A boundary Schwarz Lemma for holomorphic mappings between unit balls of different dimensions

Abstract

In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+\alpha }$ at $z_0\in \partial {\mathbb{B}}^n$ with $f(z_0)=w_0\in \partial {\mathbb{B}}^N$ for any $n,N\ge 1$, then the Jacobian matrix $J_f(z_0)$ maps the tangent space $T_{z_0}(\partial {\mathbb{B}}^n)$ to $T_{w_0}(\partial {\mathbb{B}}^N)$, and the holomorphic tangent space $T_{z_0}^{(1,0)}(\partial {\mathbb{B}}^n)$ to $T_{w_0}^{(1,0)}(\partial {\mathbb{B}}^N)$ as well.

This paper has been withdrawn by the author, since the main result of the paper is included as a lemma in a new paper of authors own