调和映射Schwarz-Pick估计和全纯映射插值问题的研究

This project deals with the Schwarz-Pick estimate for harmonic mappings and the interpolation problem for holomorphic mappings. Both are based on the classical Schwarz-Pick lemma. In recent years, Schwarz-Pick lemma has been generalized to many results on holomorphic mappings. But the Schwarz-Pick estimate for harmonic mappings is relatively few. So one of the contents we will study in this project is the Schwarz-Pick estimate for harmonic mappings. It consists of two aspects: the Schwarz-Pick estimate for harmonic mappings from the unit disk in the complex plane into itself; the Schwarz-Pick estimate for harmonic mappings between some unit balls in the real space. In addition, in 1916, by the Schwarz-Pick lemma, Pick solved an interpolation problem for holomorphic mappings from the unit disk in the complex plane into itself. The interpolation conditions are some points and their functional values. Based on the above interpolation conditions, some derivative factors will be concerned in this project. So the other content we will study in this project is the interpolation problem for holomorphic mappings. It also consists of two aspects: the interpolation problem for holomorphic mappings from the unit disk in the complex plane into itself, in which the interpolation conditions are some points, their functional values and their derivatives; the interpolation problem for holomorphic mappings from the unit ball in the complex space into itself, in which the interpolation conditions are some points, their mapping values or plus their derivatives.

本项目研究调和映射的Schwarz-Pick估计和全纯映射的插值问题。两者都基于经典的Schwarz-Pick引理。Schwarz-Pick引理在单复变及多复变全纯映射上的推广较为多见，但在调和映射上的推广结果相对较少。因而本项目研究内容之一为：复平面中单位圆盘的调和自映射的Schwarz-Pick估计；实空间中同维及不同维球之间的调和映射的Schwarz-Pick估计。另外，1916年Pick从Schwarz-Pick引理出发，解决了有关单复变中单位圆盘全纯自映射的插值问题，其插值条件为若干点及其函数值。本项目在上述插值条件的基础上增加了对导数因素的考虑，因而本项目研究内容之二为：有关单复变中单位圆盘全纯自映射的插值条件为若干点、这些点处的函数值及导数值的插值问题；有关多复变中单位球全纯自映射的插值条件为若干点、这些点处的映射值或再外加这些点处的导数值的插值问题。

在国家自然科学基金青年基金的支持下，本项目顺利完成。在国内外期刊共发表论文5篇，均为SCI收录论文。本项目主要研究调和映射的 Schwarz-Pick 估计和全纯映射的插值问题。两者都基于经典的Schwarz-Pick 引理。Schwarz-Pick 引理在单复变及多复变全纯映射上的推广较为多见，但在调和映射上的推广结果相对较少。因而本项目研究了调和映射的 Schwarz-Pick 估计，得到了实空间中球上的复值有界的调和函数的Schwarz-Pick估计式；实空间中球之间的调和映射的Schwarz-Pick估计式；实空间中球上的实值有界的调和函数的Schwarz引理；实空间中球上的正值的调和函数的Schwarz引理；复空间中球之间的多重调和映射的Schwarz引理和边界Schwarz引理。另外，1916 年Pick 从Schwarz-Pick 引理出发，解决了有关单复变中单位圆盘全纯自映射的插值问题，其插值条件为若干点及其函数值。本项目在上述插值条件的基础上增加了对导数因素的考虑，因而本项目研究了单复变中单位圆盘全纯自映射的插值条件为若干点、这些点处的函数值及导数值的插值问题，得到了单复变中单位圆盘全纯自映射的关于导数的多点的Schwarz-Pick引理。研究内容基本按计划进行，同时也超越了原定计划，还研究了复空间中若干全纯映射的模的Schwarz-Pick估计。我们得到了复空间中球之间的全纯映射的模的Schwarz-Pick估计式；复空间中多圆柱到球的全纯映射的模的Schwarz-Pick估计式；复平面中单位圆盘到复空间中某类特殊球p-球的全纯映射的模的Schwarz-Pick估计式。上述结果不仅能丰富调和映射和多复变的相关理论，且对进一步探索全纯映射与调和映射、单复变与多复变之间的联系有一定的帮助。