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. The regularity theorem for elliptic operators with smooth coefficients sates 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 $L^2$. For more details about elliptic regularity check section 10.3 of my lecture notes.