On the Boundary Schwarz lemma and the rigidity theorem for certain mappings

TL;DR

This paper characterizes specific holomorphic mappings and extends boundary Schwarz lemmas and rigidity theorems for vector-valued and pluriharmonic functions on unit balls in various $ ext{l}_p^n$ spaces.

Contribution

It provides new characterizations of mappings, simplified proofs of boundary Schwarz lemmas, and establishes boundary rigidity theorems for holomorphic self-mappings.

Findings

Characterization of holomorphic mappings from $B_{ ext{l}_p^n} imes ext{D}^m$ into $ ext{D}^m$ for $p ext{ in } ext{ extbraceleft } extbf{2,} extbf{ extbackslash{}infty} extbf{ extbracerleft }$.

A simplified proof of the boundary Schwarz lemma for vector-valued holomorphic functions, extending existing results.

Boundary Schwarz lemma for pluriharmonic self-mappings of the unit ball $B_{ ext{l}_p^n}$ for $p ext{ in } [2, ext{infty}]$.

Abstract

In this article, we characterize the holomorphic mappings from $B_{\ell_p^n}\times\mathbb{D}^{m}$ into $\mathbb{D}^{m}$ for $p\in \{2,\infty\}$. In addition, we give a simple proof for the boundary Schwarz lemma for vector valued holomorphic functions, which also extends the existing result. Also, we obtain the boundary Schwarz lemma for pluriharmonic self-mappings of the unit ball $B_{\ell_p^n}$, $p \in [2,\infty]$. Furthermore, we establish the boundary rigidity theorem for holomorphic self-mappings of $B_{\ell_p^n}$, $p \in (1,\infty)$.