I was wondering about the following:

Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$.

Can every smooth linear partial differential operator $P$ from $E$ to $F$ of finite order be written as $\sum_{i=0}^n (T_i)_* ∘ P_i$ for certain smooth linear partial differential operators $P_0, \dots, P_n$ from $E$ to $E$ and certain vector bundle homomorphisms $T_0, \dots, T_n$ from $E$ to $F$?

Can every smooth linear partial differential operator $P$ from $E$ to $E$ of finite order be written as a finite sum of compositions of smooth linear partial differential operators from $E$ to $E$ of order at most 1?

Of course, locally (i.e., on a chart domain over which the vector bundles trivialize) the answer to both questions is yes, but this does not seem to be of much help when trying to answer the questions globally.

Differential analysis on complex manifoldslooks fine). Using a smooth partition of unity subordinate to such a finite cover to patch the local expressions indeed seems to solve the problem, but I will come back to this after I have checked the details (probably tomorrow; I am quite busy right now). – Marcel de Reus Aug 7 '11 at 11:55