Mathematical Sciences: First-Order Elliptic Systems in Harmonic Analysis

这个项目为继续研究与一阶超定椭圆型微分方程组有关的边值问题提供了支持。相应的微分算子起源于欧氏对称空间和黎曼对称空间调和分析中的广义位势理论。将应用于欧几里德Hardy空间理论和显式实现半单李群的酉化例外表示。由于对群不变性的重视，多项式不变量理论将在这些研究中发挥越来越重要的作用。关于群论联系的进一步工作也将展开。系统的椭圆性确保了赋范空间的完备性，在赋范空间中可以找到解，而超定允许人们只研究所考虑的(复)函数的调和实部。

This project provides support for continuing work on boundary problems associated with families of first-order systems of overdetermined, elliptic differential equations. The corresponding differential operators originate in the generalized potential theory in harmonic analysis on both Euclidean and Riemannian symmetric spaces. Applications to Euclidean Hardy-space theory and to explicit realization of unitarizable exceptional representations of semi-simple Lie groups will be made. Because of emphasis on group invariances, polynomial invariant theory will play an increasingly important role in these investigations. Further work on the group-theoretic connections will also be carried out. The ellipticity of the systems ensures the completeness of the normed spaces in which the solutions can be found while the over-determinedness allows one to study just the harmonic real parts of the (complex) functions under consideration.