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 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 $L^2$. 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.

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.

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.

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 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 $L^2$. 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.