Hartogs-Stawski's theorem for bounded analytic elements (Q2769545)

The following non-archimedean version of a theorem of Hartogs is shown (extending the results obtained by the author in Lect. Notes Pure Appl. Mat. 207, 77-96 (1999; Zbl 0954.32017)]). NEWLINENEWLINENEWLINELet \(\mathbb{C}_p\) be the completion of the algebraic closure of the \(p\)-adic number field. Let \(f\) be a bounded \(\mathbb{C}_p\)-valued function on \(D_1\times \cdots\times D_n\), where each \(D_j\) is a domain (i.e., a `closed' disk in \(\mathbb{C}_p\), minus finitely many `open' disks). If \(f\) is holomorphic in each variable separately then, for each \(i\), \(f\) is holomorphic on \(D_1^0 \times\cdots \times D^0_{i-1} \times D_i\times D^0_{i+1} \times\cdots \times D^0_n\), where for \(j\neq i\), \(D^0_j\) are certain subsets of \(D_j\) that are slightly smaller than \(D_j\).NEWLINENEWLINENEWLINEPrevious claims of Stawski in 1983 in this direction are shown to be incorrect. In the present paper Stawki's claims are modified and the `proofs' are rectified.