Elliptic estimates on compact manifolds

Question

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it.

For example, for $L$ a linear elliptic operator, (eg. $L = \Delta$), I want to know that $$\lVert u \rVert_{H^2(M)} \leq C(\lVert Lu \rVert_{L^2(M)} + \lVert u \rVert_{L^2(M)})$$ holds where $M$ is a compact (boundaryless) Riemannian manifold.

I am interested in general nth order elliptic estimates, but a source for the above would be good as well. Thanks.

Answer 1

Lawson, Michelsohn: Spin Geometry, Princeton University Press, 1989.

The needed norm estimate is Theorem 5.2(iii) in Chapter III, §5.

Answer 2

How about this: http://math.mit.edu/~rbm/18.156-S08/Lecture-Notes.pdf

Try Proposition 9 on page 43. Anybody have any comments on this source?