I am reading Moser's paper On commuting circle mappings and simultaneous Diophantine approximations and I found it hard because it is my first time that I seriously have to read a paper. It is a local simultaneous conjugacy result for commuting diffeomorphisms in the neighborhood of rotations.
The main theorem is Theorem 1 which states that if $\phi_k$ are diffeomorphisms of $S^1$ in the neighborhood of rotations $R_{\alpha_k}$ in $C^{\infty}$-topology, $k=1,\ldots,n$, where $\alpha=(\alpha_1,\ldots,\alpha_n)$ is Diophantine vector in $\mathbb R^n$, and if $\phi_k \circ \phi_l=\phi_l \circ \phi_k$, then there is a smooth diffeomorphism of $S^1$ simultaneously conjugating each $\phi_k$ to $R_{\alpha_k}$.
Here is the set-up for the proof of the main theorem. Let $V_0=C^{\infty}_0(S^1,\mathbb R)$, $V_1=C^{\infty}_0(S^1,\mathbb R^n)$ and $V_2=C^{\infty}_0(S^1,so(n))$, where $0$ subscript means that functions have mean value $0$ and $so(n)$ is the Lie algebra of $SO(n)$, i.e. the vector space of all skew-symmetric matrices. Also, let $A \colon V_0 \to V_1$ and $B \colon V_1 \to V_2$ be linear maps given by $(Av)_k=L_kv$ and $(Bw)_{kl}=L_kw_l-L_lw_k$, where $L_kv(x)=v(x+\alpha_k)-v(x)$.
Write the formal adjoints (which I also don't know what it means) of $A$ and $B$ as $A^* \colon V_1 \to V_0$ and $B^* \colon V_2 \to V_1$ given by $A^*w=\sum_{k=1}^{n}L_k^*w_k$ and $(B^*z)_l=\sum_{k=1}^{n}L_k^*z_{kl}$, where $L^*_kv(x)=v(x-\alpha_k)-v(x)$. It was very easy to show that $BA=0$, $A^*B^*=0$ and $AA^*+B^*B=A^*A \otimes Id_{V_1}$ and also understand the proof of the Lemma 3.1 which states that the operator $M=A^*A \colon V_0 \to V_0$ is bijective if and only if $\alpha$ is Diophantine.
It is obvious that $im A \subset ker B$, since $BA=0$, but may one help me by telling me how he uses Lemma 3.1 to obtain the exactness of the sequence $0 \to V_0 \stackrel{A}{\to} V_1 \stackrel{B}{\to} V_2 \to 0$?
And one more thing. I am prety confused with a compatibility condition, since in the linearizes equation $g_k=\phi_k$ has to have zero mean value, but in the nonlinearized equation it does not. So, how he solves that mean value "obstacle"?
Thank you a lot.
