Some properties of Van Koch's curves

We investigate the properties of an integral operator $T$ with a Cauchy kernel. The operator acts from $L^\infty(\Gamma,\mu)$, where $\Gamma$ is a Van Koch curve, to the space of functions $\mathbb C\to\mathbb C$. We prove that the range of $T$ is nontrivial and lies in the space $\operatorname{AC}(\Gamma)$ of functions continuous in $\mathbb C$, vanishing at $\infty$, and analytic outside $\Gamma$. We also show that $T$ is injective and compact while satisfying some special functional equation. These results may be regarded as a natural continuation of our research on the problem of $\operatorname{AC}$-removability of quasiconformal curves whose solution was announced in [1] for the first time and supplemented later with some other properties of Van Koch's curves [2], [3]. In this paper the problem is discussed in a more general setting and, in particular, all important details lacking in [1] are given. Some open problems are formulated.