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; 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.