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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 $M$.

  1. 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 $F$?

  2. 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.

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Why not just use a partition of unity to make the local results global? –  Dmitri Pavlov Aug 6 '11 at 17:48
    
Because, as far as I can see, on noncompact manifolds using a partition of unity will not give finite sums. –  Marcel de Reus Aug 7 '11 at 7:50
1  
@Marcel: You can always cover a connected manifold with a finite set of coordinate charts, see this question: mathoverflow.net/questions/58988/… –  Dmitri Pavlov Aug 7 '11 at 8:51
    
@Dmitri: Thanks a lot for pointing this out! This looks really promising. Of course, here we need a finite cover by coordinate charts such that $E$ trivializes over these charts and $M$ is real, but this does not seem to be a problem (for example, the proof of Proposition 4.1 in Wells' Differential analysis on complex manifolds looks 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

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