Smoothness of solution to a PDE

Question

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.

Answer 1

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.