TITLE: Elliptic Regularity on Manifolds
QUESTION [10 upvotes]: Sorry if this has been asked before.
Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about regularity on a manifold be reduced to regularity on $\mathbb{R}^n$?
Thanks.

REPLY [5 votes]: Chapter 6 of Warner (Foundations of Differentiable Manifolds and Lie Groups) covers elliptic operators on manifolds. Specifically, you're probably interested in Theorem 6.5 and its proof in 6.32. The proof uses coordinate charts and elliptic regularity in $\mathbb{R}^n$ to establish smooth local solutions and then pieces the global solution together with a partition of unity.
You may also be interested in Booss-Bleeker (Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics). Additionally, here is a sketch of some of the ideas from a course at Indiana University several years ago.