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 $\mathcal{E}^n_{(p_1, \dots, p_{k})}\overset{\text{def}}{=} \{ z \in \mathbb{C}... ...\vert^2 + \vert z_{n-k+1}\vert^{2 p_1} + \dots + \vert z_n\vert^{2 p_{k}} < 1\}$, provided that $f$ is defined on a region $\mathcal{U} \subset \mathcal{E}^n_{(p)}$ such that: i) $\partial \mathcal{U} \cap \partial \mathcal{E}^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\mathcal{U}\cap\{ z_i = 0 \} \neq \emptyset$ for any $n-k +1 \leq i \leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal.