# Elliptic estimates on compact manifolds

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.

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It doesn't suffice to use equivalent estimates on bounded domains of $\mathbb{R}^n$, and then use a partition of unity to extend it to the manifold? – Ray Yang Feb 25 '13 at 2:33
@RayYang Change of variables introduces extra first-order terms (to be controlled by $Lu$), which looks like a bit of extra work. It should be written somewhere, I'm sure. – user53153 Feb 25 '13 at 3:18
@RayYang Also I am not that familiar with manifolds and PDEs and have never seen such a POI trick to extend a result. – george.s Feb 26 '13 at 9:41

## 3 Answers

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

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

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Thanks. I didn't think a book called Spin Geometry would have such a result! – george.s Feb 27 '13 at 18:53

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?

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I don't know if it is still relevant, but in Gilkeys "Invariance Theory, the Heat equation and the Atiyah-Singer Index Theorem", the results are established in full generality (for any Exponent and any elliptic (pseudo)differential opertator).

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