias had already beat me to it. Please feel free to acknowledge whatever answer you feel more appropriate.
Diferential operators of positive order and smooth coefficients are indeed unbounded with respect to the $L^2$ (Hilbert space) norm, but are certainly bounded with respect to the $C^\infty$ (Fréchet space) topology as given by the $L^2$ Sobolev norms of all non-negative orders. I'll compile the answer by tomorrow at the latest.
The remarks in my last comment above put together should yield that the harmonic decomposition is a topological isomorphism of Fréchet spaces, even though I don't really think Warner addresses this question explicitly.
