Let $a:\mathbb{C}\rightarrow\mathbb{C}$ be a fixed smooth function. Consider the equation $$\partial_{\overline{z}}f=a\overline{f}.$$ I'm interested in classifying the (local) solution $f:\mathbb{C}\rightarrow\mathbb{C}$.
I was trying to solve it by using a Banach fixed point argument, for this I was thinking on using the generalized Cauchy integral formula for smooth functions $$f(z)=\frac{1}{2\pi i}\int_{\partial\mathbb{D}}\frac{f(w)}{w-z}dw+\frac{1}{2\pi i}\int_\mathbb{D}\frac{\partial_{\overline{z}}f}{w-z}dw\wedge d\overline{w}.$$ That is, considering without lost of generality that $a$ has compact support in $\mathbb{D}$ (since local solution is enough for me), that $r^2\|a\|_{\infty}<<1$ where $r$ is the radius of $\mathbb{D}=\mathbb{D}_r$ , and the operator $P:C^0(\overline{\mathbb{D}_{r/2}},\mathbb{C})\rightarrow C^0(\overline{\mathbb{D}_{r/2}},\mathbb{C})$ (where $C^0(\overline{\mathbb{D}_{r/2}},\mathbb{C})$ is the set of continuous function) defined by
$$P(f)(z)=\frac{1}{2\pi i}\int_{\partial\mathbb{D}_{2r}}\frac{f_0(w)}{w-z}dw+\frac{1}{2\pi i}\int_\mathbb{C}\frac{a\overline{f}}{w-z}dw\wedge d\overline{w},$$
where $f_0:\partial\mathbb{D}_{2r}\rightarrow\mathbb{C}$ is some fixed "initial condition". Using this operator I'm able to find a fixed point $f:\overline{\mathbb{D}_{r/2}}\rightarrow\mathbb{C}$. However, it is not clear for me if this fixed point is smooth to conclude that this is a solution of the problem.
I think that I'm pretty close to solving the problem. Any suggestion to prove smoothness?