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 $\E^n_{(p_1, …, p_{k})} = {z \in \C^n : \sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + … + |z_n|^{2 p_{k}} < 1 }$,providedthat$f$ is defined on a region $\text{U}\subset {\text{E}}_{(p)}^n$ such that: i) $\partial \U \cap \partial \E^n_{(p)}$containsanopensetofstronglypseudoconvexpoints;ii)$\U \cap {z_i = 0 } \neq \emptyset$forany$n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.