Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-integrating against the Cauchy kernel $$ \Gamma : \mathcal O(\mathbb C) \longrightarrow \mathcal O(\mathbb C\setminus \gamma) \\ f\longmapsto \left(z\mapsto \int_\gamma \frac{f(w)}{w-z}\mathrm{d}w\right)$$

In the trivial case where $\gamma$ is closed then $\Gamma$ is injective (apply Cauchy formula to all $z$ in the bounded connected component of $\mathbb C\setminus \gamma$) and its range consists in those functions that are zero outside $\gamma$ and the restriction of an entire function inside.

**My question:** what happens if $\gamma$ is not closed (*i.e.* $\mathbb C\setminus \gamma$ connected)? Is it possible to characterize the range of $\Gamma$, besides the obvious $O(|z|^{-1})$ towards infinity?

(In this case $f$ may only be a germ near $\gamma$ of a holomorphic function, but keep it simple if necessary.)

I'd find it hard to believe that injectivity would fail (although I don't quite have an argument).

If this question has already a well-documented answer (which is fairly probable but I didn't know which keywords to look for), please accept my apologies. A pointer to a reference would be great!