Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is the (0,1) part of the connection. The integrability theorem for holomorphic structures says that if $D''\circ D''=0$, then $E$ admits a holomorphic structure such that $D''=\bar{\partial}$. I am trying to understand the proof of this statement given in the book by Donaldson and Kronheimer, Geometry of 4 manifolds, section 2.2.2, and I am facing problems understanding the following. 


Given a smooth function $\theta:\mathbb{C}\to \mathbb{C}$ which is compactly supported in a small neighborhood around 0, define the operator $L$ by 


$$ L(\theta)(w):=-\frac{1}{2\pi i}\int\_{\mathbb{C}}\frac{\theta(z)}{z-w}d\mu\_z$$ 


Then we have 


$$\frac{\partial}{\partial \bar{z}}L(\theta)=\theta$$


The authors now (after replacing $\theta$ by another function $\alpha$ which is again compactly supported and with small infinity norm) find a function $f\in L^{\infty}(\mathbb{C})$ which satisfies $f=L(\alpha + f\alpha)$ and claim that $f$ is smooth. The reason they give is that, "elliptic regularity for the $\bar{\partial}$ operator implies that any bounded solution of the equation $f=L(\alpha + f\alpha)$ is smooth". 


In Warner's book on Lie Groups, in the last chapter, one has the following statement for periodic elliptic operators : 


Let $T$ be a periodic elliptic operator and let $u\in H\_{-\infty}$ and $v\in H\_{\infty}$ such that $Tu=v$, then $u\in H\_{\infty}$. 


In the above, $H\_n$ are the Sobolev spaces. The regularity theorem for compact manifolds is then deduced by reduction to the periodic case. 


I am not able to see how to apply the regularity theorem since I do not see any equation of the type $Tu=v$ in the present setup. Could explain to me how to fill in the details.



Smoothness of solution to a PDE
The equality 


$$f= L(f+\alpha f) $$


implies $\newcommand{\pa}{\partial}$


$$\pa\_{\bar{z}} f-\alpha f =\alpha. $$


The operator with smooth coefficients


$$ T:=\pa\_{\bar{z}}-\alpha $$


is elliptic and the above equation has the form


$$Tf=\alpha. $$


The regularity theorem for elliptic operators with smooth coefficients states that if $Tf\in H\_k$, $f\in L^2$, then $f\in H\_{k+1}$, where $H\_k$ denotes the Sobolev space of functions with derivatives up to order $k$ in $L^2$. In your case $\alpha\in H\_k$, $\forall k>0$ so that $f\in H\_k$, $\forall k>0$ and invoking the Sobolev embedding theorems you conclude that $f$ is smooth. For more details about elliptic regularity check section 10.3 of [my lecture notes.](http://www.nd.edu/~lnicolae/Lectures.pdf)


