On the localization principle for the automorphisms of pseudoellipsoids

We show that Alexander's extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $f$ of a pseudoellipsoid $\E^n_{(p_1, ..., p_{k})} \= \{z \in \C^n : \sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + ... + |z_n|^{2 p_{k}} < 1 \}$, provided that $f$ is defined on a region $\U \subset \E^n_{(p)}$ such that: i) $\partial \U \cap \partial \E^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\U \cap \{z_i = 0 \} \neq \emptyset$ for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.