Maybe the following helps:
Theorem 3.12 (page 20) in the following source has such a related result, albeit for higher Sobolev spaces.
There are quite subtle requirements for the trivialising atlas and the partition of unity which are used in the proof. 


- [MR2343536](http://www.ams.org/mathscinet-getitem?mr=2343536); Eichhorn, Jürgen Global analysis on open manifolds. Nova Science Publishers, Inc., New York, 2007. x+644 pp.

You can access the beginning of the book via scholar.google.com.