Boundary analogues of Hartog's theorem (Q809229)

Let \(\Omega\) be a bounded domain in \({\mathbb{C}}^ n\). A family L of complex lines is called ``sufficient'' if from: (i) \(f\in C(\delta \Omega),\) (ii) for every \(\ell \in L\), \(f| \ell \cap \delta \Omega\) has an analytic prolongation in \(\ell \cap {\bar \Omega},\) it follows that f is analytic in \(\Omega\) and continuous in \({\bar \Omega}\). The aim of this paper is to give examples of ``sufficient'' families of complex lines.