Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:


For any $u\in H^1(M)$, and $f\in L^2(M)$ such that $\Delta u = f$ (in the sens of distributions), Then $u \in H^2(M)$.
If there is a nice reference for such regularity result It would be good.



Elliptic regularity on compact manifold without boundary
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $\Omega \subset \mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $\Omega$,
$$ \tag{\*} |u|\_{H^2} \le C|\Delta\_0 u|\_{L^2}, $$
where $\Delta\_0$ is the standard flat Laplacian.


To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p \in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$ 
\Delta u = (\delta^{ij} + a^{ij}(x))\partial^2 + b^k\partial\_ku
$$
where $|a^{ij}|, |b\_k| < \epsilon << 1$.
Therefore, if $\Delta\_g u = f$, then
$$
\Delta\_0u = -a\_{ij}\partial^2\_{ij}u - b^k\partial\_ku + f
$$
Therefore, by $(\*)$
$$
|u|\_{H^2} \le C(\epsilon |u|\_{H^2} + |f|\_{L^2}).
$$
If the neighborhood is sufficiently small, then $C\epsilon < 1$ and therefore,
$$
|u|\_{H^2} \le C|f|\_{L^2}.
$$


