On the extension of holomorphic mappings around sets with zero Hausdorff \((2n-1)\)-measure (Q928434)

Let \(D\subset\mathbb C^n\) be a domain, let \(E\subset D\) be relatively closed, and let \(X\) be a complex analytic space. The main results of the paper are the following two extension theorems. (1) Assume that \(\mathcal H_{2n-1}(E)=0\) and \(X\) is Carathéodory complete. Then every \(f\in\mathcal O(D\setminus E,X)\) extends to an \(\widetilde f\in\mathcal O(D,X)\). Moreover, the extension operator \(\mathcal O(D\setminus E,X)\ni f\longmapsto\widetilde f\in\mathcal O(D,X)\) is continuous in the topology of almost uniform convergence in \(D\setminus E\) and \(D\), respectively. (2) Assume that \(E\) is pluripolar and \(X\) is complete with respect to the pseudodistance \(p_X\) generated by the Green function. Then \(\mathcal O(D,X)| _{D\setminus E}=\mathcal O(D\setminus E,X)\).