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.
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, $$ |u|_{H^2} \le C|f|_{L^2}. $$
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
Expanding @Piotr Hajlasz's answer, the result it true, and does appear in Warner's Foundations of differentiable manifolds and Lie groups. The result for the Laplacian appears as Theorem 6.32. Theorem 6.30 describes the local theory, then in 6.32 Warner describes how to apply the local theory to get a result for the whole manifold.
Warner does the local theory for periodic functions in $\mathbb{R}^n$ (with period $2\pi$). Then he uses charts which map to subsets of the $2\pi$ cube, and extends the result to the whole cube using freezing coefficients (essentially set the coefficients in your linear operator equal to constants outside some open set).
[Side notes: (1) Theorem 6.32 generalises easily to other elliptic second-order linear differential operators. (2) Warner doesn't discuss boundary points and boundary conditions, but Taylor does in Partial Differential Equations I, chapter 5.]
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $\rho=1, \delta=0$.
